Characterizing Quantum Devices Using the Principles of
Quantum Information
by
Anthony M. Polloreno
B.A., University of California, Berkeley, 2016
M.S., University of Colorado at Boulder, 2022

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2023
Committee Members:
Graeme Smith, Chair
Joshua Combes
Oliver DeWolfe
Murray Holland
Daniel SlichterJames Thompson

ii
Polloreno, Anthony M. (Ph.D., Physics)
Characterizing Quantum Devices Using the Principles of Quantum Information
Thesis directed by Prof. Graeme Smith

The title of this thesis is “Characterizing Quantum Devices Using the Principles of Quantum
Information”, named after the NSTGRO fellowship of the same name. As with any work, the focus
has become more clear after the fact. The structure is roughly as follows:(1) Quantum computation.
(2) Quantum metrology.
(3) Quantum characterization.
This thesis outlines work pertaining to the development and characterization of quantum devices. The suggested methodologies borrow ideas and rely on techniques from quantum information
theory. Importantly, this thesis is meant to be a thesis on theoretical physics, rather than mathematics or computer science, and so we hope that the reader is able to glean physical intuition fromthis work.
Arguably the most important question in recent years is “Is this a quantum computer?” Thisanswer, in all of its various realizations, is the prerogative and impetus of the field of quantum
characterization, verification and validation (QCVV). We will generalize this question to “Is thisa quantum device,” provide theoretical motivation for a number of interesting experiments, anddevelop theories that we hope provide new insights into the field and have the potential to lead tofurther advancements in quantum technology.
Specifically, seven projects are discussed: developing single-site rotations in a Penning trap as
a scalable mode of quantum computation, enabling large-scale quantum optimization in a neutral
atom trap, benchmarking quantum computers with a quantum error detecting code, bounding
the integrated quantum Fisher information for metrological protocols, developing the theory of

iii
reservoir computing with stochastic reservoirs, rigorously deriving a theory of direct randomized
benchmarking and understanding the impact of Markovian errors in random circuits.
The first two projects are experimental collaborations working towards quantum computation
with hundreds of qubits. In recent years quantum devices appear poised to break the hundred qubit
mark, with varying levels of control and fidelity. Surely there is much work to be done in developing
these technologies if we want to perform genuine quantum computation, but the demonstration ofcontrol over a large number of quantum degrees of freedom is, in itself, a marvel of engineering anddemonstrates a growth of our mastery of physics. In these two projects, we propose a new methodof single-qubit control in an ion trap which routinely has hundreds of qubits, and investigate theimpact of measurement rate of neutral atom traps on our ability to execute quantum algorithms.
The next three projects work towards developing techniques to demonstrate genuine quantum
information theoretic phenomena. The first work is simple, but was done at a time when many
results were coming out suggesting the demonstration of fault tolerance, when they were instead
implementing fault tolerant circuits, above threshold. We propose a simple Bacon-Shor code, overa family of circuits, that should be easy to implement on existing hardware, with current error
rates, that should give a clear demonstration of actual fault-tolerance - a break-even of the logicalerror rate and physical error rate. While quantum computation has stolen the scene, quantum
metrology has already been demonstrated to provide significant improvement over classical methods,
if only in the context of scientific experiments. In particular, the probing of quantum gravitational
effects and dark matter are only enabled by considering quantum devices. We first explore anapplication of sensing that relates to both dark matter detection and quantum computation, andthen consider a theoretical investigation of a computational technique that may be able to leveragethese metrological advantages.
Finally, the last two chapters work towards the more general goal of characterizing quantum
circuits. Despite significant research into the characterization of quantum states and gates, it isrelatively poorly understood how errors impact the performance of structured circuits such as thoseused in error correction. By using techniques from the theory of randomized benchmarking we firstiv
establish an extremely general result about the decay rate of a large class of random circuits, andthen use this result to understand how certain physical error rates impact the success probabilitiesof those circuits.
This work was done in collaboration with others. With the exception of the last chapter, allchapters have included with them a reference to the published article or preprint. The sections ofthese papers that were done primarily by another author have been elided from this document. Theinterested reader should refer to the referenced works to see the elided results.
Finally, this work was supported by a variety of funding agencies that I would like to acknowledge support from. This work was supported by a NASA Space Technology Graduate Research Opportunity award, the U.S. Department of Energy, Office of Science, National Quantum Information
Science Research Centers, Quantum Systems Accelerator (QSA), the Defense Advanced Research
Projects Agency (DARPA) under Contract No. HR001120C0068, the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research, Quantum Testbed Pathfinder,
the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects
Activity (IARPA), the NSF JILA PFC grant 1734006 and NSF award DMR-1747426.

Dedication

This thesis is dedicated to my family.

Acknowledgements

My thesis is the culmination of four years of work, but of course, I myself am currently the
culmination of 28 years of work, and that is, emphatically, a collaborative piece!
Naturally, the first thanks go to my family - Mom, Dad, Jana, Nana, and Pa - thank you all
for supporting me and loving me. Briefly:
Dad - you frequently told me, "follow your passion, discover your bliss," and looking back,this has been apt advice and has paid dividends.
Jana - thank you for growing with me, being supportive, speaking your mind, being true toyourself, and being a great friend and sister.
Pa - besides imparting on me your sense of humor, you’ve always instilled in me a passion for
science. There is a lot more I could say, but it suffices to say that a passion for science has been alarge part of my life.
Nana - thank you for always being there for me, looking out for me, and helping me in every
and any way you can.
Mom - thanks for being my daily hype squad, support group, team captain, and coach. It
looks like I "got that Ph.D." Especially - especially - thank you for being a grounded source ofstability nearly every day. You have had an effect on my life that is hard to put into words. Surely,
for the five of you, there is much more to say, but perhaps in brevity we can find abundance.
Besides my family, I’ve had the enormous blessing of a large number of friends and mentors
who have significantly shaped not only my career, but also who I am.
Alexander Papageorge - naturally, I must begin with you! How different my life would havevii
been if I hadn’t met you at Rigetti! Your advice, friendship, support, and patience, among otherthings, have been essential to my development. I wouldn’t be the scientist I am today, nor the man,without having met you - and both of those are of equal importance. Your influence on me as ayoung physicist at Rigetti was substantial. Your insistence on precision, intuition, and general bias
towards pedantic pedagogy grounded me in a very important way. I look forward to our future
ventures together, and I am happy beyond words that I have had the opportunity to both be taught
by you and be your friend.
Nikolas Tezak - while working with you was where our friendship began, the past few years in
Boulder have been where it has blossomed. Your advice, both technical and personal, has profoundly
influenced me. You are a truly fantastic and modern scientist, and your style of problem solving has
molded me into a more sensible and committed physicist. Fittingly, it was from you that I heard thesuggestion to choose who you work with, rather than what you work on. Following this advice, I am
extremely happy we were able to work on the quantum reservoir computing work together. Most
importantly for this thesis, I have had the opportunity to be exposed to your physical intuition,
understanding, artistry, analytic skills, coding abilities, pragmatism, and company.
Eric Peterson - you’re not only a great and patient teacher of mathematics, but also a truly
wonderful friend. I am incredibly happy that we have spoken as regularly as we have, both because
it means I get to learn about algebraic topology, but also because it means that we have had a chance
to further our friendship. To the first point, you have maintained in me a feeling of mathematicalcompetency and eagerness that I hope to capitalize on. To the second point, my life has been
substantially richer due to our conversations. I say with confidence that you have had a profoundinfluence on me, both as a mentor and a friend.
Prasahnt Sivarajah - you’ve been a constant friend since we met at Rigetti. You are surely amentor, but most importantly a colleague. Thank you for reading with me, thank you for always
expressing interest in my personal and professional life and success, and thank you for being a
constant source of scientific camaraderie. While it is unrelated to my thesis, thank you also forbeing a very accommodating neighbor at Rigetti. It was surely on the red team that my career asviii
a scientist began to blossom.
Reuben Wang - in addition to being a lovely person, thank you for being "that-other-student"
in the program that I got to do science with. You made Boulder feel more like a home and community
to me.
Graeme Smith - naturally my advisor should find a place on this list! Thank you for advocating
for me, for providing interesting problems, and helping me achieve my goals. I look forward to
crossing paths with you again in the future. I’m not sure I could have had a better advisor thanyou. There is surely more I could say, but, as with my family above, perhaps in the brevity of mythanks, we can find depth in the space around it.
Ana Maria Rey - thank you for working very much as a second local advisor to me. You
taught me a lot about being a diligent scientist, and I feel extremely fortunate for having had theopportunity to work with you.
Robin Blume-Kohout - you’ve provided something that is quintessential but hard to communicate. A long time ago, I tried to burn off any kind of mystical romanticism about doing sciencethat I had. You rekindled that in me and helped me double down on a project. Thank you forlistening, investing, and contributing. As a child, I found a lot of joy in believing in magic andreading fantasy. You’ve convinced me that it’s tenable to continue finding that magic in science.
Furthermore, you have pushed me to develop my scientific communication, something that I am
grateful for.
Tim Proctor - I think you are the first example of an early-late stage career scientist I’ve had
the pleasure of working with. You already do, and will continue to, serve as an excellent archetypeof how I should push through the next few years of my life. Thank you for always being jovial andfor taking me under your wing. You are wonderful to work with, and I am grateful for the learning
opportunities, friendship, and projects we’ve been able to share.
Kevin Young - thank you for working with me in 2015! That’s really when this all started forme. I’m glad we finally got our paper published. But most importantly, you’ve been a great mentor
and friend, and I’m glad to have you in my life.

ix
Eleanor Rieffel - thank you for supporting my research goals, providing helpful advice, andfurthering my interest in the mathematics of quantum computing. I look forward to our futurecollaborations.
To you all, a lesson I learned at Rigetti, and relearned in graduate school, was that working
with people you like is extremely important. I will be forever grateful that I have managed to find
you all.
It takes a village, and I am so happy to have found one.
And beyond the matter at hand, there is the fact that growing through your 20s and living
through a global pandemic is hard. I’ve leaned heavily on my friends and they must be called out.
Chadwick Casper - I’m glad that we’ve remained in touch, kept skiing, and stayed in eachother’s lives. I’m looking forward to seeing where life takes both of us in the coming years.
André Melo and Genya Crossman - a special shout-out to the two of you. While you both havedeveloped as scientists alongside and with me, the more substantial and crucial contribution to mylife during these years has been through our lofi chat. One might say that matters of the heart are
separate from the trials and tribulations of graduate school, but I think anyone who experienced the
pandemic can attest to the fact that productivity and progress in work are strongly dependent on
mental state. You two, despite being across the world, have had one of the most profound impacts
on these past four years of my life. Thank you for being a fantastic support network and fantastic
people.
Bryan Berggren - thank you for keeping me sane! Despite not having the honor of working
with you, I did have the honor of having you in my life as a grounded, personable, cheery scientist
who was always willing to push me to my best climbing, go on crag adventures with me and play
guitar with me. You kept me happy and healthy in times that I otherwise surely would have beenless than that. Thanks for being my friend, neighbor, colleague and accountability buddy. I lookforward to staying in touch, and seeing where you go.
The Worts - thank you for being a daily community to share life with. You all have played anximportant role in my life and time in graduate school, and are exceptional friends and colleagues.
Chloe Rossin - you’ve provided a bright light to the end of the graduate school tunnel, a warmcompanionship, and I am excited to start our life together in San Francisco.
I truly love you all dearly.
And of course, I must thank all of my collaborators - every piece of work in this thesis has
been collaborative, in the literal sense. So, I would like to thank Jacob L. Beckey, Robin BlumeKohout, John Bollinger, Arnaud Carignan-Dugas, Michael Foss-Feig, David Hayes, Jordan Hines,
Joshua Levin, Timothy Proctor, Ana Maria Rey, Ariel Shlosberg, Graeme Smith, Nikolas A. Tezak,
James Thompson, Reuben R.W. Wang, and Kevin Young.
And finally, I must thank Michael Perlin for his help with compiling this thesis and for being
yet another great colleague.

Contents

Chapter
1 Individual qubit addressing of rotating ion crystals in a Penning trap [1]2

1.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Experimental Set-up and Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.5

Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.5.1

Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.5.2

Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.5.3

Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

1.6.1

Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1.6.2

Elliptical Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.6.3

Displaced Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

1.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

1.8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.8.1

35

1.6

The RWA is Exact for Integer Multiples of the Trap Rotation Frequency . . .

2 The QAOA with Slow Measurements [34]2.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36
36

xii
2.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.3

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.4

Choice of Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.5

Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3 Towards Demonstrating Fault Tolerance in Small Circuits Using Bacon-Shor Codes [77]

48

3.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.3

Bacon-Shor Subsystem Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.4

Site-Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.5

Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4 Opportunities and Limitations in Broadband Sensing [97]

60

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.2

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.3

Ramsey and π-pulse protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.4

B=0 Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.5

Entangled probe advantage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.6

Quadratic scaling of IQFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.A.1 Derivation of Average IQFI for Instantaneous π-Pulses . . . . . . . . . . . . .

73

4.A.2 Perturbative Expansion to O(B 2 ) . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.A.3 Derivation of the IQFI under the Rotating Wave Approximation . . . . . . .

74

4.A.4 Bound on IQFI of Continuous Protocols . . . . . . . . . . . . . . . . . . . . .

74

xiii
5 Noisy Reservoir Computation [144]

79

5.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.2

Reservoir Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.3

Noisy Reservoir Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.3.1

82

Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 A Theory of Direct Randomized Benchmarking [147]

86

6.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

6.3

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

6.3.1

Gates, gate sets, and circuits . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

6.3.2

Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Direct randomized benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

6.4.1

The direct RB protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.4.2

A simple numerical demonstration of direct RB . . . . . . . . . . . . . . . . .

98

6.4.3

Direct RB’s standard sampling parameters . . . . . . . . . . . . . . . . . . . .

99

6.4.4

Selecting the gate set to benchmark . . . . . . . . . . . . . . . . . . . . . . . 100

6.4.5

Selecting the sampling distribution . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4.6

The reason for randomized 2-design state preparation and measurement in

6.4

direct RB circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4.7

The reason for conditional compilation: improved scalability . . . . . . . . . . 106

6.4.8

The reason for randomizing the success outcome . . . . . . . . . . . . . . . . 108

6.4.9

The error rate convention: choosing the decay rate scaling factor . . . . . . . 108

6.4.10 Comparison to other methods for native gate randomized benchmarking . . . 109
6.5

Understanding direct RB using error scrambling . . . . . . . . . . . . . . . . . . . . . 110

6.6

Sequence-asymptotic unitary designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.6.1

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xiv

6.7

6.6.2

Ordinary unitary designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.6.3

Sequence-asymptotic unitary designs . . . . . . . . . . . . . . . . . . . . . . . 113

A theory of Direct RB with gate-dependent errors . . . . . . . . . . . . . . . . . . . . 119
6.7.1

Assumptions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.7.2

Background: gauge-invariant metrics . . . . . . . . . . . . . . . . . . . . . . . 121

6.7.3

An exact theory of direct randomized benchmarking . . . . . . . . . . . . . . 122

6.7.4

Modelling the direct RB decay with twirling superchannels

6.7.5

Modelling the direct RB decay using the spectrum of twirling superchannels . 127

6.7.6

The direct RB decay is approximately exponential . . . . . . . . . . . . . . . 129

6.7.7

The direct RB measures a version of infidelity . . . . . . . . . . . . . . . . . . 132

6.7.8

Validating the theory using numerical simulation . . . . . . . . . . . . . . . . 135. . . . . . . . . . 124

6.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.9

Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.10 Sampling distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 The Impact of Markovian Errors on Random Circuits

142

7.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.3

Errors, Gate Sets and Gauge in Quantum Circuits . . . . . . . . . . . . . . . . . . . 144

7.4

Error Factorization in Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.5

7.4.1

Factoring Errors out of Random Circuits . . . . . . . . . . . . . . . . . . . . . 148

7.4.2

DRB as Random Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

The Impact of Markovian Errors on the Depolarization Rate . . . . . . . . . . . . . . 151
7.5.1

Active Errors Don’t Contribute to the Error Rate . . . . . . . . . . . . . . . . 152

7.5.2

The Ground Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.5.3

Gauge Action Determines Lower Eigenvalues . . . . . . . . . . . . . . . . . . 155

7.5.4

Example: RX(π/2), RY(π/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

xv
7.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8 Summary and conclusions

163

References

165

Figures

Figure1“A Quantum Computer in the Style of Van Gogh” - DALL·E 2 . . . . . . . . . . . .

1.1(a) Ions in a Penning trap. The ions are axially confined by a set of cylindrical

1⃗electrodes and rotate with frequency ω. In the presence of a strong magnetic field B
along the ẑ direction, the induced Lorentz force radially confines the ions. A moving
1D optical lattice formed by interfering two beams with angular frequencies ωodfand ωodf + µ respectively, generates an optical dipole force. (b) Imprinting a phasepattern with a DM. By reflecting one of the laser beams off of a deformable mirror(grey) surface, we can imprint a phase pattern on the wavefront (blue) of δu (xL , zL ),where xL , yL and zL are the beam-centric coordinates. The phase pattern δu (xL , zL )can be decomposed into a basis of Zernike polynomials. (c) To generate a distortion
δ(x, y) at the ion crystal, one must generate a distortion in the upper beam given byδu (xL , zL ) ≡ δ[x, zL / sin θ] with θ the angle between the beam propagation direction
and the y−axis. The ions (blue) are at z = 0. . . . . . . . . . . . . . . . . . . . . . .

6

xvii
1.2

(a) Error from the Rotating Wave Approximation. Starting in the state|ψ(0)⟩ =√1 (|↑⟩ + |↓⟩),
2

we plot ⟨σX ⟩ as a function of time under evolution by

Eq. (1.37) with µ = ω and δ(ρ, ϕ) = Aρ cos (ϕ) for A = 0.25, and different values of ω at (ρ, ϕ) = (1, 0). These expectation values are compared with those comingfrom the rotating wave approximation given by Eq. (1.13) with m = 1. We see thatby increasing the ion rotation frequency from ω = 2π ×43.8 kHz to ω = 2π ×180 kHz,the correction terms in Eq. (1.40) get suppressed so that the maximum infidelity is nolarger that 10−2 . (See Fig. 1.2b). The pink dots mark times that are positive integer
multiples of 2π/43.8kHz where the RWA error vanishes for the slower trap rotation
frequency. (b) Distribution of errors from RWA. Panel (a) samples ⟨σX ⟩ at 1000
different points in time. Panel (b) shows the histogram of the log infidelities between
the exact evolution and evolution under Eq. (1.13) evaluated at the same points intime. Choosing ω = 2π × 43.8 kHz results in most of the infidelities being larger
than 10−2 . Increasing the rotation frequency to ω = 2π × 180 kHz, the maximum
infidelity is decreased to smaller than 3 × 10−3 (orange). By sampling many different
values of evolution time, we can be confident that the infidelity contributed by the
RWA for arbitrary angles of rotation are sufficiently small for high-fidelity generationof |ψ(T )⟩ [Eq.(1.33)]. Red and blue lines have been added to the red and blue distributions, respectively, to highlight the mean and median, which are indistinguishable
at this scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3

17Absolute value of the error E = |F − F̃ |/A in reconstructing an annulus. With
nmax = 24 and m = 0 we can reconstruct an annulus with an error no larger than
0.05. This is sufficiently low error to reproduce a high fidelity state. (See Fig. 1.5.)
The red dots represent the ion positions and are shown for reference. . . . . . . . . .

1.4

23

⟨σX ⟩ for an annulus after following the protocol in Sec. 1.4. Using nmax = 24 and m =0, we see that ⟨σX ⟩ is −1 on the annulus and 1 outside the annulus, as desired.Thepink pattern illustrates the targeted AC Stark shift pattern F .

. . . . . . . . . . . .

24

xviii
1.5

Infidelity I [see (Eq.

1.35)] for preparing an annulus following the protocol ofSec. 1.6.1. (a) For n ≤ 24 and m = 0, the maximum infidelity is smaller than
10−2 . (b) For n ≤ 54 and m = 0, the maximum infidelity is smaller than 10−3 . (c)A histogram over the ions’ infidelities, comparing (a) and (b). . . . . . . . . . . . . .
1.6

24

Absolute value of the error E = |F − F̃ |/A in reconstructing an elliptical Gaussian.
By using nmax = 26 and mmax = 10 with m always positive we can reconstruct anelliptical Gaussian pattern with a maximum error of less than 0.09. Nevertheless the
maximum error at an ion is approximately 0.035 since there are no ions sitting at the
maximum of the phase pattern. The periodicity in the error pattern is mainly set by
the mmax value considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7

26

⟨σX ⟩ for an elliptical Gaussian. We use n ≤ 26, 0 ≤ m ≤ 10, and A = 0.5. Ions
along the center of the elliptical phase pattern are rotated towards |−⟩, while those
far away from the center remain in |+⟩. . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

27

Infidelity for preparing an elliptical Gaussian with the serial protocol. (a) For n ≤ 26,
0 ≤ m ≤ 10, and A = 0.5, the maximum infidelity is smaller than 10−2 . (b) Forn ≤ 32, 0 ≤ m ≤ 12, and A = 0.5, the maximum infidelity is smaller than 10−3 . (c)A histogram over the ions’ infidelities, comparing (a) and (b). . . . . . . . . . . . . .

1.9

28

Infidelity for preparing an elliptical Gaussian with the parallel protocol. (a) Forn ≤ 26 and 0 ≤ m ≤ 10. By applying all polynomials and beatnotes simultaneously,with A = 0.4, the resulting maximum infidelity is less than 10−2 . (b) With n ≤ 32,
0 ≤ m ≤ 12, but and with A = 0.2, the maximum infidelity is no more than 3 × 10−3
with all polynomials and beatnotes applied simultaneously. (c) A histogram over theions’ infidelities, comparing (a) and (b). . . . . . . . . . . . . . . . . . . . . . . . . .

29

xix
1.10 Absolute value of the error E = |F − F̃ |/A in reconstructing a displaced Gaussian.
With nmax = 40 and |mmax | = 9 we get a maximum error smaller than 0.06. The
error is worse around the targeted ion since this is the sharpest feature trying to bereconstructed with 20 dark fringes corresponding to the extrema of the next omitted
m = 10 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

1.11 ⟨σX ⟩ for a displaced Gaussian after following the protocol in Sec. 1.4. Using nmax = 40
and |mmax | = 9, we see that a single ion spin is rotated to a very good approximation. 31
1.12 Infidelity for preparing a displaced Gaussian with the serial protocol. (a) For n ≤ 40,−9 ≤ m ≤ 9, and A = 3.0 the maximum infidelity is smaller than 10−2 . (b) Forn ≤ 40, −20 ≤ m ≤ 20, and A = 3.0 the maximum infidelity is smaller than 10−3 .
(c) A histogram over the ions’ infidelities, comparing (a) and (b).

. . . . . . . . . .

32

1.13 Infidelity for preparing a displaced Gaussian with the parallel protocol. (a) For n ≤ 40
and −9 ≤ m ≤ 9. By applying all polynomials and beatnotes simultaneously, with
A = 0.3, the maximum infidelity is smaller than 10−2 . (b) For n ≤ 40, −20 ≤ m ≤ 20,
and still with A = 0.3, the maximum infidelity is less than 10−3 . (c) A histogram
over the ions’ infidelities, comparing (a) and (b). . . . . . . . . . . . . . . . . . . . .
2.1

33

An example of a circuit implementing the QAOA from Eq. 2.1, using the MAX-CUT
Hamiltonian in Eq. 2.2. In this example, we’ve assumed that there are only edges
between nearest neighbors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

39

Example Graph from Section 2.4. Above we see a graph on v = 20 vertices,
generated according to the distribution described in Sec. 2.4. Edgeless vertices areomitted from the figure. We sample e = 3v/5 edges at random, and edge weights arechosen uniformly at random from [−1, 1].

. . . . . . . . . . . . . . . . . . . . . . . .

44

xx
2.3

Optimization using dual annealing. Each white point corresponds to a single
shot measured by dual annealing. The kernel density estimates show peaks corresponding to the local maxima, demonstrating that optimizer preferentially samples
from higher energies. Moreover, the red star shows the final sampled point which lies
near a global maximum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

46

Cumulative maximum statistics. The orange line plots all 326 points sampled
by the dual annealing optimizer, while the blue is the maximum cut seen so far.
While Fig. 2.3 shows that the optimizer preferentially samples from higher energies,
this figure shows that the single shot data can vary substantially. Despite this, theoptimizer quickly finds a good cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

46

Comparison of the encoded fault-tolerant circuit to the bare circuit. a) A physical,
fault-tolerant circuit encoding the logical circuit shown in Fig. b. Following state
preparation and each logical gate, a syndrome measurement is applied, denoted by
S. This corresponds to the case of one gate per stabilizer measurement round. Finalmeasurement outcomes are post-selected on obtaining trivial error syndromes. The
logical operators XL and ZL in Fig. b are implemented via the physical operators
XIXI and ZZII. b) Qubit ℓ0 is acted on by logical gates XL and ZL , and thenmeasured. We say an experiment demonstrates fault tolerance if a family of logicalcircuits (one member of which is shown in b)) has a higher success probability on
average when implemented with encoded circuits (one example shown in a)), rather
than with the corresponding unencoded circuits (a single X followed by a single Z, in

3.2

this example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Logical zero (|0L ⟩) preparation circuit for the [[4, 1, 2]] Bacon-Shor code. . . . . . . .

54

xxi
3.3

(a) Error probability thresholds computed through site-counting. While the typicalvalue is an order of magnitude lower than in the case of the numerics, the shape of thebound is qualitatively similar, showing an improved threshold with a nontrivial rateof intermediate measurements. (b) Post-selection probabilities computed throughsite-counting, evaluated at the threshold. We see that the results approach a fixedvalue when only one measurement is performed at the end.

4.1

. . . . . . . . . . . . . .

57

IQFI (K(T )) for using Xπ/2 pulses. Depending on the magnetic field strength we
observe a more rapid accumulation of IQFI. The dashed lines are ∼ T and ∼ T 2
scalings to guide the eye. For non-zero magnetic field, we see that a crossover from
quadratic to linear scaling occurs when when BT ζ ≪ 1 (where the perturbative
results are valid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

68

QFI as a function of frequency for some protocols considered in this paper, for different protocol times T [(a) T = 2, (b) T = 4, (c) T = 6, and (d) T = 8 seconds], withBζ = 1 Hz. Ramsey has a large DC QFI, as expected, and very little ability to detect
any AC signal. π-pulses, on the other hand, can be used to measure signals at higher
frequencies, as might be expected from spectroscopy techniques such as CPMG. The
gX protocol, with g = 2π × 14 Hz in this example, is seen to be sensitive near 2g.
Moreover, it is seen to be sensitive over a broad bandwidth. The π-protocol shown
here consists of π rotations about the X axis, at each integer value of time. The
π/2-protocol consists of π/2 rotations about the X axis, every half second. Similar
to the π-protocol, the π/2-protocol has AC sensitivity, as well as more broadband
sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

xxii
6.1

An illustration (adapted from [167]) comparing the varied-length random circuits
used in the standard RB protocol of Magesan et al. [154, 155] (“Clifford group RB”)
and the streamlined direct RB protocol that was introduced in [167] and that weinvestigate further here. Direct RB can be implemented on almost any n-qubit gateset (a set of circuit “layers” or “cycles”) that generates a unitary 2-design. The caseshown here is when that gate set is a set of Clifford gates (cnot, Hadamard, and thephase gate) that generate the n-qubit Clifford group. The circuits of direct RB canbe shallower than those of Clifford group RB (e.g., consider the case of d = 0 in eachcircuit), allowing for RB on more qubits. . . . . . . . . . . . . . . . . . . . . . . . . .

88

xxiii
6.2

Simulated n-qubit direct RB and Clifford group RB for n = 2, 4, 6, . . . , 14. Here,
direct RB is benchmarking n-qubit gates that consist of a single layer of parallelone- and two-qubit gates. Clifford group RB is benchmarking the n-qubit Cliffordgroup generated out of these layers. In this simulation, an imperfect n-qubit layer
is modelled by the perfect unitary followed by independent uniform depolarization
on each qubit at a rate of 0.1%. The points and violin plots are the means and the
distributions of the estimated circuit success probabilities versus benchmark depth(d), respectively, and the lines are fits of the means to Sd = A + Bpd . The estimatedRB error rates (r̂), which are reported in the legends, are obtained from the fit
decay rates, using Eq. (6.15). We observe that the direct RB average circuit successprobabilities (Sd ) decay exponentially for all n, and that the direct RB error rate isgiven by r̂ ≈ 1−(1−0.001)n ≈ n×0.001. Direct RB is therefore accurately estimating
the error rate of the n-qubit gates it is benchmarking. The Clifford group RB error
rate grows very quickly with n—as it is benchmarking the n-qubit Clifford group,
whose elements must be compiled into the available layers of parallel one- and twoqubit gates. The Clifford group RB d = 0 intercept quickly decays to approximately
12n

as n increases, at which point r is close to 100% and it cannot be estimated toreasonable precision with a practical amount of data. This demonstrates that direct
RB is feasible on substantially more qubits than standard Clifford group RB. . . . .

99

xxiv
6.3

A comparison of the mean number of two-qubit gates in circuits for implementing
a uniformly random n-qubit Clifford gate either unconditionally (green connected
points) or conditionally on the input of |0⟩⊗n (blue line). In the latter case, thisis a circuit mapping |0⟩⊗n to a uniformly random n-qubit stabilizer state. This
demonstrates the significant increase in the number of qubits on which direct RB
is feasible—for a fixed two-qubit gate error rate—compared to Clifford group RB.
This data was generated using the same open-source compilers that we use for all
our direct RB and Clifford group RB simulations [224], and here we compiled into
one-qubit gates and cnot gates between any pair of qubits (i.e., all-to-all connectivity).107

6.4

The eigenvalues of the LG̃,G,Ω matrix for the single qubit gate set G = {X( π2 ), Y ( π2 )}without errors [red circles], i.e., G̃ = G, and with a small coherent over-rotation
error on each gate [black crosses], i.e. G̃ = {X( π2 + ϵ), Y ( π2 + ϵ)}, with ϵ = 0.1.
There are two unit eigenvalues for the error-free gate set, and all other eigenvalues
have absolutely value strictly less than 1. Here the upper spectral gap—meaning thedifference in the magnitudes of the second and third largest eigenvalues of L—is 12
for the error-free gate set, and it is still close to 21 for the imperfect gates. . . . . . . 131

6.5

Verifying the accuracy of our L matrix theory for direct RB, using numerical simulations. We show the theory’s prediction for the direct RB error rate (rγ ) versus anestimate r̂Ω of the true direct RB error rate (rΩ ) obtained by simulating direct RB.
Each data point shows rγ versus r̂Ω for a different randomly generated Markovian
error models (see main text for error model details). We find that rγ and r̂Ω areconsistent, but note that small systematic deviations between rγ and rΩ would not
be visible due to the finite sample error on r̂Ω . . . . . . . . . . . . . . . . . . . . . . . 133

xxv
6.6

Verifying the accuracy of our L matrix theory for direct RB, using our exact R matrix
theory. We show the L matrix theory’s prediction for the direct RB error rate (rγ )versus rΩ computed from our exact R matrix theory (the R matrix theory enables
computing Sd exactly, and we then fit Sd to Sd = A+Bpd ). Each data point shows rγ
versus rΩ for the same randomly generated Markovian error models as in Figure 6.5.
The inset shows a histogram of the relative error [(rΩ − rγ )/rΩ ], demonstrating that
our L matrix theory is extremely accurate. We observe extremely close agreement
between rΩ and rγ , with no more than ±1% relative error (see inset histogram). . . . 135

7.1

On the left, the circuit has three layers, with errors (denoted as E1 and E2 ) after eachof the first two layers. It is possible to factor the errors to the right hand side of
the circuit (the third layer), but the errors will in general be transformed. This is′ after the third layer of the circuit. . . . . . . 147
denoted on the right with the error E12

7.2

A simple gate set with a single relational error [247]. Intrinsic errors on either gateincrease the decay rate in proportion with the fraction of the gate being introduced.
Relational errors, on the other hand, can form an “inverse arch” - they contribute themost when ratio of gates is uniform. This is evidence that the errors are truly properties of the gate set, rather than of any individual gate. This figure also demonstrates
that the L−matrix theory is descriptive in this case despite not being “benchmarkingcompatible” [246] near p = 0 and p = 1, giving evidence that the theory in that work
is more reliable than proven in the previous chapter, and is applicable to this work. . 153

7.3

We sample 500 random Markovian error models, with leading order errors of size ϵ,for the RX(π/2), RY (π/2) gate set, at Ω = (1/2, 1/2), and demonstrate that ourformula is correct. The residuals are quadratic, demonstrating that we have thecorrect formula to first order. The fit line is proportional to x3/2 , demonstrating thatthe error in our estimate is O(ϵ3/2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

xxvi
7.4

Interestingly, in the limit of p = 0 or p = 1, each eigenvector has support on a single
FOGI direction. The largest eigenvalue diverges at the extremal sampling distributions, since in these limits coherent over-rotations on a single gate can coherently
amplify to very large errors. The smallest eigenvalue, conversely, is seen to havesupport only on the other gate in the extremal limits, since these errors appear vanishingly often. Finally, the intermediate eigenvalue is completely supported by the
relational error at the extremal points. . . . . . . . . . . . . . . . . . . . . . . . . . . 160

1Figure 1: “A Quantum Computer in the Style of Van Gogh” - DALL·E 2

Chapter 1

Individual qubit addressing of rotating ion crystals in a Penning trap [1]1.1

Abstract
Trapped ions boast long coherence times and excellent gate fidelities, making them a usefulplatform for quantum information processing. Penning traps offer the possibility of trapping large
two-dimensional crystals of several hundred ions. The ions are confined by controlling the rotationof the ion crystal in the presence of a strong magnetic field. However, the rotation of the ion crystal
makes single ion addressability a significant challenge. We propose a protocol that takes advantage
of a deformable mirror to introduce AC Stark shift patterns that are static in the rotating frameof the crystal. Through numerical simulations we validate the potential of this protocol to perform
high-fidelity single-ion gates in crystalline arrays of hundreds of ions.

1.2

Introduction
Ions confined in RF traps are one of the leading platforms for quantum information processing[2–5]. They hold the record for the highest fidelity entangling gates [6–10], and have exceptional
coherence times [11]. However, scaling to larger numbers of ion qubits, potentially solved by photonic
interconnects between Paul traps [12], or by shuttling ions in the quantum charge-coupled device(QCCD) architecture [2], remains a key challenge. In this work, we consider Penning ion traps
which offer the possibility of quantum information processing with samples of as many as 500
trapped ions self-assembled in a large two-dimensional (2D) Coulomb crystal. They use a set ofcylindrical electrodes and static voltages to generate axial confinement. The radial confinement is3provided by the Lorentz force experienced by the ions as they undergo a controlled rotation about
the trap symmetry axis in the presence of a strong axial magnetic field, typically generated by asuperconducting magnet [see Fig. 1.1a]. Following the standard approach of encoding a qubit in
two internal levels of the ions, Penning traps have the potential to perform quantum information
processing with hundreds of qubits. These qubits typically have transition frequencies from 10’s to
100 GHz, making microwaves suitable for global addressing [13]. We note that other ion trapping
technologies also allow for the generation of 2D crystals of ions. For instance, it is possible to use theradial degree of a freedom in a Paul trap, or to introduce arrays of closely spaced individual traps[14–17]. To date the number of ions used for quantum information processing in these experiments
has been modest (< 20).
Two-qubit gates for entangling the ions in Penning traps have been engineered via spindependent optical dipole forces [18]. These forces are generated by interfering two lasers with adifference frequency (sometimes called the beatnote) adjusted to excite phonons in the crystal [seeFig. 1.1a]. Virtual excitation of the center-of-mass mode, for example, generates collective spin-spin
interactions across the ion array.
Single-site rotations along with global rotations and a global entangling operation form auniversal set of operations for quantum computation—that is, they can be used to implement every
unitary operation [19]. Thus, an important step in enabling general quantum information processing
in a Penning trap is the ability to perform individual-qubit rotations.
A well-known technique for introducing single-site qubit rotations with one-dimensional (1D)
ion strings is through AC Stark shifts with off-resonant, focused laser beams [20]. AC Stark shifts
produce σ Z rotations, which can be turned into more general rotations through the application of
global rotations.
The analogous implementation of variable AC Stark shifts in a Penning trap is feasible, but
requires introducing focused co-rotating laser beams. The fast rotation frequency used in typicalexperiments [21–25], ranging from tens to a few hundred kHz in recent NIST experiments, makes
this task challenging. Variable AC Stark shifts can also be implemented with spatially fixed, focused

4
beams directed at the correct radius so that an ion experiences a time-varying AC Stark shift as
it rotates through the off-resonance laser beam. Similarly a fixed, focused pair of laser beams ina stimulated Raman configuration can be used to generate a spin rotation as an ion qubit passes
through the laser beam waists. However, such approaches would in general require some sequential
addressing of the ions, which is inherently slower than parallel addressing.
Here we propose another path for introducing variable AC Stark shifts that are static in therotating frame of the ion crystal by using the same optical dipole force that is used for implementing a
global entangling operation [18, 21]. This can be done by introducing distortions (or, more precisely,
spatially dependent phase offsets) to the wavefront of the optical dipole force while setting thebeatnote frequency to be a multiple of the rotation frequency [see Figs. 1.1b and 1.1c]. As we willsee, a nice feature of this technique is that the rotations of the ions’ spins can be conducted inparallel.
Spatially dependent wave front deformations can be implemented with a deformable mirror(DM). DM technology has been under development for many decades, motivated by the desire ofadaptive wavefront control in astronomy [26, 27]. For the application discussed in this article, DMs
with good wavefront control that employ many physically small actuators with fast switching speeds
are desirable. Devices which employ more than 4000 actuators that are 300-micrometer in sizewith a mechanical response time of order 50 microseconds are commercially available. Wavefront
control of better than 10 nm has been documented [26]. The surface of the DM can be coatedwith aluminum providing good reflectivity down to near ultraviolet wavelengths (< 300 nm). The
technology continues to rapidly develop with anticipated improvements in the actuator size andspeed.
Any wavefront distortion on the unit disk can be decomposed into the basis of Zernike polynomials [28]. Such functions are generally expressible as5Znm (ρ, ϕ) =Rn|m| (ρ) cos (mϕ), for m ≥ 0,
Rn|m| (ρ) sin (mϕ),

.

(1.1)

for m < 0,|m|

Here n ≥ |m|, Rn (ρ) are radial polynomials defined on the unit disk. For example, the first few
Zernike polynomials are

Z00 (ρ, ϕ) = 1(1.2)

Z1−1 (ρ, ϕ) = 2ρ sin(ϕ)(1.3)

Z11 (ρ, ϕ) = 2ρ cos(ϕ).

(1.4)

Through the decomposition of wavefront distortions in the basis of Zernike polynomials, wemotivate two protocols in Sec. 1.3 for imprinting an AC Stark shift pattern across the crystal ofions. A numerical simulation of the protocols is outlined in Sec. 1.4, followed by a discussion of three
primary sources of error in Sec. 1.5. These errors are from ignoring off-resonant terms, consideringa finite number of Zernike polynomials, and applying distortions that are too large. However, inSec. 3.5 we show with numerics that the errors can be controlled to have maximum infidelities as
small as 10−3 . In particular, we demonstrate the faithful reconstruction of an annulus, an elliptical
Gaussian, and a displaced Gaussian using parameters that are representative of typical Penning trap
conditions [18, 21, 22]. The annulus and elliptical Gaussian patterns produce initial states that areinteresting for quantum simulation. For example, both patterns are 2D analogs of a domain wall,and might allow us to study non-trivial spin transport phenomena [29]. The displaced Gaussian
is chosen to rotate a single qubit in the crystal. The high-fidelity reconstructions suggest that our
protocol provides a path forward for implementing high-fidelity single- and multi-site qubit rotations
in a Penning trap.

6

(a)

(b)

(c)

Figure 1.1: (a) Ions in a Penning trap. The ions are axially confined by a set of cylindrical⃗ along the ẑ
electrodes and rotate with frequency ω. In the presence of a strong magnetic field B
direction, the induced Lorentz force radially confines the ions. A moving 1D optical lattice formed
by interfering two beams with angular frequencies ωodf and ωodf + µ respectively, generates anoptical dipole force. (b) Imprinting a phase pattern with a DM. By reflecting one of the laser
beams off of a deformable mirror (grey) surface, we can imprint a phase pattern on the wavefront(blue) of δu (xL , zL ), where xL , yL and zL are the beam-centric coordinates. The phase patternδu (xL , zL ) can be decomposed into a basis of Zernike polynomials. (c) To generate a distortion
δ(x, y) at the ion crystal, one must generate a distortion in the upper beam given byδu (xL , zL ) ≡ δ[x, zL / sin θ] with θ the angle between the beam propagation direction and they−axis. The ions (blue) are at z = 0.

7

1.3

Experimental Set-up and Protocols
In a Penning trap, a system of N ions is confined axially by voltages applied to a set ofcylindrically symmetric electrodes. Radial confinement is implemented by the Lorentz force experienced by the ions as they rotate around the trap symmetry axis in the presence of a static
B0 ẑ field [see Fig. 1.1a]. The ion crystal rotation frequency is precisely controlled by a rotating electric field [30]. In work at NIST [18, 21, 22], the 2 S 1/2 ground-state valence electron spin|↑⟩ ≡ |mJ = +1/2⟩ (|↓⟩ ≡ |mJ = −1/2⟩) in each trapped 9 Be+ ion encodes a spin-1/2 degree of freedom, which can be globally controlled by external microwaves resonant with the 124 GHz frequency
splitting of the electronic spin states in the B0 = 4.5 T magnetic field of the trap.
In a frame rotating with the ion crystal, the motion of the ions can be decomposed into
in-plane modes, which describe ion motion in the plane of the ion crystal, and axial or drumheadmodes, which describe ion motion transverse to the plane. For performing quantum simulations [18,21, 22], the drumhead modes are coupled to the spin degree of freedom by a spin-dependent optical
dipole force (ODF) produced by a pair of off-resonant laser beams far detuned from the nearest
optical transitions. The beams generate a 1D traveling-wave lattice potential at a frequency µ [seeFig. 1.1a]. The system can be well described by the Hamiltonian
ĤODF /ℏ = U

Xicos (δkẑi − µt + ψ)σ̂iZ .

(1.5)

Here U is the zero-to-peak AC Stark shift, δk the wave vector of the moving lattice potential, andψ is an initial phase. For clarity, we will use capital letters to denote directions in spin space, e.g.,σ̂ Z , and lowercase letters to denote directions in real space, e.g., ẑ.
In what follows we show how we can modify the ODF interaction in Eq. (1.5) to introduce an
AC Stark shift pattern that is static in the rotating frame of the crystal through the introductionof wave front deformations. We assume Eq. (1.5) does not excite spin-dependent motion, which isreasonable if µ is far off-resonant with any modes.
It is convenient to assume that without a DM the two ODF beams have flat wavefronts. Thenposition-dependent phase offsets (or wavefront distortions) δ(x, y) can be implemented in Eq. (1.5)

8with a single DM that is inserted into one of the ODF laser beam lines, say the upper beam. The
DM generates a position-dependent phase offset δu (xL , zL ) where xL , zL are the distances fromthe center of the beam in a coordinate system perpendicular to the k-vector of the laser beam
[beam-centric coordinates, see Figs. 1.1b and 1.1c]. We can choose the xL coordinate in the upper
beam-centric coordinate system to be the same as the laboratory frame x coordinate (xL = x) at thesingle-plane crystal. Because the ODF beam crosses the ion crystal with an angle θ, an ion locatedat the laboratory frame coordinates (x, y) samples the DM generated phase offset δu (x, y sin θ).
Therefore, to generate a wavefront offset pattern δ(x, y) at the ion crystal requires generating a
wavefront pattern in the upper beam given by δu (xL , zL ) ≡ δ(xL , zL / sin (θ)). In general, the DM
will be located at some distance from the ion crystal and an optical imaging set-up is required toimage the DM surface to the ion crystal with different demagnification ratios in the zL and xL
directions.
Suppose, as sketched in the previous paragraph, we introduce waveform deformations to the
1D optical-dipole lattice potential [Eq. (1.5)] so that the interaction is described by
ĤODF /ℏ = U

Xicos (δkẑi − µt + ψ + δ(xi , yi ))σ̂iZ .

(1.6)

lab
Here xi = ρi cos [ϕlabi (t)] and yi = ρi sin [ϕi (t)] are the laboratory frame coordinates of ion i in thez = 0 plane, and δ(x, y) is a local phase shift that describes the distortions of the optical dipole
force wavefronts. The azimuthal angle in the laboratory frame is related to the azimuthal angle in= ϕi − ωt where ϕi is independent of time for a stable crystal and ω isthe rotating frame by ϕlabi
the ion crystal rotation frequency.
Our goal is to employ wave front deformations δ(x, y) described in Eq. (1.6) to generate aneffective Hamiltonian given by
ĤEF F /ℏ = U

X

F (ρi , ϕi )σ̂iZ .

(1.7)

i

Here F describes an AC Stark shift pattern that is static in the rotating frame of the crystal.
To do this, we will consider two different protocols with slightly different choices of δ(x, y). Before

9
introducing these choices, we motivate a convenient decomposition of F (ρ, ϕ) by expanding in termsof Zernike polynomials, as described in Eq. (1.1). This gives a decomposition of the form

F (ρ, ϕ) =

∞
∞
X
X

m
Acm
n Zn (ρ, ϕ)

n=0 m=−∞
∞
X

=




−m
ARnm (ρ) cm
n cos (mϕ) + cn sin (mϕ)

n,m=0
∞
X

AP m (ρ) cos (mϕ) + AQm (ρ) sin (mϕ),

=

(1.8)

m=0

where we included A as an overall amplitude scaling of the pattern F (ρ, ϕ), and set max[F (ρ, ϕ)]/A =mm
1.0, which provides a normalization condition for the cm
n . The radial functions P (ρ) and Q (ρ)|m|are weighted sums of the appropriate Rn (ρ) for n ≥ |m|. Specifically, in expanding F (ρ, ϕ) in thisway, we gathered all terms with the same cos (mϕ) or sin (mϕ) azimuthal dependence.
We explore using the deformable mirror to generate each term in the sum of Eq. (1.8). Because
F in Eq. (1.7) can, in general, have arbitrarily high spatial frequency components (and the protocols
we consider necessarily generate AC Stark shifts up to some maximum order mmax ) truncation error
will be introduced by considering a finite number of terms in Eq. (1.8). This error will be consideredin Secs. 1.5 and 3.5 where we carry out a numerical analysis.
We first consider generating an AC Stark shift pattern proportional to the term
AP m (ρ) cos (mϕ)(1.9)

in Eq. (1.8). Suppose the DM is set to generate a distortion in Eq. (1.6) of the formδ̃(ρ, ϕlab ) = δem (ρ) cos (mϕlab ).

(1.10)

We now set ẑi = 0 (the ions are located in the z = 0 plane) and substitute ϕlab
= ϕi − ωt. Theni
Eq. (1.6) can be written
ĤODF /ℏ =U X  i[−µt+ψ+δ̃(ρ,ϕi −ωt)]e+ c.c σ̂iZ .
2
i(1.11)

10
The phase modulation term can be expanded in terms of Bessel functions, using the Jacobi–AngerPℓiℓθ
expansion [eiz cos θ = ∞ℓ=−∞ i Jℓ (z)e ],∞U X  i[−µt+ψ] X ℓ
ĤODF /ℏ =ei Jℓ [δem (ρi )]eiℓ(mϕi −mωt) + c.c. σ̂iZ .
2
i(1.12)ℓ=−∞

By setting µ = mω only the ℓ = −1 term is static in the rotating frame. All other terms are rapidly
oscillating and can be ignored. It is also possible to get static terms by choosing µ to be a higher
integer multiple of mω. However, these terms will be scaled by a higher-order Bessel function, andtherefore produce a smaller static AC Stark shift, assuming sufficiently small arguments δem (ρi ).
Furthermore, we will show in Sec. 1.5.1 and Appendix A that the contribution of the fast rotating
terms is exactly zero if we choose to apply HODF for a duration T satisfying ωT = 2πr, with r an
integer value.
After some algebra we obtain,
ĤODF /ℏ ≈ U

Xi

J1 [δem (ρi )] sin (mϕi − ψ)σ̂iZ .

(1.13)

Therefore by choosing a distortion of the formAδem (ρ) = J1−1 ( P m (ρ)),
2(1.14)

Eq. (1.6), under the approximations discussed above, reduces to
ĤODF /ℏ ≈UX
AP m (ρi ) sin (mϕi − ψ)σ̂iZ .
2(1.15)iWith ψ = −π/2 this is exactly the targeted AC Stark shift pattern of Eq. (1.9) (up to a factor of
1/2 which we introduce for convenience as we will discuss later).
In Eq. (1.14), we denote the choice of the wave front pattern described by J1−1 [ A2 P m (ρ)] rather
than A2 P m (ρ) as precompensation. Precompensation is always possible as long as | A2 P m (ρ)| ≲ 0.58,
where 0.58 is the approximate maximum value that the J1 Bessel function can take. This conditionis always possible to satisfy by choosing A small enough in Eq. (1.14). Note that this limits thephase offset δem (ρ) to be |δem (ρ)| ≤ 1.84.

11
An identical derivation for odd Zernike polynomials reveals that a deformation of the form,A
δ(ρ, ϕlab ) = J1−1 ( Qm (ρ)) sin (mϕlab )2(1.16)

gives,ĤODF /ℏ ≈ −UX
AQm (ρi ) cos (mϕi − ψ)σ̂iZ .
2(1.17)

i

Additionally, for m = 0 corresponding to a circularly symmetric pattern, Eq. (1.11) reduces to
ĤODF /ℏ = U

X

cos (δe0 (ρi ) + ψ)σ̂iZ ,(1.18)

iso that choosing δe0 (ρ) = cos−1 [AP 0 (ρ)] − ψ for the m = 0 terms reproduces the desired phasepattern Eq. (1.9). By sequentially setting the DM to generate each even [AP m (ρ) cos (mϕ)] andodd [AQm (ρ) sin (mϕ)] term in Eq. (1.8) for each m, the above derivation shows that one can apply
any AC Stark shift pattern F (ρ, ϕ) [see Eq. (1.8)]. However, sequential application can take a longtime if there are many terms and the reset time of the DM is slow. Thus, it would be good to havea technique for applying all azimuthal phase patterns in parallel.
Applying in parallel means applying even and odd orders at the same time and applying
different beat note frequencies at the same time. First, we show that we can apply both even andodd terms simultaneously, by considering a distortion of the form
δ(ρ, ϕlab ) = δem (ρ) cos (mϕlab ) + δom (ρ) sin (mϕlab ) .

(1.19)

Using the Jacobi-Anger expansion, and setting µ = mω, we findĤODF /ℏ =U X  i(−µt+ψ)e
exp{(i[δem (ρi ) cos(mϕi − mωt) + δom (ρi ) sin(mϕi − mωt)])} + c.c. σ̂iZ ,2i

ĤODF /ℏ =(1.20)∞∞U X  i(−µt+ψ) X X aei Ja (δem (ρi ))Jb [δom (ρi )]ei[(a+b)(mϕi −mωt)] + c.c. σ̂iZ , (1.21)
2
a=−∞
i

b=−∞

12
Neglecting the fast rotating terms or operating with application times T where ωT = 2πr with r
an integer value (a condition at which the contribution of all non-static terms vanish), we obtain

ĤODF /ℏ ≈ U

X XiJa [δem (ρi )]Jb [δom (ρi )] cos (aa+b=−1a,b∈Z

π− mϕi + ψ) σ̂iZ .
2(1.22)For a = −1, and δom (ρ) = 0, this agrees with the expression in Eq. (1.13) ,ĤODF /ℏ ≈ U

X

=U

X

J−1 (δem (ρi )) cos (−ii

π− mϕi + ψ)σ̂iZ
2(1.23)

J1 (δem (ρi )) sin (mϕi − ψ)σ̂iZ .

However, for both δem (ρ) and δom (ρ) non-zero, there are now terms given by higher-order Bessel
functions that are static and non-zero.

ĤODF /ℏ ≈ U

X
i

J1 (δem (ρi ))J0 (δom (ρi )) sin(mϕi − ψ)

(1.24)−J1 (δom (ρi ))J0 (δem (ρi )) cos (mϕi − ψ) +(1.25)

J1 (δem (ρi ))J2 (δom (ρi )) sin (mϕi − ψ)

(1.26)

−J2 (δem (ρi ))J1 (δom (ρi )) cos (mϕi − ψ) − . . . σ̂iZ .
Fortunately, for small arguments, Jℓ (x) ≈ ℓ!1 ( x2 )n and J0 (x) ≈ 1 − ( x2 )2 , so that if we canchoose δem (ρi ) = AP m (ρi ) and δom (ρi ) = AQm (ρi ) to be small (by choosing A to be small), the first
two terms will reduce to the desired results [Eqs. (1.15) and (1.17)]. Note that we introduced thefactor of 1/2 in the precompensation step [Eqs. (1.14) and (1.16)] to make the outcome of the serial
and parallel protocols the same. The remaining terms will also be made small. Explicitly,

ĤODF /ℏ ≈U XAP m (ρi ) sin(mϕi − ψ) − AQm (ρi ) cos (mϕi − ψ)) + O(A2 ) σ̂iZ . (1.27)
2i

To first order in the arguments of the Bessel function, we see the even (AP m (ρ) cos (mϕ)) andodd (AQm (ρ) sin (mϕ)) terms in Eq. (1.8) can be treated additively, and the Hamiltonian consideredin Eq. (1.20) can be used to apply both the even and odd m components in parallel.

13
A similar analysis can be applied to show that it is possible to apply all different orders m at
the same time. For instance, consider the simplified case of two different nonzero even orders, m1
and m2 , and suppose we set the beatnote frequency to µ = m1 ω. Then our wavefront deformation
is given byδ(ρ, ϕlab ) = AP m1 (ρ) cos (m1 ϕlab ) + AP m2 (ρ) cos (m2 ϕlab )(1.28)and our Hamiltonian is(1.29)ĤODF /ℏ =U X  i(−µt+ψ)e
exp{(i[AP m1 (ρi ) cos(m1 ϕi − m1 ωt) + AP m2 (ρi ) cos(m2 ϕi − m2 ωt)}]) + c.c. σ̂iZ ,2
i(1.30)ĤODF /ℏ =∞∞U X  i(−µt+ψ) X X a+b
i Ja (AP m1 (ρi ))Jb (AP m2 (ρi ))ei(a(m1 ϕi −m1 ωt)+b(m2 ϕi −m2 ωt)) +c.c. σ̂iZ
e
2
a=−∞
i

b=−∞

As before, neglecting the fast rotating terms, or operating with application times T where ωT is apositive integer multiple of 2π (a condition at which all non-static terms vanish) we get
ĤODF /ℏ ≈ U

X

X

i

am1 +bm2 =−m1a,b∈Z

Ja (AP m1 (ρi ))Jb (AP m2 (ρi )) cos ((a + b)π
− m1 ϕi + ψ) σ̂iZ .
2(1.31)The lowest order terms occur when a = −1 and b = 0, resulting in

ĤODF /ℏ ≈U XAP m1 (ρi ) sin (m1 ϕi − ψ) + O(A2 ) . . . σ̂iZ .
2(1.32)i

For small A this is approximately the desired AC Stark shift pattern. If we set the ODF beatnoteµ = m2 ω we select an AC Stark shift pattern described by the second term in Eq. (1.28).

14
When considering terms with m = 0 and setting ψ = −π/2, the leading order contribution
is instead U AP 0 (ρi ), so that there is an additional factor of 2 multiplying the radial polynomial.
Note that the precompensation for m = 0 [see Eq. (1.18)] was chosen to also make the outcome ofthe serial and parallel protocols the same.
The above analyses support two experimental procedures for generating an AC Stark shift
pattern F (ρ, ϕ) that is static in the rotating frame of the crystal. The first is sequential: fora phase pattern with terms of at most order mmax , we sequentially set the DM to 2mmax + 1
different azimuthal phase patterns, applying the appropriate beatnote frequency at each step andthe corresponding precompensation in the applied waveform. This has the advantage of allowing
for larger amplitudes A and higher accuracy.
The second procedure is a parallel application: we set the DM once to a phase patternproportional to F (ρ, ϕ) at the ion crystal, and simultaneously (or in rapid succession) apply all
beatnote frequencies µm = mω, for 0 ≤ m ≤ mmax . The beatnote at µm will imprint an AC Stark
shift in the rotating frame of the ions proportional to P m (ρ) cos(mϕ) + Qm (ρ) sin(mϕ), rotating theion’s spins according to the mth-order component of F (ρ, ϕ). This has the benefit of being faster
if the pattern has a large number of frequency components, but at the cost of lower accuracy andrequiring smaller amplitudes. We note that small amplitudes [i.e., A in Eq. (1.8) or δ in Eq. (1.6)]
can be offset through the use of large U or long application durations T .

1.4

Numerical Simulation
In this section we outline a numerical study whose results are presented in Secs. 1.5 and 3.5

for preparing arbitrary qubit rotation profiles, F (ρ, ϕ), across the crystal. In the work that follows,
we set U = 2π × 10 kHz and ω = 2π × 180 kHz, which are typical experimental parameters [21, 22].
Our goal will be to prepare the ions in the state|ψ(T )⟩ =where T is the gate duration.

Oiie−iU F (ρi ,ϕi )T σ̂Z |+⟩i ,(1.33)

15
First, we will prepare all of the ions in the |+⟩ = √12 (|↑⟩ + |↓⟩) state, which can be easily done
by preparing all ions in |↓⟩ and then applying a global rotation around the Y axis. We will thendetermine a maximum m and n, based on the desired fidelity of the state preparation, such that wen for all |m| ≤ mmapproximately reconstruct F as F̃ , using only Zm
max and n ≤ nmax . Writing αnfor the coefficients of F in the Zernike basis, we haveX

F =A−n≤m≤n0≤n≤∞αnm Znm ≈ A

X

αnm Znm = F̃ .

(1.34)−mmax ≤m≤mmax
0≤n≤nmaxExperimentally, nmax could be constrained by the available resolution of the deformable mirror—
since an nth order polynomial is determined by n + 1 points, a mirror with N actuators in adimension can only hope to parametrize a family of polynomials of degree N − 1. In our analysis we
will assume that the DM has a sufficiently large number of actuators and prioritize minimizing mmax ,which sets the number of terms in the decomposition of F [see Eq. (1.8)] that will be included in
the reconstruction F̃ . Larger mmax in general requires a longer gate duration or higher laser power.
As discussed in Sec. 1.3, there are two ways to apply the full phase pattern—sequentially,
and in parallel. In the analysis that follows we will consider both of these approaches. Although
in principle one could consider using these techniques to apply arbitrary qubit rotations, for the
purposes of assessing the performance of our protocols we consider the experimentally useful exampleof π rotations. That is, in the examples considered in Secs. 1.5 and 3.5 the time evolution will be set
for a time T such that in the final state |ψ(T )⟩ the ion located at the maximum of the phase-pattern
F will be rotated by π radians in the XY -plane.

1.5

Sources of Error
We will now discuss three sources of error that can occur in the protocols described in Sec. 1.3.

To quantify the error and to analyze the performance of our protocol we will use the single-spin
infidelity
Ij = 1 − | ⟨Pj ψ̃(T )|Pj ψ(T )⟩ |2 ,(1.35)

16
where Pj traces out all but the jth ion. The quantity Ij is the infidelity of spin j in the simulated state|ψ̃(T )⟩ with respect to the target state |ψ(T )⟩. State-of-the-art quantum information processing
platforms often have single-qubit gate infidelities of 10−3 – 10−2 [31, 32] or less, and so this will be
the standard of comparison in our analysis. While trapped ion computers often have substantiallybetter single-qubit infidelities [6, 7, 33], our goal here is to demonstrate that Penning traps are acompelling candidate for quantum information processing, for which it is sufficient to study theselarger infidelities.

1.5.1

Rotating Wave Approximation

The first type of error comes from the rotating wave approximation, where we ignore therapidly oscillating terms in Eq. (1.12). As we will show below these errors nevertheless can be
avoided if one chooses the gate duration to be commensurate with the ion crystal rotation frequency:
ωT = 2πr, with r an integer.
As a simple example, we consider a phase pattern with a single angular order m and with anadditional amplitude parameter A that will be chosen sufficiently small so that precompensation is
not necessary. Thus our phase function in Eq. (1.6) is given asδ(ρ, ϕ) = AP m (ρ) cos (mϕ).

(1.36)For our simple Hamiltonian, we can analytically express the expectation value ⟨σX ⟩ of the spin after
evolving for time T . Writing f (τ ) as the time-dependent coefficient in Eq. (1.12), i.e., ĤODF /ℏ =PZ
j fj (t)σ̂j , and setting µ = mω, we find (for m ̸= 0)j⟨ψ̃(T )|σX|ψ̃(T )⟩ = cos

 Z T

2
fj (τ )dτ .

(1.37)

0

We can write
Z Tdτ fj (τ ) =
0

∞
X
ℓ=−∞

sj (ℓ),(1.38)

17

(a)

(b)

Figure 1.2: (a) Error from the Rotating Wave Approximation. Starting in the state|ψ(0)⟩ = √12 (|↑⟩ + |↓⟩), we plot ⟨σX ⟩ as a function of time under evolution by Eq. (1.37) withµ = ω and δ(ρ, ϕ) = Aρ cos (ϕ) for A = 0.25, and different values of ω at (ρ, ϕ) = (1, 0). These
expectation values are compared with those coming from the rotating wave approximation given
by Eq. (1.13) with m = 1. We see that by increasing the ion rotation frequency from
ω = 2π × 43.8 kHz to ω = 2π × 180 kHz, the correction terms in Eq. (1.40) get suppressed so thatthe maximum infidelity is no larger that 10−2 . (See Fig. 1.2b). The pink dots mark times that are
positive integer multiples of 2π/43.8kHz where the RWA error vanishes for the slower trap
rotation frequency. (b) Distribution of errors from RWA. Panel (a) samples ⟨σX ⟩ at 1000 different
points in time. Panel (b) shows the histogram of the log infidelities between the exact evolution
and evolution under Eq. (1.13) evaluated at the same points in time. Choosing ω = 2π × 43.8 kHz
results in most of the infidelities being larger than 10−2 . Increasing the rotation frequency toω = 2π × 180 kHz, the maximum infidelity is decreased to smaller than 3 × 10−3 (orange). By
sampling many different values of evolution time, we can be confident that the infidelity
contributed by the RWA for arbitrary angles of rotation are sufficiently small for high-fidelity
generation of |ψ(T )⟩ [Eq.(1.33)]. Red and blue lines have been added to the red and blue
distributions, respectively, to highlight the mean and median, which are indistinguishable at thisscale.

18
where
sj (ℓ) =U J1 [AP m (ρj )] sin[mϕj − ψ]T,if ℓ = −1,
4U Jℓ [AP m (ρj )] sin( 21 m(ℓ+1)T ω ) cos(mℓϕ+ 12 [πℓ−m(ℓ+1)T ω]+ψ )m(ℓ+1)ω

if ℓ ̸= −1.

(1.39)

Therefore, if U Jℓ [AP m (ρj )]/(mω) ≪ 1 then,j⟨ψ̃(T )|σX|ψ̃(T )⟩ ≈ cos (Θ) − sin (Θ)∞
Xaℓ [T (ℓ + 1)mω]ℓ=−∞,ℓ̸=−1

Θ = 2U J1 [AP m (ρj )] sin[mϕj − ψ]T,U,(ℓ + 1)mω

(1.40)(1.41)

for some function an . For m = 0 we have simply thatZ T

fj (τ )dτ = U cos [AP 0 (ρj ) + ψ]T,

(1.42)

0which has no time-dependent corrections.
As an example, choosing m = 1 and P 1 (ρ) = ρ, we will evaluate Eq. (1.37) at the point thatwill have the largest infidelity, ρ = 1 and ϕ = 0, with A = 0.25. This choice of A allows us to focus
primarily on effects from the RWA, and ignore the other sources of error discussed below. This
gives superimposed oscillations around a cosinuisoidal evolution, with corrections proportional toUmω . Thus, by increasing ω for fixed U , the RWA becomes more accurate. This is shown in Fig. 1.2a,

where the evolution is sampled at 1000 points in time. As ω is increased from 2π × 43.8 kHz (blue)
to 2π × 180 kHz (orange), the oscillations become smaller. We see that the ion completes a full π
rotation in nearly 200 µs, as expected from Eq. (1.13). Additionally, we see that at evolution times
that are positive integer multiples of 2π/ω the difference between the exact evolution and the RWA
is zero [see pink points in Fig. (1.2)]. This can be seen in Eq. (1.39), as the terms with n ̸= −1 arezero at these points. See also Appendix A for a more generic case. In Sec. 3.5 we choose evolution
times that take advantage of this fact. Note that for this simple case the fast-rotating terms also
vanish at T = π/ω.
Figure 1.2b shows a histogram of the log infidelity obtained at the different evolution times
sampled in Fig. 1.2a. By increasing the rotation frequency from 2π ×43.8 kHz (blue) to 2π ×180 kHz(orange), the maximum single-spin infidelity is decreased from approximately 5 × 10−2 to 3 × 10−3 .

19This implies that setting ω = 2π × 180 kHz is sufficient for obtaining infidelities of 3 × 10−3 orsmaller.

1.5.2

Truncation

The second source of error comes from the fact that in practice we apply a finite number ofbeatnote frequencies, truncating the basis expansion at finite order. This will produce imperfectreconstructions of the desired phase pattern. While choosing finite nmax is also a possible source oferror, as discussed earlier we choose nmax large enough that it is not the limiting factor—effectively
assuming that the DM has enough actuators to give good resolution.
The Zernike polynomials form an orthogonal set of polynomials on the disk, D, and therefore
arbitrary functions can be decomposed into these polynomials. The inner product on the space offunctions on the disk is given by:⟨F, G⟩ =ZDρdρdθF · G(1.43)

We can write the coefficients of F from Sec. 1.4 as
αnm =2n + 2 F m⟨ , Z ⟩,ϵm π A n(1.44)

where ϵm is 2 if m = 0, and 1 otherwise. The prefactor in Eq. (1.44) is due to the fact that thepolynomials are not normalized,′⟨Znm (ρ, ϕ)Znm′ (ρ, ϕ)⟩ =ϵm π
δn,n′ δm,m′ .
2n + 2(1.45)

Truncating the number of terms we include will give us a different phase function, F̃ , fromwhich we can define the error from truncation as
E = maxD (|F − F̃ |/A).

(1.46)

Note that max[F (ρ, ϕ)] = A [see discussion after Eq. (1.8)], so E is the truncation error normalized
to the maximum value of F .

20
The amount that this truncation contributes to the infidelity will vary depending on theparticular phase function F being considered, as we will see in Sec. 3.5. Here, to provide a roughestimate, we consider the maximum single-spin infidelity ε across the crystal, defined as
ε = maxj (Ij )= maxj (1 − | ⟨Pj ψ̃(T )|Pj ψ(T )⟩ |2 )
!
2 2
1
≈ maxj 1 − 1 −U AT [F (ρj , ϕj ) − F̃ (ρj , ϕj )]/A
2
≈ EU AT )2 ,(1.47)

where Pj traces out all but the jth ion.
We consider rotations where the ion located at the maximum of F is rotated by π radians,
corresponding to U AT = π/2. An infidelity requirement of ε, therefore, in general necessitates a√truncation error E ⪅ π2 ε. For an infidelity requirement of 10−2 (10−3 ) the maximum truncation
error should be less than 0.064 (0.02).

1.5.3

Linear Approximation

The final source of infidelity, which is only relevant for the parallel application discussed inSec. 1.3, is in assuming that A in Eqs. (1.27) and (1.32) is small enough so that ignoring higherorder terms is justified. By increasing the product U T and decreasing the amplitude A, the linear
approximation can be made arbitrarily good. However, increasing U T will also increase decoherence
due to off-resonant light scattering from the ODF beams [Uys2010] during the qubit rotations.
To estimate the contribution of higher-order terms in A on the infidelity for the target caseof a π-rotation on a single ion, we consider the parallel application of two P m1 , P m2 terms [see(Eq. 1.28)]. It can be shown that leading corrections, of order O(A2 ), arise when 2m1 = m2 . In
this case the leading-order static terms generate a Hamiltonian of the form

ĤODF /ℏ ≈


A2
U X
AP m1 (ρi ) sin (m1 ϕi − ψ)− P m1 (ρi )P 2m1 (ρi ) cos (m1 ϕi − ψ)+O(A3 ) (1.48)
2
2
i

21
We now compute the infidelity between the state |ϕ⟩ = e−iĤODF T /ℏ |+⟩ and |ϕ̃⟩ =e−i

U AT P m1 (ρ0 )σ̂Z2|+⟩ obtained by considering only the desired first order term. We choose ψ = −π/2

in the rest of this discussion. By Taylor expanding, we find the infidelity to beI(ϕ̃, ϕ) ≈2
U T 2 m1
A P (ρi )P 2m1 (ρi ) sin (m1 ϕi )4(1.49)

Thus, for P m1 (ρi )P 2m1 (ρi ) sin (m1 ϕi ) < 1, and U T A/2 ∼ π, the infidelity reduces toπI(ϕ̃, ϕ) ≲ ( A)2 .
2(1.50)We see that if A ≤ 0.02 the infidelity can be constrained to be less than 10−3 . If we relax our
infidelity requirements to 10−2 we can choose A as large as A = 0.06. While these amplitude
requirements may seem strict, we note that this estimate is pessimistic — we considered the worstcase situation when 2m1 = m2 , which gives leading order error contributions of size O(A2 ). As wewill see in Sec. 3.5, the amplitude can often be made larger. In fact, as we will see in Sec. 1.6.3
where we look at the case of flipping the spin of a single ion, the amplitude can be taken more thanan order of magnitude larger while achieving the same infidelity goals.

1.6

Numerical Results
Following the discussion of Secs. 1.4 and 1.5, we now numerically demonstrate a few interest-

ing examples of implementing different AC Stark shift patterns across a circular crystal with our
protocols from Sec. 1.3. The preparation of initial states with targeted spatial profiles can be ofgreat utility for investigating propagation of quantum information and entanglement. With that
purpose in mind, here we consider a range of geometries including an annulus, an elliptical Gaussian,and a displaced Gaussian. For these patterns results for ⟨σX ⟩ as well as the log infidelity across acrystal of 91 ions are shown. This number was chosen to have inter-ion spacings of 0.1 of the crystal
diameter. For the phase patterns considered, Figs. 1.3, 1.6, and 1.10 show the truncation error,

22|E|, from considering a finite number of Zernike polynomials. Next, we study the error generated
during the dynamical evolution. First we apply the protocol in series, evolving under each even[AP m (ρ) cos (mϕ)] and odd [AQm (ρ) sin (mϕ)] term in Eq. (1.8) that constitutes F̃ (ρ, ϕ) one-byone, for two different choices of target maximum infidelity. The infidelities of the final state areshown in Figs. 1.5, 1.8, and 1.12. We discuss contributions to the infidelity arising from the RWA
and truncation errors (see Sec. 1.5). Next, we apply the protocol in parallel for all m such that0 ≤ m ≤ mmax in Eq. (1.34). In Figs. 1.9 and 1.13 we show the corresponding infidelities for theelliptical and displaced Gaussians. (The annulus only requires implementing a single m = 0 term.)

1.6.1

AnnulusAs a first example, we consider preparing ions in an annulus. To make the problem ofreconstruction in a basis of continuous functions easier, we will smooth the edges with sigmoid
functions, givingg(ρ) =

11−.
1 + e−κ(ρ−r1 ) 1 + e−κ(ρ−r2 )(1.51)

Scaling this function to be one at its maximum, we have a targeted normalized AC Stark shift
pattern given by:1
F (ρ, ϕ) = Ag(ρ)/g(r1 + (r2 − r1 )),2(1.52)

which corresponds to the phase functionδ(ρ, ϕ) = cos−1 [F (ρ, ϕ)](1.53)

in Eq. (1.6). For our numerical experiment we will set r1 = 0.45, r2 = 0.55, and κ = 10. This valueof κ was chosen to avoid sharp rising and falling edges for the annulus.

1.6.1.1

Reconstruction

Because the phase pattern is azimuthally symmetric, the only nonzero coefficients have m = 0
and only the application of a single beatnote with frequency µ = mω = 0 is required. We see that
all terms are static, and therefore, incur no error from the RWA. We choose nmax to be sufficiently

23

Figure 1.3: Absolute value of the error E = |F − F̃ |/A in reconstructing an annulus. With
nmax = 24 and m = 0 we can reconstruct an annulus with an error no larger than 0.05. This issufficiently low error to reproduce a high fidelity state. (See Fig. 1.5.) The red dots represent theion positions and are shown for reference.

large (nmax = 24) so that the reconstruction error E, presented in Fig. 1.3, is less than 0.064. From
the discussion in Sec. 1.5 this should enable a single-spin infidelity of less than 10−2 . The next
contributing error term is radially symmetric, since the pattern itself is radially symmetric, which
can be seen clearly in Fig. 1.3.

1.6.1.2

Evolution

Given that all the terms in the Zernike expansion have the same value of m, they can be applied
simultaneously using the precompensation technique [Eq. (1.53]) above, incurring no errors fromthe linear approximation. Setting A = 1.0, we obtain a gate duration of 25 µs for U = 2π × 10 kHz,
which is significantly faster than typical decoherence times in trapped ions.
The expectation values ⟨σX ⟩ after performing the precompensation protocol is shown in
Fig. 1.4, and the infidelity to the target state is shown in Fig. 1.5. Here the infidelity is due
to finite n truncation. Figure 1.5a shows an infidelity better than 10−2 for nmax = 24. An infidelityof 10−3 can be obtained with nmax = 54 as shown in Fig. 1.5b. A histogram of the infidelities forthe two different nmax values is shown in Fig. 1.5c. The presented analysis shows that the protocol
can produce a faithful reconstruction of the annulus pattern with a small state infidelity (< 10−3 ).

24Figure 1.4: ⟨σX ⟩ for an annulus after following the protocol in Sec. 1.4. Using nmax = 24 andm = 0, we see that ⟨σX ⟩ is −1 on the annulus and 1 outside the annulus, as desired.The pink
pattern illustrates the targeted AC Stark shift pattern F .

(a)

(b)

(c)

Figure 1.5: Infidelity I [see (Eq. 1.35)] for preparing an annulus following the protocol ofSec. 1.6.1. (a) For n ≤ 24 and m = 0, the maximum infidelity is smaller than 10−2 . (b) For n ≤ 54
and m = 0, the maximum infidelity is smaller than 10−3 . (c) A histogram over the ions’infidelities, comparing (a) and (b).

25
1.6.2

Elliptical Gaussian

The next example we consider is an elliptical Gaussian,F (ρ, ϕ) =We choose ηx =√

A
exp (−(ρ cos (ϕ))2 /(2ηx2 ) − (ρ sin (ϕ))2 /(2ηy2 )2

2/10 and ηy =√(1.54)

2 to generate an elliptical pattern with a “narrow" and “wide"

distribution covering the 91-ion crystal. Because there is no longer azimuthal symmetry, we must
consider m > 0.

1.6.2.1

Reconstruction

We use a numerical integrator to compute the Zernike basis coefficients in Eq. (1.44). The
reconstruction error for mmax = 10 and nmax = 26 is shown in Fig. 1.6. Interestingly, there areradial bands, set by the value of mmax . For mmax = 10, the next higher-order omitted term has 12
full periods of oscillation corresponding to the 12 pairs of dark fringes in Fig. 1.6. Since the error
is minimized at the bright fringes, we can, in principle, achieve a higher fidelity reconstruction byadjusting mmax so that ions near the edge of the crystal in the y-direction are positioned near theminima of the reconstruction error. At any ion in the crystal, the reconstruction error E shown in
Fig. 1.6 is less than 0.035, which should enable an implementation of the elliptical Gaussian AC
Stark shift pattern with an infidelity less than 10−2 .
The error in the reconstruction is seen to be smaller near the center of the disk, since including
lower-degree radial polynomials can match the behavior there. To capture the phase pattern far
from the center of the disk requires higher-order radial polynomials. In fact the error is observed
to be maximum at the vertical wings of the distribution since to capture the decay of the Gaussian
closer to the ρ = 1 boundary requires higher-order terms.

1.6.2.2

Series Application

Figure. 1.7 shows ⟨σX ⟩ at the end of a series evolution with mmax = 10, nmax = 26, and
A = 0.5. Note that because of the elliptical symmetry, m is restricted to non-negative even values,

26

Figure 1.6: Absolute value of the error E = |F − F̃ |/A in reconstructing an elliptical Gaussian. By
using nmax = 26 and mmax = 10 with m always positive we can reconstruct an elliptical Gaussian
pattern with a maximum error of less than 0.09. Nevertheless the maximum error at an ion isapproximately 0.035 since there are no ions sitting at the maximum of the phase pattern. The
periodicity in the error pattern is mainly set by the mmax value considered.

27

Figure 1.7: ⟨σX ⟩ for an elliptical Gaussian. We use n ≤ 26, 0 ≤ m ≤ 10, and A = 0.5. Ions along
the center of the elliptical phase pattern are rotated towards |−⟩, while those far away from thecenter remain in |+⟩.

corresponding to six terms in total. We choose a gate duration of T = 18 × (2π/ω) = 100 µs andhence we remove the errors from the rotating wave approximation. The protocol generates spinrotations along the three vertical columns of ions in the center of the disk, while very nearly leaving
all other ions in the |+⟩ state. Figure 1.8a shows a maximum single-spin infidelity of 10−2 . With
mmax = 12, corresponding to seven terms, and making nmax = 32, Fig. 1.8c shows clearly that the
maximum infidelity is reduced to 10−3 .
Each application of one even [AP m (ρ) cos (mϕ)] term in Eq. (1.8) takes 100 µs. When donein series, the six different values of m employed to obtain an infidelity of 10−2 sets a gate durationof 600 µs, neglecting the reset time of the DM. A gate duration of approximately 700 µs is required
for the seven terms that enable an infidelity of 10−3 . The reset time of the DM can significantly
increase the gate duration. For example, a DM reset time of 50 µs increases the gate duration by50%.

1.6.2.3

Parallel Application

The protocol where all terms are applied in parallel has the advantage that the DM is only set
once. This can possibly lead to shorter gate durations. The targeted AC Stark shift [Eq. (1.54)] andchoices of parameters for the parallel application are the same as those considered in the previous
paragraphs with the exception of the choice of A and therefore T . The parameter A is chosen

28

(a)

(b)

(c)

Figure 1.8: Infidelity for preparing an elliptical Gaussian with the serial protocol. (a) For n ≤ 26,
0 ≤ m ≤ 10, and A = 0.5, the maximum infidelity is smaller than 10−2 . (b) For n ≤ 32,
0 ≤ m ≤ 12, and A = 0.5, the maximum infidelity is smaller than 10−3 . (c) A histogram over theions’ infidelities, comparing (a) and (b).

29

(a)

(b)

(c)

Figure 1.9: Infidelity for preparing an elliptical Gaussian with the parallel protocol. (a) Forn ≤ 26 and 0 ≤ m ≤ 10. By applying all polynomials and beatnotes simultaneously, with A = 0.4,
the resulting maximum infidelity is less than 10−2 . (b) With n ≤ 32, 0 ≤ m ≤ 12, but and with
A = 0.2, the maximum infidelity is no more than 3 × 10−3 with all polynomials and beatnotes
applied simultaneously. (c) A histogram over the ions’ infidelities, comparing (a) and (b).

sufficiently small so that the linear approximation discussed in Sec. 1.3 holds, and T is picked sothat the spins at the maxima of the phase pattern experience a full π rotation, as discussed inSec. 1.4. By applying the different orders in parallel, we incur all of the errors from the previous
section and additional errors from the linear approximation discussed in Sec. 1.5.3. Because then and m chosen in the previous section were minimally large to meet our fidelity requirements,
we keep them the same. For a maximum infidelity of 10−2 , Fig. 1.9a demonstrates that choosing
A = 0.4 is sufficient. This changes the gate duration to 250 µs, obtained with T = 45 × (2π/ω). If
we increase our infidelity requirements to 3 × 10−3 , we see in Figs. 1.9b and 1.9c that we can choose
A = 0.2. For the gate duration to be commensurate with the crystal rotation frequency we find
T = (90 × 2π/ω) giving a longer gate duration of 500 µs. We note that we chose 3 × 10−3 as thetarget infidelity rather than 10−3 as in the other examples that we show.

1.6.3

Displaced Gaussian

Finally, we consider an AC Stark shift pattern described by a displaced Gaussian,F (ρ, ϕ) =Aexp −{[ρ cos (ϕ) − δx ]2 + [ρ sin (ϕ) − δy ]2 }/(2η 2 )2(1.55)

30

Figure 1.10: Absolute value of the error E = |F − F̃ |/A in reconstructing a displaced Gaussian.
With nmax = 40 and |mmax | = 9 we get a maximum error smaller than 0.06. The error is worsearound the targeted ion since this is the sharpest feature trying to be reconstructed with 20 dark
fringes corresponding to the extrema of the next omitted m = 10 term.

√√with standard deviation η = 0.1/ 2, displaced by δx = 0.3 in x and δy = 0.1 3 in y. The
displacement was chosen to coincide with an ion in the crystal, and the width was chosen to achievea single-spin rotation. In particular, this choice of η corresponds to a Gaussian profile that decays bya factor of 1/e at a diameter of the inter-particle spacing. In this case, we no longer have azimuthal
symmetry, so we must include m ≥ 0 and m < 0 as well.
1.6.3.1

Reconstruction

In Fig. 1.10, we included up to m = ±9 and nmax = 40 terms. We see that the reconstruction
has a maximal error in a region surrounding the targeted ion. The maximum truncation error isapproximately 0.06, which should enable a single-spin infidelity of 10−2 . We note that choosing an
ion closer to the boundary of the crystal will require a large m value to reach similar truncation
error.

1.6.3.2

Series Application

In this example we consider A = 3.0, motivated by our infidelity goal of 10−2 . We picked
this value of A to be as large as possible while still requiring that each term being applied in the
expansion [Eq. (1.8)] can be inverted as discussed in Sec. 1.3 for precompensation. Additionally,

31

Figure 1.11: ⟨σX ⟩ for a displaced Gaussian after following the protocol in Sec. 1.4. Using nmax = 40
and |mmax | = 9, we see that a single ion spin is rotated to a very good approximation.

we chose T = 3 × (2π/ω), to remove the error from the RWA. For A = 3.0, the application timerequired for each beatnote is approximately 16.66 µs. When the polynomials are applied in series
for 19 different values of m, we get a gate duration of approximately 316.66 µs, again assuminga DM with zero reset time. If we increase our infidelity goals to 10−3 , we can choose |m| ≤ 20
and A = 3.0, giving a gate duration of 683.33 µs. As in the previous two cases, we see excellent
agreement with the desired phase pattern in the evolution of ⟨σX ⟩ shown in Fig. 1.11. The protocol
very nearly rotates a single spin, as all of the ions surrounding the desired ion are very nearly in
the |+⟩ state. This behavior leads us to conclude that single ion addressability is feasible with thistechnique, requiring about a factor of 2 more terms than in the other cases, due to the additionalodd (AQm sin (mϕ)) terms. The infidelities in Figs. 1.12a and 1.12b are worse in a radial bandof ions containing the ion being flipped. This is intuitive — suppressing that error requires theintroduction of high-order angular terms, while we used only up to m = 9.

1.6.3.3

Parallel Application

The parameters in this section are the same as those considered in the previous paragraphs,
with the exception of the choice of A and therefore T . Figures 1.13a and 1.13b demonstrate thatchoosing A = 0.3 is sufficient for the 10−2 and 10−3 infidelity requirements. Consequently, the gate
duration in both cases is approximately 333.33 µs, which is given as T = 60 × (2π/ω) and thus theerror incurred from the rotating wave approximation is zero. This reduces the gate duration for the

32

(a)

(b)

(c)

Figure 1.12: Infidelity for preparing a displaced Gaussian with the serial protocol. (a) For n ≤ 40,−9 ≤ m ≤ 9, and A = 3.0 the maximum infidelity is smaller than 10−2 . (b) For n ≤ 40, −20 ≤m ≤ 20, and A = 3.0 the maximum infidelity is smaller than 10−3 . (c) A histogram over the ions’infidelities, comparing (a) and (b).

33

(a)

(b)

(c)

Figure 1.13: Infidelity for preparing a displaced Gaussian with the parallel protocol. (a) For n ≤ 40
and −9 ≤ m ≤ 9. By applying all polynomials and beatnotes simultaneously, with A = 0.3, themaximum infidelity is smaller than 10−2 . (b) For n ≤ 40, −20 ≤ m ≤ 20, and still with A = 0.3,
the maximum infidelity is less than 10−3 . (c) A histogram over the ions’ infidelities, comparing (a)and (b).

parallel application compared to the serial application only for the case of a targeted infidelity of
10−3 . However, this neglects the reset time of the DM. For a targeted infidelity of 10−2 , the serial
application needed mmax = 9, which would require the DM to be set 18 times. With a reset time ashigh as 50 µs, this incurs an overhead of 900 µs. This is substantially larger than the gate duration
itself, and highlights a potential reason to instead consider the parallel protocol.

1.7Conclusion
Penning traps are promising candidates as platforms for quantum information processing,

due to their ability to control hundreds of qubits, and perform non-local entangling operations.
However, existing experiments lack the ability to address individual ions, and therefore, fail to meetthe criteria for universal quantum information processing. In this paper we discussed a method for
implementing programmable Z-rotations in a Penning trap, thus providing a path forward for more
complex quantum simulations and general large-scale quantum information processing.
By employing a DM in the path of one of the laser beams that creates the optical dipole
force in Refs. [18, 21], we showed how wave front deformations introduced by the DM can be used
to generate AC Stark shift patterns that are static in the rotating frame of the crystal. A pattern34
of azimuthal order m [P m (ρ) cos(mϕ), Qm (ρ) sin(mϕ)] is introduced by setting the frequency µof the optical dipole force to the mth harmonic of the rotation frequency, µ = mω. General
AC Stark shift patterns are obtained through the introduction of different azimuthal orders. We
analytically and numerically demonstrated the feasibility of this approach for generating singlesite rotations. Choosing a Gaussian phase pattern with a 1/e diameter equal to the interparticle
spacing, we demonstrated that we can resolve single ions in a crystal of about 100 ions under typicalexperimental conditions [18, 21]. Moreover, by applying the required beatnote frequencies mω inparallel, one can obtain 99% fidelity single-qubit gate durations of 333.33 µs, which is faster thantypical single particle decoherence times, ∼ 10 ms, in current Penning trap experiments.
In this paper we assume a perfectly performing DM and analyze the sources of infidelity
for two different protocols where patterns of different azimuthal order are introduced serially or inparallel. In particular, we assume that the number of DM actuators is large compared to the numberof ions and that the surface of the DM can be set with arbitrary precision. Deformable mirrors
with greater than 4000 actuators, surface figures of less than 10 nm, and mechanical response timesof less than 50 µs are available commercially. The desired AC Stark shift pattern as well as theperformance of the DM will impact whether the serial or parallel protocol should be employed. In
general, the serial protocol enables the implementation of larger amplitudes and therefore higher
accuracy phase patterns. However, for patterns that require introducing many azimuthal orders m,
the reset time of the DM can add significant overhead in the time required to implement the desired
AC Stark shift pattern. The parallel protocol removes any overhead due to the reset time of theDM, but the restriction on the amplitude of the phase pattern can impact the accuracy with which
the desired phase pattern can be implemented.

35

1.8

Appendix

1.8.1

The RWA is Exact for Integer Multiples of the Trap Rotation Frequency

As discussed in Sec. 1.5.1, choosing the evolution time T such that ωT = 2πr with r a positive
integer can remove all error from the rotating wave approximation. In Sec. 1.5.1 we showed that
this was the case when we applied different Pm patterns. We now argue that this can be generalized
to arbitrary order. The Hamiltonian we consider will have both even [AP mi cos (mi θ)] and odd[AQmi sin (mi θ)] terms, up to mNf , so that i ≤ Nf . From the Jacobi-Anger expansion each of theseterms will introduce a new sum, as in Eq. (1.21), with an index ai or bi , respectively. In general,
we then get phase factors in the sum, f (t), of the form()
XX
f (t) = exp (−iµt − iϕ(ai + bi )mi − i(ai + bi )mi ωt) .
i(1.56)i

Because this is the only time dependence, we can integrate over time from 0 to 2πr/ω to get
Z 2πk/ω0PP
exp{[−iµt − iϕ i (ai + bi )mi − i i (ai + bi )mi ωt)]} 2πr/ωP= 0.
dtf (t) =−i[µ + i (ai + bi )mi ω]
0(1.57)

This is true except when the denominator vanishes, which is also the case where the term in
Eq. (1.56) is static. If we have a drive µ = mj ω the denominator vanishes when
Nf
X
mj +
(ai + bi )mi = 0,(1.58)

i=1

as we have seen previously in the case of two terms. The case in Sec. 1.5.1 was special in that it
repeats every π/ω. The general argument is given above, and shows that in general a multiple of
2π/ω is needed.

Chapter 2

The QAOA with Slow Measurements [34]2.1

Abstract
The Quantum Approximate Optimization Algorithm (QAOA) was originally developed tosolve combinatorial optimization problems, but has become a standard for assessing the performance of quantum computers. Fully descriptive benchmarking techniques are often prohibitively
expensive for large numbers of qubits (n ≳ 10), so the QAOA often serves in practice as a computational benchmark. The QAOA involves a classical optimization subroutine that attempts tofind optimal parameters for a quantum subroutine. Unfortunately, many optimizers used for the
QAOA require many shots (N ≳ 1000) per point in parameter space to get a reliable estimate of
the energy being minimized. However, some experimental quantum computing platforms such as
neutral atom quantum computers have slow repetition rates, placing unique requirements on theclassical optimization subroutine used in the QAOA in these systems. In this paper we investigate the performance of a gradient free classical optimizer for the QAOA - dual annealing - anddemonstrate that optimization is possible even with N = 1 and n = 16.

2.2

Introduction
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classicalalgorithm that uses a subroutine run on a quantum computer together with a classical optimizer tofind approximate solutions to combinatorial optimization problems [35, 36]. While many results havedemonstrated that the QAOA faces substantial challenges if it is to outperform classical algorithms

37[37–45], it has nevertheless become a standard technique for benchmarking quantum computers[46–48], serving as a holistic test.
The classical optimization routine in the QAOA is used to select a collection of angles thatparameterize a quantum circuit. While [35] discusses the feasibility of an open-loop grid search for
the QAOA, in practice classical optimizers are closed-loop and require accurate estimates of theenergy expectation value for any choice of angles. Previous work [46, 47, 49, 50] has often chosenclassical optimizers that require a large number of measurement repetition rates [35, 46, 51–54].
Furthermore, theoretical works also often assume that the optimizer has access to the expectation
value at each point in parameter space [55, 56].
However, this assumption is not always justified - some quantum computing platforms, suchas neutral atoms, have substantially slower circuit repetition rates. In the case of neutral atoms,
this is due in part to atom reloading [57, 58]. Modern neutral atom quantum computers use
lossy measurements which remove atoms from the trap, requiring atoms to be reloaded between
measurements. Moreover modern systems such as [48], can have measurement durations as long as
30ms. These facts together can give total shot-to-shot measurement rates of around 5Hz [59].
This places substantial difficulties on the classical optimizer used in the QAOA. While noise
is commonly modeled as coming from decoherence and miscalibration, it can also be a result of shotnoise in sampling from the quantum computer a limited number of times. This noise can cause
unreliable estimates of quantities such as the gradient and the energy.
In this paper we explore a classical optimization routine for circumventing these difficulties –dual annealing. Dual annealing is a form of simulated annealing, which explores the search space
while annealing a “temperature” that parameterizes how likely the algorithm is to jump to higher
energy configurations. This can be useful for avoiding local minima. We first discuss modifications
that can be made to the algorithm to accommodate the error from taking a limited number ofshots. Next, we find empirically that this optimizer performs well even without modification. In
particular, we find that dual annealing can find global maxima with a single shot at each point inparameter space that it evaluates.

38
To assess the efficacy of this optimization technique, we propose a family of graphs and use
the QAOA to find the maximum cut of a member of this family. We provide complexity theoretic
arguments that suggest these graphs have maximum cuts that may be hard for classical computers
to approximate. While several families of graphs are known to be poor choices for showing quantum
advantage [35, 37, 40, 60], it is an open problem to find families of graphs on which the QAOA canbe shown to outperform classical algorithms.

2.3

TheoryThe quantum circuit used by the QAOA is described by 2p parameters, in a layered ansatz.

Specifically, the unitary evolution of the quantum computer has input angles ⃗γ , β⃗ and is given as⃗ =U (⃗γ , β)pYe(−iβj X̂⊗n )e(−iγj H) ,

(2.1)

j=1

where H is called the driver and σx⊗n is called the mixer. An example of this circuit is shown in
Fig. 2.1 and the details of H for the problem considered in this paper are given in Eq. 2.2. H is a
Hamiltonian whose ground state encodes the solution an optimization problem. In this paper weconsider the MAX-CUT problem, with associated HamiltonianH=X 1
ωij (1 + Ẑi Ẑj ).
2

(2.2)

(i,j)∈E

Here E is the set of edges in a graph, and ωij are the weights for the associated edges.
The goal of the QAOA is to find optimal values of ⃗γ and β⃗ that minimizeDE⃗ = +|U (⃗γ , β)⃗ † HU (⃗γ , β)|+⃗C(⃗γ , β).

(2.3)⃗ we equivalently maximize −C(⃗γ , β).
⃗In this paper, instead of minimizing C(⃗γ , β)We will only
consider the p = 1 QAOA, and therefore only have γ1 and β1 . Because there is only one of each angle
in the examples we consider, we will refer to them as γ and β for the rest of the paper. Numerous
descriptions of the QAOA and MAX-CUT exist in the literature [35, 47], and further discussion isomitted here. MAX-CUT is a common problem to consider for demonstrations of QAOA due to

39

|+⟩
|+⟩

RZ(ω1,2 γ1 )RX(β1 )...

RX(β1 )

...

..
.
|+⟩
|+⟩

RZ(ωn−1,n γ1 )RX(βp )
RZ(ω1,2 γp )RX(βp )

..
.

..
.

RX(β1 )...

RX(βp )

RX(β1 )...

RZ(ωn−1,n γp )RX(βp )

Figure 2.1: An example of a circuit implementing the QAOA from Eq. 2.1, using the MAX-CUT
Hamiltonian in Eq. 2.2. In this example, we’ve assumed that there are only edges between nearest
neighbors.

40
the locality of the Hamiltonian, requiring only two-body terms. This makes it attractive for early
demonstrations of hardware such as that in [48].
To find the optimal angles, γ and β, we consider simulated annealing algorithms. Simulated
annealing is traditionally introduced in analogy to the metallurgical process, wherein the optimizer
has a temperature which controls random fluctuations and an energy function that is used forpreferentially exploring certain configurations. The system has “thermal kinetic energy”, with avisiting temperature which allows it to probabilistically explore the search space while avoiding
local minima. The transition between any two points has an associated hopping probability thatcharacterizes how likely a transition between the two states is. As the temperature is lowered,exploration becomes more difficult in the presence of energy barriers.
Simulated annealing over a discrete set (such as that achieved by discretizing γ and β) can beshown to converge to the set of global minima [61]. For each temperature, the hopping probabilities
form a Markov chain, whose stationary distribution is given by


1
J(i)
πT (i) =exp (−
)
ZT
T

(2.4)

[61], with ZT a normalizing constant. In the limit that T → 0, we see that this distribution
concentrates around the collection of minima of J.
We further note that if the energy estimate is noisy, this convergence is not guaranteed. Thiscan easily be seen by considering the space with two elements, i1 and i2 , and function K such that
K(i1 ) = 0 andK(i2 ) =





2,

with probability 1/2

.

(2.5)




−1, otherwise
Taking J = ⟨K⟩ to be the energy function, we see that minimum is i1 , however the stationary pointof the Markov chain will be the uniform distribution over i1 , i2 .
The algorithm can, however, be modified to maintain its convergence properties. We canmodel finite shot noise as a random variable for the k th step in the annealing chain that depends

41
on the state being measured,
N

N

ℓ=1

ℓ=1

1 X
1 X
Xk = (fℓ (xj ) − J(xj )) − (fℓ (xi ) − J(xi )),
N
N

(2.6)

where J(m) = ⟨m|C|m⟩ and fℓ is the ℓth evaluation on the quantum computer - i.e. the ℓth shot.
One of the conditions for convergence given in [62] is that σk ∈ o(Tk ), where σk is the standard
deviation of Xk , Tk is the annealing temperature and o(·) is the standard asymptotic notation fora strict upper bound. Likewise in what follows, O(·) denotes an asymptotic upper bound withpossible equality, ω(·) denotes a strict asymptotic lower bound, Ω(·) denotes an asymptotic lower
bound with possible equality and Θ(·) means both O(·) and Ω(·).
From [35] we know that the values of the sampled cost function are concentrated about theirmean C(γ, β), with variance, σ 2 upper bounded byσ 2 ≤ 2e(v − 1)(2p+2) − 1,(v − 1) − 1(2.7)

for a graph with e edges and v vertices. Thus
(2.8)

σµ̂2 = σ 2 /N ∈ O(e/N ),where σµ̂2 is the variance of sample mean. This shows that we should expect the number of shots
required to be at worst proportional to the number of edges in the graph.
p
From [62], we expect the modified annealing algorithm to work if N/e ∈ ω(1/Tk ). We willshow, however, in Sec. 3.5 that even without modifying dual annealing, it can find the local maxima
with a single shot per point in parameter space.
The implementation details of the dual annealing algorithm can be found in [63] and theoretical details can be found in [64]. We reproduce some relevant details here. In particular, thealgorithm uses a distorted Cauchy-Lorentz distribution for the “visiting temperature”
D− 3−q

gqv (∆x(t)) ∝[Tqv (t)][1 + (qv − 1)v(∆x(t))2
2[Tqv (t) ] 3−qv

1

] qv −1

+ D−1
2

,

(2.9)

42
where x(t) is a jumping distance, and D is the dimension of the optimization problem [64]. For thep = 1 QAOA D = 2, since there are only two values to optimize, γ and β. This temperature isannealed according to
Tqv (t) = Tqv (1)2qv −1 − 1,(1 + t)qv −1 − 1(2.10)and new points are accepted with probability1pqa = min{1, [(1 − qa )∆E/Tqv ] 1−qa }.

(2.11)

qa and qv are called the acceptance and visiting parameters, and are hyperparameters used
to control how exploratory the annealing algorithm is. This annealing routine is accompanied by
local search at the end of each annealing attempt and a number of restart attempts. Simulated
annealing, and hence dual annealing, benefits from a wealth of analysis and convergence guarantees
[65].

2.4

Choice of Graph
While the primary purpose of this paper is to discuss the choice of classical optimizer in the

QAOA, we need to choose a problem instance. So as to avoid solving a problem that is known tobe trivial, we give a heuristic argument for our choice of graph. The particular problem we choose
due to its simple-to-implement Hamiltonian and abundance in the quantum computing community
is MAX-CUT. The difficulty of MAX-CUT depends on the family of graphs being considered, andthe general problem has been well-studied classically - the Goemans-Williamson (GW) algorithm
gives an optimal approximation ratio assuming the unique games conjecture [66–68]. Therefore one
would like to identify a collection of graphs that classical computers perform worst on.
In this work we choose random graphs G(v, e) on v vertices with e edges chosen at random
from all possible edges. MAX-CUT on these graphs has a phase transition when e/v = 1/2, in thefollowing sense: below 1/2 there are efficient classical algorithms for solving the problem, however
above 1/2 there seem to be hard instances [69, 70]. Furthermore, if we consider signed graphs,√
then if the cover number of the positively signed edges is near v, the problem becomes strongly

43
NP hard [71]. A problem is strongly NP hard if the problem remains NP hard when its numericalparameters are bounded by a polynomial in the input size. This means that hard instances are,roughly speaking, easy to generate, and so in particular randomly generating instances should resultin graphs that may be more challenging for classical algorithms.
The cover number of the positively signed edges is the minimum size of a subset V of thevertices, such that every edge with positive weight has at least one endpoint in V . In particular,[72] shows this family is strongly NP hard for graphs with cover number Ω(v −k ), with k a positive
integer. We have already specified e ∈ Θ(v) so that k = 2 is the smallest k that is guaranteed tobe compatible. This suggests that algorithms like the GW algorithm may perform less favorably onthese graphs. An example of this graph is shown in Fig. 2.2. Similar graphs have been consideredin [56].

2.5

Numerics
We simulate the p = 1 QAOA on a graph sampled from the family of graphs discussed inSec. 2.4 using quimb [73]. We emphasize that the purpose of this paper is to demonstrate theperformance of the optimizers with low shot rates. The p = 1 QAOA has been well-studied,and its performance is well-understood analytically [35, 74, 75]. Thus, our emphasis is that thesetechniques allow for the demonstration and benchmarking of a quantum computer - for the case of
p = 1 optimization, one should use the known-optimal choice of angles if the goal is produce a best
approximate solution to the problem being studied. In certain problem instances of MAX-CUT,
we can reduce the search space for both γ and β to the range [0, π] [51]. In general we see this is
not possible, since scaling the edge weight by a factor of 1/c moves the maximum in γ by a factor
of c. Nevertheless, we restrict our search space, since we are interested primarily showing that
optimization is possible. For our example, we choose a graph on v = 20 vertices, which has n = 16
qubits that interact with at least one other qubit. We note that n = 16 is a non-trivial example for
initial demonstrations of new quantum computing architectures [48].
The particular graph considered in this section is shown in Fig. 2.2, and is sampled from

44

Figure 2.2: Example Graph from Section 2.4. Above we see a graph on v = 20 vertices,
generated according to the distribution described in Sec. 2.4. Edgeless vertices are omitted fromthe figure. We sample e = 3v/5 edges at random, and edge weights are chosen uniformly at random
from [−1, 1].

45
the distribution described in Sec. 2.4. The graph is chosen from a distribution with e = 3v
5 edges,with v = 20, and therefore only contains 16 vertices with edges. Such a graph would be useful fordemonstrating the QAOA on an architecture of a 4 × 4 grid of atoms. As mentioned in Sec. 3.3,
rather than minimizing Eq. 2.3, we maximize −C(γ, β).
In Fig. 2.3 we see the performance of dual annealing on the graph in Fig. 2.2. We use one shot
per point in parameter space (N = 1) and allow the dual annealing algorithm ten opportunities
to restart and re-anneal. In total the optimizer sampled 326 times. For a clock rate of 5Hz thiswould take just over a minute to run. Fig. 2.3 also provides two kernel density estimates of the
distribution of sampled points which highlights that there are two distinct attractors over β, andone over γ, corresponding to the optima of the landscape. Kernel density estimates use Gaussian
kernels to produce a non-parametric estimate of the sampling distribution that is given by
1 X
fˆh (x) =exp (−(x − xi )2 /h2 ) ,nh(2.12)

where the bandwidth h given by
h = n−1/(d+4) ,(2.13)and n is the number of samples (326) and d is the dimension (1) as suggested in [76]. This demonstrates that the dual annealing optimizer not only can find a good cut, but also is taking advantageof the structure of the optimization landscape.
Another interesting feature of single-shot optimization, which we leave to future work, is the
behavior of the single-shot performance statistics. While many optimizers attempt to estimate andmaximize the QAOA energy, the goal of the QAOA is to optimize the single best observed bitstring.
The shot to shot energy is plotted against the cumulative best energy in Fig.2.4. While the optimizer
quickly finds a good cut and Fig. 2.3 shows that the optimizer samples preferentially from thebasins of attraction of the optima, it can be seen that the single shot data varies substantially. The
maximum value in the grid of values produced in Fig. 2.3 is 3.78, however the optimum reported by
the optimizer is 4.13. This is in fact the sum of the positive weights, and hence the maximum cut.
We see that dual annealing is able to perform with just a single shot per choice of angle pair γ

46Figure 2.3: Optimization using dual annealing. Each white point corresponds to a single shotmeasured by dual annealing. The kernel density estimates show peaks corresponding to the local
maxima, demonstrating that optimizer preferentially samples from higher energies. Moreover, thered star shows the final sampled point which lies near a global maximum.

Figure 2.4: Cumulative maximum statistics. The orange line plots all 326 points sampled by
the dual annealing optimizer, while the blue is the maximum cut seen so far. While Fig. 2.3 showsthat the optimizer preferentially samples from higher energies, this figure shows that the single shot
data can vary substantially. Despite this, the optimizer quickly finds a good cut.

47
and β. For quantum computers with low shot rates this provides a tractable method for performing
closed-loop quantum-classical optimization, with noisy estimates of the energy at each point.

2.6

Conclusion
Due to its easy implementation and interpretation, the QAOA is an appealing holistic bench-

mark for quantum computers. However, different quantum computing architectures place different
requirements on the classical optimization subroutine of the QAOA. In particular, neutral atom
quantum computers often have substantially slower measurement rates than conventional superconducting quantum computers. To use the QAOA as a benchmark for these platforms thus requires
choosing a classical optimizer that can use noisy estimates of the QAOA energy.
In this paper, we have demonstrated an optimization technique that is well-suited for thisproblem, dual annealing. Dual annealing searches the parameter space in a thermal way, accepting
transitions if the cost is lower or if the annealing temperature is sufficiently high to allow jumps toworse points. We have shown that this algorithm can use a single shot per point in parameter space
while still finding good cut values for the MAX-CUT problem on as many as n = 16 qubits.
Additionally, we have suggested a new family of graphs which have a phase transition in
their known classical hardness. There are several straightforward analyses to extend this work including realistic noise models, scaling to larger numbers of qubits and extending the algorithm
considered here to model the shot uncertainty explicitly [62], as described in Sec. 3.3. Finally, while
the intention of this work was to discuss the optimizer choice for demonstrating the performanceof NISQ hardware, and thus focused on p = 1, exploring the increased dimensionality of the search
space for higher p is an interesting direction for future research.

Chapter 3

Towards Demonstrating Fault Tolerance in Small Circuits Using Bacon-Shor
Codes [77]3.1

Abstract
Quantum error correction is necessary to perform large-scale quantum computations in thepresence of noise and decoherence. As a result, several aspects of quantum error correction havealready been explored. These have been primarily studies of quantum memory[78, 79], an importantfirst step towards quantum computation, where the objective is to increase the lifetime of theencoded quantum information. Additionally, several works have explored the implementation oflogical gates[80–82]. In this work we study a next step - fault-tolerantly implementing quantum
circuits. We choose the [[4, 1, 2]] Bacon-Shor subsystem code, which has a particularly simple
error-detection circuit. Through both numerics and site-counting arguments, we compute pseudothresholds for the Pauli error rate p in a depolarizing noise model, below which the encoded circuits
outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as
p = 3% for short circuits, and p = 0.6% for circuits of moderate depth. Additionally, we see thatmultiple rounds of stabilizer measurements give an improvement over performing a single roundat the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of aquantum algorithm that could be accessible with existing hardware.

49

3.2

Introduction
In order to perform useful quantum computational tasks that require many qubits or deep

circuits, it is necessary to use quantum error-correcting codes that have the ability to handle faulty
physical hardware[83]. Historically, fidelities of quantum devices have been too far below the necessary thresholds to warrant any attempts to demonstrate fault-tolerant quantum computation.
However, slow and steady progress has pushed both single and two-qubit gate infidelities as low
as 10−3 [6, 7, 32]. It has therefore become an important theoretical challenge to develop initialsmall-scale quantum error correction (QEC) experiments to validate the theory.
The aim of error correction is to make logical qubits and gates that have lower effective error
rates than bare physical systems. Over the past decade, there have been multiple non-fault-tolerant
QEC code demonstrations using platforms such as superconducting qubits, trapped-ion systems, andmicrowave modes[84–87]. While these have all been important building blocks towards large-scalequantum computation, we are more generally interested in fault-tolerant schemes, which effectively
mitigate the errors that occur throughout the operation of a QEC scheme. In addition to faulttolerant quantum circuits, other hardware-efficient fault-tolerant schemes have been proposed, suchas performing stabilizer extraction through coupling to nonlinear-oscillators[88]. In this paper, however, we will restrict our attention to the quantum circuit model. In recent years, experiments havestudied the near-threshold performance of quantum error-correcting and error-detecting schemes.
This has included exploring multiple rounds of error-detection/correction in small surface codes[79]and a large repetition code[78]. Recent trapped ion experiments have additionally yielded promising
results. For example, Egan et. al implemented circuits designed to be fault-tolerant and studied theimplementation of logical gates in a [[9, 1, 3]] Bacon-Shor code[80], although the logical gates werestill outperformed by the physical gates. In another experiment, Ryan-Anderson et. al prepared
codewords of the [[7, 1, 3]] color code and demonstrated several rounds of error correction, also withcircuits designed to be fault-tolerant[89].
These experiments have been primarily quantum memory experiments, which aim to preserve

50
quantum information, rather than process it. However, the ultimate goal of QEC codes is to perform
computations on faulty physical hardware in addition to preserving quantum memory. Therefore,
we must consider QEC schemes that can handle errors that occur in the logical manipulation ofquantum data.
In this paper, inspired by Gottesman[90], we explore a small quantum error-detecting scheme
to demonstrate a limited form of fault-tolerant quantum computation. The advantage of considering
quantum error-detection, as opposed to error-correction, is that the physical circuit we take consistsof only 5 physical qubits, greatly simplifying the experimental procedure. In particular, the [[4, 2, 2]]code requires only one additional ancilla to encode two logical qubits. By treating one of those logicalqubits as a gauge qubit, we can construct a Bacon-Shor subsystem code, which has fault-tolerant
error-detection circuits. These error-detection circuits can be repeated intermittently throughout
a computation to detect errors as they occur and to significantly reduce the likelihood of multiple
undetectable errors.
Gottesman defines fault tolerance in small experiments as being when any encoded circuit,from a given family of circuits C, outperforms its unencoded counterpart[90]. That is to say thatevery encoded circuit in C must have a lower error rate than the error rate of the equivalent barecircuit. Given the computational difficulty of testing every element of C, which is exponentiallylarge for any given circuit depth, we loosen the notion of fault tolerance to only requiring that the
average encoded circuit from a uniform random sample collected from C have increased performance.
For our family we have chosen circuits which consist of X, Z and H gates, that are transversally
implementable in the code we consider, and thus fault tolerant. An example of such a circuit isshown in Fig. 3.1, consisting of a single X and Z gate.
In what follows, we use threshold to mean the pseudo-threshold for the particular code being
considered. We take two approaches to estimate the fault tolerance threshold for the four-qubit
Bacon-Shor code. First, we study depolarizing noise numerically. By simulating circuits of varying
depth, we find a threshold for the depolarizing parameter as high as 0.6% for deep circuits. Next,
we consider a site-counting argument which uses a stochastic noise model, where errors occur at

51
each site within a circuit with some probability[91]. This site-counting argument is more pessimistic
than the depolarizing model - it fails to account for benign sets of errors, and assumes that any
error that occurs is the worst possible one. Even so, this analysis yields a threshold of 0.2%. These
analyses show that the threshold for seeing increased performance over the unencoded circuit could
be achievable using existing hardware.

3.3

Bacon-Shor Subsystem Codes
The fundamental idea of QEC is to encode the state of a given set of qubits, with Hilbert

space HS , into a larger physical system. Generically this entails mapping states on HS into thecodespace, Hc , of a higher-dimensional Hilbert space that can be decomposed as H = Hc ⊕ Hc⊥ .
Physical errors that move the state of the qubits out of the codespace, and into Hc⊥ , can bedetected and in some cases corrected. A subsystem code breaks apart the codespace into a productof subsystems(Hc = HL ⊗HG )[92, 93]. One of the subsystems, HL , is taken as the logical subsystemand the remaining subsystem, HG , contains gauge degrees of freedom. In this construction, the stateon HG does not affect the encoded logical state.
A large class of QEC codes are the stabilizer codes, where logical qubits are encoded as
codewords that are in the shared +1-eigenspace of a set of commuting Pauli operators[94]. The
two-dimensional Bacon-Shor code is a subsystem code with a codespace defined by an associatedstabilizer group on a square lattice[93]. The stabilizer framework provides a method of detecting
whether errors occur in the process of computation, and the division of the codespace into logicaland gauge subsystems allows for a simplified scheme of error syndrome extraction. In the case of the
Bacon-Shor code, stabilizer measurements can be performed by measuring a set of spatially-local,
two-qubit gauge operators, S, whose products contain the stabilizers. The gauge operators do notaffect the logical state, but may act non-trivially on the gauge subsystem.
In this work, we consider the four-qubit Bacon-Shor subsystem code, in which there are twoelements in the tensor factorization, giving a single logical qubit and a single gauge qubit. We choose
the set of gauge operators to be S = IIXX, XXII, ZIZI, IZIZ. The logical space of the code

52

Figure 3.1: Comparison of the encoded fault-tolerant circuit to the bare circuit. a) A physical,
fault-tolerant circuit encoding the logical circuit shown in Fig. b. Following state preparation andeach logical gate, a syndrome measurement is applied, denoted by S. This corresponds to the caseof one gate per stabilizer measurement round. Final measurement outcomes are post-selected on
obtaining trivial error syndromes. The logical operators XL and ZL in Fig. b are implemented via
the physical operators XIXI and ZZII. b) Qubit ℓ0 is acted on by logical gates XL and ZL , andthen measured. We say an experiment demonstrates fault tolerance if a family of logical circuits(one member of which is shown in b)) has a higher success probability on average when implemented
with encoded circuits (one example shown in a)), rather than with the corresponding unencoded
circuits (a single X followed by a single Z, in this example).

53
is given as the simultaneous +1-eigenspace of the center of S, forming the stabilizer group of thecode - in this case IIII, XXXX, Y Y Y Y, and ZZZZ. The remaining operators will in general act
non-trivially on the gauge qubit, meaning that each logical state has in general a family of physicalstates associated with it.
To see how this works, we will identify the logical operations and states of the code being
considered. A logical zero state is given by |0⟩L = √12 (|0000⟩+|1111⟩). However as we are consideringa subsystem code, then there is another orthogonal state that can be associated with logical zero,
given by |0̃⟩L = √12 (|0011⟩+|1100⟩). Any state in the subspace spanned by |0⟩L and |0̃⟩L corresponds
to the same logical zero state, and each of these states only differ by the state of the gauge qubit.
The associated [[4, 2, 2]][95, 96] code has the same stabilizers as the Bacon-Shor code discussed inthis paper. The codewords of the [[4, 2, 2]] code are
1|00⟩ → √ (|0000⟩ + |1111⟩)
2
1|01⟩ → √ (|1100⟩ + |0011⟩)
2
1|10⟩ → √ (|1010⟩ + |0101⟩)
2
1|11⟩ → √ (|0110⟩ + |1001⟩),
2

(3.1)

and so we see that |0̃⟩L is generated by acting on |0⟩L with the gauge operator XXII ∈ S. The
operator ZIZI similarly performs a Z rotation on the second logical qubit in the [[4, 2, 2]] code
(which is taken as the gauge qubit), but does not act on the logical codespace. This is because
gauge operators are logical operators on only one of the qubits in the [[4, 2, 2]] code, and therefore
don’t affect the logical state of the qubit encoded in the [[4, 1, 2]] Bacon-Shor code. In addition togauge operators, there are the logical operators XL = XIXI and ZL = ZZII, which transition thestate between distinct logical codewords (|0⟩L to |1⟩L in the case of XIXI).
While the number of logical qubits is fewer than that associated with the [[4, 2, 2]] code, we
get several good properties from using a Bacon-Shor code. First, a convenience of working withthis code is that the stabilizer operators are all weight-two, which are generally easier to measurefault-tolerantly. As shown in Fig. ??, a measurement of all the error syndromes only requires eight54
two-qubit gates. While these gates may require additional SWAPs in a superconducting architecture,
trapped-ion architectures may incur no overhead, as they typically have all-to-all connectivity. Thissuggests that the required threshold for being able to fault-tolerantly measure the stabilizers will
be lower for trapped-ion systems.
Finally, the preparation and stabilizer circuits in Fig. 3.2 and Fig. ?? can be implemented
fault-tolerantly. In fact, while the state preparation circuit for the [4, 2, 2] code requires an ancilla,
Fig. 3.2 shows that the Bacon-Shor code does not, since a single error can at worst propagate toerrors on the first and second qubit, or the third and fourth qubit. In both cases, these correspondto performing a gauge operation that changes the state of the gauge qubit, which is irrelevant forthis code. The logical codeword |0L ⟩ can thus be prepared through the initialization circuit shownin Fig. 3.2.

Figure 3.2: Logical zero (|0L ⟩) preparation circuit for the [[4, 1, 2]] Bacon-Shor code.
While a universal gate set is not transversally implementable on the Bacon-Shor subsystemcode (meaning that not all single-qubit logical gates can be decomposed entirely into combinationsof single-qubit phyical gates), it is possible to transversally implement the logical gates XL = X1 X3 ,ZL = Z1 Z2 , and HL = SWAP23 H1 H2 H3 H4 . SWAP23 , for our purposes, can be implemented justby updating the labels of the qubits.

55

3.4

Site-Counting
For another estimate of the threshold of this code, we will follow the site-counting argument

in [90, 91] closely. We will derive a bound on the logical error rate for an analytic prediction ofwhen we can expect an encoded algorithm to outperform its unencoded counterpart. As discussed inboth Sec. 3.2 and Sec. 3.5, we consider circuits consisting uniformly of XL , ZL and HL gates. These
require two, two and four physical gates (as noted previously, SWAP gates can be implemented byrelabeling qubits) respectively, so a circuit with T logical gates, will on average contain (2 + 2 +4)T /3 = 8T /3 physical sites where an error can occur.
We assume perfect initialization of both the |0⟩ and |+⟩ states, perfect measurements andno idle noise. While in principle these will contribute more infidelity to the logical circuit, thecontribution will be small. Using the initialization circuit in Fig. 3.2 with 3 CNOTs, we get NA = 6
locations in the logical state preparation circuit where errors can occur.
The stabilizer measurements require measuring IIXX and XXII, which both require 2
CNOTs and 1 Hadamard, contributing 10 locations for error, and measuring IIZZ and ZZII
requires 2 CNOTs, giving 8 locations for errors. Thus we have NB = 18 locations per stabilizer
measurement. A logical error only arises if more than one error occurs between measurements.
Using site-counting we find that the probability for a logical error, pl , in a circuit with M stabilizer
measurements, and N = 38 T /(M + 1) physical gates between measurements is bounded as
1

−

pl

≥

(1

−



NA + N 2
p )(1
2


−


NB + N 2 Mp ) , (3.2)
2where p is the physical error rate. Additionally, we can bound the post-selection probability. For
each block of the circuit between stabilizers, a bound on the post-selection probability is given by
1 − Sp where S is the number of sites per block, giving
pps ≥ (1 − (NA + N )p)(1 − (NB + N )p)M .

(3.3)

For the unencoded circuit, we have that pu ≈ T p. Finally, to compare the two, we will consider theconditional probability of success. The resulting threshold for when the encoded circuit begins to56
outperform the unencoded circuit is shown in Fig. 3.3a. We can additionally use site-counting toestimate the rejection probabilities as shown in Fig. 3.3b. In both cases, the result is qualitatively
similar - by repeatedly measuring stabilizers, the post-selection probability decreases exponentially,
however by measuring the stabilizers fewer times the post-selection probability increases, due tothe potential for errors to collude and become undetectable. In both cases, site-counting gives amore conservative threshold than the numerics. This is due to the looseness of the union bound andbecause our argument fails to account for benign errors. We could improve our bound by considering
benign errors, as in [91].
As in Sec. 3.5, we can estimate the optimal number of stabilizer measurements to perform foreach circuit depth. Taking the maximum of each curve in Fig. 3.3a we get (1, 2, 5, 10, 20, 15, 20,
16, 20, 28, 25) gates between stabilizer measurements for depths (1, 2, 5, 10, 20, 30, 40, 48, 60, 84,100) respectively. The site-counting argument is overly pessimistic about the errors contributed by
the stabilizer measurements, and so is more sparing in their use, which can be seen by comparing
to the 15 logical gates between stabilizer measurements suggested by the numerics in Sec. 3.5.

3.5

Numerics
Elided.

3.6

Conclusion
Quantum error correction is a necessary step towards the development of large-scale quantum

computers. Quantum fault tolerance is the demonstration of a quantum error correction schemethat is robust to errors in the error correction schemes themselves. Experimental platforms haverecently achieved impressive fidelities that put us near the range that we should consider small
quantum codes for both quantum memories and small demonstrations of fault tolerance. Indeed,
we have yet to see an unambiguous demonstration of improvement from fault tolerance. In thispaper, we have proposed a simple demonstration of fault tolerance that can be used to perform
encoded circuits.

57

(a)

(b)

Figure 3.3: (a) Error probability thresholds computed through site-counting. While the typicalvalue is an order of magnitude lower than in the case of the numerics, the shape of the bound isqualitatively similar, showing an improved threshold with a nontrivial rate of intermediate measurements. (b) Post-selection probabilities computed through site-counting, evaluated at the threshold.
We see that the results approach a fixed value when only one measurement is performed at the end.

58
Through both numerics and a site-counting argument, we have shown that Pauli error rates for
short circuits (with a depth of five) as high as p = 0.03 are capable of demonstrating an improvementof the encoded circuit over the unencoded circuit. Moreover, for deeper circuits (with a depth of100), we find that performing more than one round of gauge measurements (one round per 15 logical
gates) provides a threshold error rate as high as p = 0.005, making experimental observation of thethreshold potentially realizable on superconducting and trapped ion platforms.
In our analysis, we have assumed that single- and two-qubit gates have similar error rates,modeling two-qubit gates as being subjected to two single-qubit errors. However, state-of-the-art
quantum computing platforms have single-qubit error rates that are often smaller by at least anorder of magnitude. Nevertheless, the code considered in this paper encodes a single logical qubit.
Thus the natural quantity to compare against in a discussion of fault tolerance, and the one wechose to compare against, is another single qubit physical error rate.
There are at least two natural extensions to this work. The first is considering two logicalqubits, each encoded in their own four-qubit Bacon-Shor code. In this case, the natural quantity tocompare against is the two-qubit gate infidelity, and in the analysis of the logical circuit one could
consider single-qubit gates as free. Then one would only count the two-qubit gates used in the state
initialization circuit, stabilizer measurements, and to perform entangling operations between thetwo encoded qubits. This model makes the more realistic assumption of different error rates forgates involving different numbers of qubits. This assumption is particularly relevant in trapped ion
systems where single-qubit gate infidelities are exquisitely small[6, 7].
The other straightforward extension of this work is to consider the [[9, 1, 3]] Bacon-Shor code.
This code also encodes a single logical qubit, but is distance three and therefore can correct one error.
Thus, rather than simply rejecting circuits as discussed in this paper, we can consider implementing
error-correcting circuits which should improve the threshold for deeper circuits.
While this work focused on an uncorrelated depolarizing noise model, an additional directionof research is to consider more realistic noise models for trapped-ion and superconducting architectures. However, we are finally reaching an exciting point in hardware where quantum codes can be

59
implemented and have the ability to outperform bare physical qubit schemes, and are therefore on
the brink of scalability.

Chapter 4

Opportunities and Limitations in Broadband Sensing [97]

Abstract
Detecting a signal at an unknown frequency is a common task, arising in settings from dark
matter detection to magnetometry. For any detection protocol, the precision achieved depends on
the signal’s frequency and can be quantified by the quantum Fisher information (QFI). To study
limitations in broadband sensing, we introduce the integrated quantum Fisher information and derive inequality bounds that embody fundamental trade-offs in any sensing protocol. Our inequalities
show that sensitivity in one frequency range must come at the cost of reduced sensitivity elsewhere.
For many protocols, including those with small phase accumulation and those consisting of π-pulses,
we find the integrated quantum Fisher information scales linearly with T . We also find protocols
with substantial phase accumulation that can have integrated QFI that grows quadratically withT , and prove that this scaling is asymptotically optimal. These protocols may allow the very rapid
detection of a signal with unknown frequency over a very wide bandwidth. We discuss the implications of these results for a wide variety of contexts, including dark matter searches and dynamical
decoupling. Thus we establish fundamental limitations on the broadband detection of signals andhighlight their consequences.

4.1

Introduction
Quantum systems, from atoms to SQUIDs, can be excellent sensors. Indeed detecting weaksignals requires the consideration of quantum effects. Even better, entangled detectors are well-

61
known to be more sensitive than their unentangled counterparts [98]. In practice, quantum sensors
have been used for dark matter searches, entanglement-enhanced magnetometry, microwave clocks,
and matterwave interferometers. [99–108].
The quantum Fisher information (QFI) captures the performance of a parameter estimation
protocol [109]. For a pure state parameterized by θ, ρθ := |ψθ ⟩ ⟨ψθ |, the QFI is
J(θ) = 2Tr (∂θ ρθ )2 ,(4.1)[110]. The QFI tells us, via the Cramer-Rao bound [111], how well an unbiased estimator of θ
can approximate its true value. In particular, given m copies of |ψθ ⟩ the variance of any unbiased
1
estimator θ̂ must satisfy Var(θ̂) ⩾ mJ(θ). This bound can be saturated, so the more quantum Fisher

information a protocol has, the better one can estimate θ [111].
We consider a coupling between the qubit and the signal as
H(t) = µB cos(ωt + φ)Z,

(4.2)

where µ is the magnetic moment of the qubit. For instance, we may wish to estimate the strengthof an AC magnetic field [112–120]. While we have included φ in the analysis in Eq. (4.7), we haveomitted it for convenience elsewhere.
To gather information about this Hamiltonian, we need to establish a protocol, which generically consists of preparing the sensing qubit in an initial state, applying a time-dependent control
sequence, and finally performing a measurement. The performance of a protocol will depend on thefrequency of the signal ω. For example, preparing |+⟩ followed by free evolution for time T andmeasurement in the |±⟩ basis is optimal for ω = 0 but performs poorly for ω ≫ 1/T . In fact, we willshow that trade-offs in sensitivity at different frequencies are inevitable. We make this quantitativeby considering the integrated QFI (IQFI).
While this may be different than the integrated sensitivity of a single protocol, it provides
an analytically tractable method of analysis which is tight in many cases of interest, namely weak
fields and protocols consisting only of π−pulses. In general, a choice of protocol includes a choice62of measurement basis, as defined above. In the majority of examples we consider in this paper, theoptimal measurement basis is frequency independent, so that the the optimization over the final
measurement in the definition of QFI can be ignored. By bounding this integral we formalize theidea that there is a fundamental tension between having sensitivity in different frequency bands.
The longer we observe a signal, the more we can expect to learn about it. Thus, it is no surprisethat the IQFI will typically grow with the duration T of a protocol. In fact, we find a number of
constraints on how IQFI grows with T . First, we find that any protocol starting on the equator of
the Bloch sphere that involves only π-pulses has an IQFI of 2πζ 2 T , where ζ ≡ µ/ℏ is the inverse
gyromagnetic ratio of the system being used as a sensor [121]. Second, for an arbitrary protocol
with ζBT ≪ 1, the IQFI is close to 2πζ 2 T . Then we study a particular protocol that significantly
exceeds 2πζ 2 T — by continuously driving a spin with a transverse gX term, our protocol has a peaksensitivity around 2g, with IQFI scaling quadratically with time. We further show that the IQFI
can not exceed quadratic scaling with time, so that this protocol is in a sense optimal. However,the practical restriction to the small signal regime ζBT ≪ 1 is not uncommon, due in part to phaseambiguities that may arise in the accrued phase if one begins to leave this regime.
Beyond signal estimation and detection, our results can be applied to better understand theperformance of dynamical decoupling [122–125]. Here we find that for many dynamical decoupling
protocols, the average (over the initial state) IQFI is at least linearly proportional to T . As a result,dynamical decoupling can at best move the noise sensitivity of a qubit around in frequency space,rather than eliminating sensitivity at all frequencies. This is reminiscent of the filter functions
discussed in [126] which considers CPMG sequences [127–129], a subset of the protocols consideredin this work.
We note that the setting under consideration is different from the waveform estimation studied in [130]. That work studied how to simultaneously estimate a large number of parameters
representing the full time series of a waveform. We consider the sensing problem of estimating asingle Fourier amplitude, i.e. the systems we consider couple to a monochromatic signal B cos (ωt).
The relative simplicity of this setting admits a global analysis of the performance of an arbitrary

63
protocol at different frequencies.

4.2

Preliminaries
We consider Hamiltonians of the form Eq. (4.2), motivated by a spin- 12 particle in a magnetic

field. We first consider estimation protocols composed of instantaneous, arbitrary unitary rotations
Pi followed by periods of free evolution. The choice of measurement at the end of the protocol isassumed to be optimal in the sense that it maximizes the Fisher information of the resulting classicalprobability distribution. For a state starting in the +1 X eigenstate, we have the final state|ψ(T, ω, B)⟩ = UN −1 (tN , tN −1 )PN −1 ...U0 (t1 , t0 )P0 |+⟩ ,(4.3)

where tN = T and U (ti+1 , ti ) is the time evolution operator under the Hamiltonian in Eq. (4.2)
between times ti and ti+1 .
Given Eq. (4.3), the QFI tells us how well we can estimate B. We write J(B|ω) to indicatethat Fisher information with respect to B will in fact depend on the signal frequency ω. Writing|ϕ⟩ := ∂B |ψ(T, ω, B)⟩ and |ψ⟩ := |ψ(T, ω, B)⟩ the QFI can be expressed as [110, 131]
J(B|ω) = 4(⟨ϕ⟩ ϕ + Re{⟨ϕ⟩ ψ 2 }).

(4.4)

To understand the total sensitivity of a protocol across all frequencies we define the integrated QFI(IQFI) for a protocol with total evolution time T as a
Z ∞K(T ) =dωJ(B|ω).

(4.5)

0

Statistically, one way of understanding the IQFI is as an unnormalized expected quantum
Fisher information (EQFI) for estimating a magnetic field amplitude at an unknown frequency
with a uniform prior. While there is no uniform distribution over the positive real line, because
the QFI vanishes at high frequencies, it is possible, in principle, to establish a cutoff at whichaNote that the integral in Eq. (4.5) always exists since the QFI is bounded and asymptotically we have J(ρω ) ∼
P −1
2
2
( N
i=0 Θi ) ∼ 1/ω64
the IQFI is approximately an unnormalized expected quantum Fisher information. As we show in
Appendix 4.A.4, this cutoff is independent of time, and so can be varied independently of protocol
duration. While the EQFI is an interesting statistical quantity, because this equivalence is only
approximate, and because we are interested in a measure of how the total (unnormalized) bandwidth
sensitivity of a protocol changes with time, we consider the IQFI rather than the EQFI. One wayof understanding this distinction is that the former has units of time, while the latter has units oftime squared, and so they measure different physical quantities.
Nevertheless, it is possible to give an interpretation of our results in the Bayesian setting.
For the choice of a delta function prior, it is known that EQFI grows at most quadratically withtime [131]. In this work, however, we are interested in understanding the optimal performance of a
protocol given an arbitrary prior. We will show that the optimal performance for the uniform prioris also quadratic in time, and thus for any prior quadratic scaling is also optimal.
To see this, consider the following integral where µ is the uniform prior, ξ is any other prior,and Ω is some cutoff frequency beyond which the QFI and ξ are both negligible,Z ΩZ Ωdξ(ω)J(B|ω) ⩽ M

0dµ(ω)J(B|ω).

(4.6)

0

In this equation M = max(ξ(ω))/µ0 where µ0 is the value of the uniform prior. Any error in thisexpression is independent of time and can be suppressed by increasing Ω, and so we see that theasymptotic behavior of the EQFI with a uniform prior provides an upper bound on the EQFI withany prior.

4.3

Ramsey and π-pulse protocols
Within the family of control sequences consisting of instantaneous rotations interleaved with

free-evolution, we now consider Ramsey spectroscopy, where a qubit is prepared on the equator of
the Bloch sphere and allowed to freely precess. Defining Θ(ti+1 , ti ) = [sin (ωti+1 − sin (ωti ]/(ℏω),
we find |ϕ⟩ = −iζΘ(T, 0) |ψ⟩ and Eq. (4.4) gives J(B|ω) = 4ζ 2 Θ2 (T, 0). Defining ζ ≡ µ/ℏ, the IQFI

65
follows as
K(T ) = 4ζ

2

Z ∞
dω
0

(sin (ωT + φ) − sin (φ))2
,
ω2

(4.7)
(4.8)

= 2ζ 2 T (π − ln(4) sin (2φ)).

If φ is unknown and therefore random in each experiment, averaging over φ gives K(T ) = 2πζ 2 T ,but if φ is known and we wish to maximize the IQFI, we would lock the experimental sequence to
φ = 3π/4 to get K(T ) = 2ζ 2 T (π + ln 4). For convenience, in what follows we assume φ = 0.
Now consider a protocol applying π-pulses at times t1 ,. . . , tN = T . At time ti we apply either
X, Y or Z. Additionally, we can apply any unitary that leaves the expectation value of Z invariant
b . Then we haveK(T ) = 4ζ

2

Z ∞

N
−1
X

dω(
0

Θ(ti+1 , ti ))2
(4.9)

i=0

= 2πζ 2 T.
K(T ) was derived assuming that the system was initialized in the |+⟩ state. If we instead2

integrate over all initial states we find that the average IQFI is given as 4πζ3 T . This average has
important implications for dynamical decoupling protocols based on π-pulses. It shows that for suchprotocols a qubit must maintain a sensitivity to environmental noise over a substantial frequency
range. Indeed, in the presence of white noise any π-pulse protocol will leave the qubit equallydegraded by the noise. Specifically, we imagine that a qubit is subjected to a background power
spectrum of magnetic field noise fluctuations whose noise spectrum is flat. A lower bound on the
IQFI implies a lower bound on how much of this noise spectrum the qubit will experience, andtherefore on its decoherence. For noise with more structure, these protocols do not allow sensitivity
to noise to be eliminated, but can simply move that sensitivity to a frequency range where theenvironmental noise is fairly low.
bnoπConsider a π−rotation about the axis halfway between the X- and Y- axes, given by exp (−i 2√(X + Y ) )
2

66

4.4

B=0 Bound
We now present an argument to bound K(T ) at B = 0, and approximately bound K(T )for short times and weak magnetic fields. Consider protocols with initial state |ψ0 ⟩ and a control
Hamiltonian H0 (t) in addition to the signal Hamiltonian H(t). We can then write |ψ(T, ω, B)⟩ =U U |ψ0 ⟩, where U is the time evolution due to H0 and U is the interaction picture time-evolution,
given by
U = 1 − iζBZ Tcos(ωτ )ZI (τ )dτ + O(B 2 ),(4.10)

0where we have used ZI (τ ) to express Z in the interaction picture. Then we have
∂B |ψ(T, ω, B)⟩

B=0= −iζUZ T
0cos(ωτ )ZI (τ )dτ |ψ0 ⟩ .

(4.11)

Substituting into Eq. (4.4) and integrating, we find that the IQFI is at most 2πζ 2 T . For small B
and T , the next term on dimensional grounds should be O(B 2 T 3 ), since we can show the term linear
in B is zero. This dimensional analysis assumes that there are no other dimensionful quantities for instance, if the interpulse spacings are not functions of T , then there may be other terms. Thus
we findK(T ) ⩽ 2πζ 2 T + O(ζ 4 B 2 T 3 )(4.12)

The full proof is provided in Appendix 4.A.2. This shows that for small magnetic fields and shorttimes we should expect a roughly linear scaling of the IQFI.

4.5

Entangled probe advantage
From standard results [132, 133] we expect that entangled inputs can outperform this bound.

Indeed an n−qubit GHZ state
1|GHZ⟩n = √ (|0⟩⊗n + |1⟩⊗n ).
2(4.13)

accumulates phase n times more rapidly, so that an analogous argument gives an IQFI at B = 0 ofK(T ) = 2πn2 ζ 2 T.

(4.14)

67
Conversely, product input states can be reduced to the single qubit example, since J(ρ⊗n ) = nJ(ρ),
so that for n qubits starting in a product state, again with B = 0, we haveK(T ) ⩽ 2πnζ 2 T.

(4.15)

So, while entanglement allows us to increase the coefficient in front of T , IQFI still increases linearlywith time.

4.6

Quadratic scaling of IQFI
We have seen that π-pulse protocols and protocols with BT ζ ≪ 1 have IQFI that scales nofaster than 2πζ 2 T . Even an entanglement-enhanced protocol gives linear scaling of IQFI with T ,albeit with an improved coefficient. We now study a simple protocol with IQFI scaling quadraticallyin T .
We consider a continuous-time protocol which applies a transverse field, ℏgX, to the sensing
qubit. This gives a full Hamiltonian ofH(T ) = ℏgX + µB cos (ωT )Z.

(4.16)

Assuming ω ∼ 2g, BT ζ ≫ 1 and using the rotating wave approximation [134], we findJ(B|ω) ∼(ζT )22 2(1 + ( ω−2g
Bζ ) )

which we can integrate from ω = g to ω = 3g to get a lower bound of




g
g 
K(T ) ≳ ζ 2 T 2 + ζB tan−1.
g2
ζB
1+ 2 2(4.17)(4.18)

ζ B

A detailed derivation is given in Appendix 4.A.3. Representative dynamics in this regime are shownin Fig. 3 and 4 of [135].
In Fig. 4.2 we numerically study this protocol. We first approximate it by a discrete pulsebased protocol, described by Eq. (4.3) and given by Trotterization, that intersperses instantaneousrotations around the X-axis by periods of free evolution under the magnetic field. For time T , we

68
Bζ =0
Bζ =2π *(0.16) Hz

K(T )ζ -2 Hz-1

100
50

10
5
0.5

1

5

10

T (seconds)

Figure 4.1: IQFI (K(T )) for using Xπ/2 pulses. Depending on the magnetic field strength we observea more rapid accumulation of IQFI. The dashed lines are ∼ T and ∼ T 2 scalings to guide the eye.
For non-zero magnetic field, we see that a crossover from quadratic to linear scaling occurs when
when BT ζ ≪ 1 (where the perturbative results are valid).

T
can approximate evolution under Eq. (4.16) by m periods of free evolution of duration m
separatedby rotations of angle Tmπ about the X-axis. Indeed, in Fig. 4.1, we see quadratic scaling choosing
m = 2T . In Fig. 4.2 we compare the QFI of the m = T and m = 2T cases with the gX protocol
and a Ramsey protocol, where it is evident that both the m = 2T protocol and the gX protocol
accumulate IQFI more rapidly than the other two protocols. These discrete protocols with quadratic
scaling of IQFI have the property that the number of pulses scales with the total time of the protocol.
In fact, this is necessary, as we will now see.
Consider a protocol with N pulses Pi applied between periods of free evolution Ui . Because⟨ψ⟩ ψ = 1, ∂B ⟨ψ⟩ ψ = 0, we have ⟨ϕ⟩ ψ = − ⟨ψ⟩ ϕ. This means that Re(⟨ϕ⟩ ψ) = 0, and so ⟨ψ⟩ ϕ2
has real part that is non-positive. Thus from Eq. (4.4) we seeJ(B|ω) ⩽ 4 ⟨ϕ⟩ ϕ.
We see that ⟨ϕ⟩ ϕ = ζ 2

PNi,j Θi Θj Vij(4.19)= ζ 2 ΘT V Θ, where Θ is a vector whose ith entry isΘi ≡ Θ(ti+1 , ti ). V is an N × N complex matrix with entries of norm at most 1, so its eigenvalues
have norm at most N . We thus findK(T ) =R∞
0R∞
dωJ(ω) ⩽ 4 0 dω ⟨ϕ⟩ ϕ

R∞R∞= 4ζ 2 0 dωΘT V Θ ⩽ 4ζ 2 0 dωN |Θ|2 = 2πN ζ 2 T.

(4.20)

69

(a)

(b)

(c)

(d)

Figure 4.2: QFI as a function of frequency for some protocols considered in this paper, for different
protocol times T [(a) T = 2, (b) T = 4, (c) T = 6, and (d) T = 8 seconds], with Bζ = 1 Hz.
Ramsey has a large DC QFI, as expected, and very little ability to detect any AC signal. π-pulses,
on the other hand, can be used to measure signals at higher frequencies, as might be expected fromspectroscopy techniques such as CPMG. The gX protocol, with g = 2π × 41 Hz in this example, isseen to be sensitive near 2g. Moreover, it is seen to be sensitive over a broad bandwidth. The
π-protocol shown here consists of π rotations about the X axis, at each integer value of time. The
π/2-protocol consists of π/2 rotations about the X axis, every half second. Similar to the π-protocol,
the π/2-protocol has AC sensitivity, as well as more broadband sensitivity.

70
We will now see that the bound Eq. (4.20) can be used to show that the IQFI can scale at mostquadratically with time, so that up to a constant the IQFI of the gX protocol grows as rapidly as
possible in time.
A continuous-time protocol involves a control Hamiltonian G that gives a total Hamiltonianof the system
H(t) = ℏG(t) + µB cos(ωt)Z.

(4.21)

We can Trotterize this evolution into a discrete sequence [136] like those considered in proving
Eq. (4.20). Then, by linearity of the derivative we can constrain the derivative of the Trotterized
evolution to be close to the actual derivative. Using U (t) to refer to the continuous time protocol,and ϵ̂′′ (t) = ∂B U (t) − ∂B U ′ (t), we see from Eq. (4.19),

DE
J(ω) ⩽ Jd (ω) + 8Re( +|ϵ̂′′ (T )† (∂B U ′ (T ) + 4ϵ̂′′ (T ))|+ ),where we have used Jd (ω) to denote the QFI of the protocol given by U ′ (t). As we show in
Appendix ??, this gives a bound ofK(T ) ⩽

2πζ 2 T 2δt

+ α(δt, Ω, µB, ℏ||G||)T 2 ,where ||G|| is maximum spectral norm of the control, maxt⩽T ||G(t)|| and α is a function that controls
the error in approximating the IQFI of the continuous protocol by the IQFI of the Trotterization.
Crucially it does not depend on T . This proves that K(T ) scales at most quadratically in time.
Thus, our examples with quadratic scaling are asymptotically optimal in the amount of IQFI they
accumulate. This bound can be extended to consider both entangled probes and separable probes
as shown in Appendix 4.A.4, and thus establishes a fundamental limit on the broadband sensitivityof detectors.

4.7

Conclusions
The QFI provides an ultimate bound on how well a quantity can be estimated, in our casethe amplitude of a sinusoid with fixed frequency. Integrating the QFI over all frequencies, we found71
fundamental limits on the broadband performance of quantum sensors. For tasks such as axion
detection [137], this implies that spectral sensitivity is a scarce resource that needs to be carefully
considered when designing metrological protocols. While conventional spectroscopy protocols suchas Ramsey interferometry and CPMG [127, 128] consist only of π-pulses, and therefore linearly
accumulate IQFI, we found both continuous and discrete protocols that quadratically increase thisaccumulation. Moreover, we have shown that this is asymptotically the largest scaling one canachieve.
We have seen that there are protocols with IQFIs that scale as both O(ζ 2 T ) and O(ζ 3 BT 2 ),but which is better? It depends—if the goal is sensitivity to a wide range of frequencies, O(ζ 3 BT 2 )may allow the protocol to work over a wider frequency band. If the goal is sensitivity to a very
narrow frequency range, O(ζ 2 T ) protocols may have support over a small band. Thus, we may see
enhancements when searching for a weak signal over a wide frequency range. In such a setting,long integration times could give a quadratic enhancement of the accumulated QFI compared to
the O(ζ 2 T ) protocols.
Additionally if Td is a characteristic decoherence time of the system, then we are practicallyconstrained to T < Td . Thus for B ≪ 1/(ζTd ) and fixed Td , we also find ζBT ≪ 1. In thesecontexts, our results show that sensing should be expected to be limited by the bound of 2πζ 2 T .
Physically, this corresponds to when the peak angular excursion of the Bloch vector is much lessthan π. For weak fields, if we can sense for long times so that ζBT ≪ 1, then we can accumulatequadratically more IQFI.
The gX protocol is sensitive to frequencies around 2g, making it an excellent candidate forbroadband detection around a particular frequency. It is an open question how to design optimal
metrological protocols with sensitivity spread evenly over wide bands. Techniques like GRAPE[138] may be useful for this task [139].
Many dynamical decoupling protocols consist solely of π-pulses (e.g., [140]). Such techniques
may be described by Eq. (4.3). Consequently, our results show these decoupling strategies arefundamentally limited - while they can move noise sensitivity, they cannot remove it. We leave

72
open whether such bounds apply to arbitrary protocols.
Our key conceptual contribution is the idea that IQFI can be used to understand the tradeoffs inherent in broadband sensing. In some cases, this metric provides a conservation law thatcan be summarized by the slogan “no free QFI”. In particular, in the case where the interaction
picture operator being sensed (ZI (t)) commutes with itself at all times and in the small signal limit(ζBT ≪ 1) we showed that QFI at one frequency ultimately comes at the cost of less QFI at anotherfrequency. This is also true for sequences consisting only of π pulses, when the sensor state begins
on the equator. Moreover we have shown that for any protocol there is a limited amount of IQFI
that can be accumulated. This demonstrates that while broadband sensing is possible, there is anupper limit on how wide the bandwidth of a given protocol can be if one desires a certain sensitivity.
We do not currently know if other classes of control protocols yield strict conservation laws, and weleave this to future work.
The bounds on IQFI that we have found on single qubit initial states can be extended, via
Eq. (4.15), to arbitrary separable states due to the additivity and convexity of the QFI [141, 142]. As
Eq. (4.15) applies only to separable probe states, we can think of it as a kind of standard quantum
limit that cannot be exceeded without entanglement. Indeed, we see that an n-qubit cat state cansignificantly exceed the 2πnζ 2 T performance of unentangled π-pulse-based protocols. This points
to the possibility of using IQFI as a form of entanglement witness, so that the quantum Fisher
information at any particular frequency may be consistent with a separable state but the breadthof such sensitivity can only be explained by an entangled state. In general, however, we find thatthe quadratic-in-time asymptotic upperbound of the IQFI holds for both separable and entangled
probes. Finally, another interesting open question is if other transformations of QFI spectra might
generate new insights into broadband sensing limitations.

73

4.A

Appendix

4.A.1

Derivation of Average IQFI for Instantaneous π-Pulses

Using the identity[A, U BU † ] = U [U † AU, B]U †(4.22)

we find the N-pulse QFI is given by



  
 
2
X
X
† † † †††U0 P0 U1 P1 ...UN −1 PN −1 ZPN −1 UN −1 ...P0 U0 ,J(ω) = −2 Tr
ΘN −1 + ... + Z,Θ02
2
Integrating over all frequency and using the orthogonality conditionZdωΘi Θj = 2π(ti+1 − ti )δij ,(4.23)

we see that when this square is integrated this will sum to be 2πT , so that for a protocol startingin the |+⟩ state and using only π-pulses we findK(T ) = 2πT.

(4.24)

We can also average over the input state, denoted with angle brackets ⟨·⟩, to find1
⟨K(T )⟩ =
4π

Z 2π

Z π
dβ

0

dα sin (α)2πζ 2 T sin2 α,(4.25)

0where one copy of sin comes from the integration measure, and two copies come from the off-diagonal
element of the density matrix. This gives the result from the main text,⟨K(T )⟩ =4.A.2

Perturbative Expansion to O(B 2 )Elided.

4ζ 2 πT.
3(4.26)

74
4.A.3

Derivation of the IQFI under the Rotating Wave Approximation

Elided.

4.A.4

Bound on IQFI of Continuous Protocols

As in the main text, consider a sensing protocol defined by a time-dependent control G(T ).
In particular, the full Hamiltonian we are considering isH(T ) = ℏG(T ) + µB cos (ωT )Z.

(4.27)

We can Trotterize the evolution into a discrete sequence [136], which will look like those consideredin the proof above, with some step size δt. The evolution will be
U ′ (T ) = T e

R N δti (N−1)δt dtG(t) iζBΘN Z

e

R δt

· · · T ei 0 dtG(t) eiζBΘ1 Z + O(δt2 )(4.28)

with Θk = [sin (ωkδt) − sin (ω(k − 1)δt)]/(ℏω), N δt = T , and where T denotes the time-ordering
operator, which is necessary because in general G(t) will not commute with itself at all times. The
number of pulses, N , in the discrete protocol described in Eq. (4.28) gives, to zeroth order,ZdωJc (ω) ⩽2πζ 2 T 2,
δt

(4.29)

for all δt. There are however error terms from the Trotter expansion that we need to propagate
through the IQFI - this is what we will do now. By linearity of the derivative operator we canalso constrain the derivative of the Trotterized evolution to be close to the derivative of the actual
evolution. In particular, we have that
U (T ) = U ′ (T ) + ϵ̂′ (T )(4.30)∂B U (T ) = ∂B U ′ (T ) + ϵ̂′′ (T )(4.31)

75
where ϵ̂′′ (T ) = ∂B ϵ̂′ (T ) is the B derivative of the error term in the Trotter expansion. Then we see
from Eq. (4.19),
D
E D
E
Jc (ω) ⩽ Jd (ω) + 4( +|ϵ̂′′† (T )∂B U ′ (T ) + ∂B U ′† (T )ϵ̂′′ (T )|+ + +|ϵ̂′′ (T )† ϵ̂′′ (T )|+ )DEDE= Jd (ω) + 8Re( +|ϵ̂′′ (T )† ∂B U ′ (T )|+ ) + 4 +|ϵ̂′′ (T )† |ϵ̂′′ (T )|+ ).

(4.32)(4.33)

We will analyze the integrals of the error terms piecewise, first up to a frequency Ω > 0. The
derivative of the first order Trotterized evolution defined above is′∂B U (T ) = −iζN−1XΘj P0 U0 ...Pj ZUj ...PN −1 UN −1 .

(4.34)

j=0

Noting that |Θk | ⩽ 2T /N for a uniformly spaced pulse sequence, this gives
||
⩽

Z Ω

N −1

ϵ̂′′ (T )(−

0

iµ X
)
Θj P0 U0 ...Pj Uj ...Pn−1 Un−1 ||ℏ(4.35)

j=0µ
maxω<Ω ||ϵ̂′′ (T )||ℏZ Ω NX
−1
0

j=0

⩽

|Θj |||P0 U0 ...Pj Uj ...Pn−1 Un−1 ||µmaxω<Ω ||ϵ̂′′ (T )||ℏZ Ω NX−1(4.36)

2T /N(4.37)

2µ=maxω<Ω ||ϵ̂′′ (T )||ΩT,ℏ(4.38)0

j=0

Similarly,Z Ω40||ϵ̂′′ (T )† ϵ̂′′ (T )|| ⩽ 4maxω<Ω ||ϵ̂′′ (T )||2 Ω(4.39)

where the maximum is over ω on which the error term implicitly depends. Then we have, forthe last two terms in Eq. (4.33)
Z ∞
8||
0

16µ
Re(ϵ̂ (T )∂B U (T ))|| ⩽ΩT maxω<Ω ||ϵ̂′′ (T )|| + ||
ℏ
′′†

′

Z ∞

8Re(ϵ̂′′† (T )∂B U ′ (T ))||,(4.40)||ϵ̂′′ (T )† ϵ̂′′ (T )||.

(4.41)

Ω

and
Z ∞
4
0

||ϵ̂′′ (T )† ϵ̂′′ (T )|| ⩽ 4maxω<Ω ||ϵ̂′′ (T )||2 Ω + 4

Z ∞
Ω

76
ϵ̂′′ (T ) is at worst proportional to T , since the error in a time step δt is independent of T (it’sproportional to ||[ℏG(t), µB cos (ωt)Z]||δt2 ), and there are N = T /δt times steps. Furthermore||∂B U ′ (T )|| = ||−iµ X
Θj P1 U1 ...Pj Uj ...Pn Un || ⩽ µN/(ℏω) = µT /(δtℏω),ℏ(4.42)

jso we need only understand ||ϵ̂′′ (T )|| at high frequency, where || · || is the spectral norm. To thisend, consider the error in time step δt given in [143] as U ′ (tk , tk−1 )F (tk , tk−1 ), where U ′ (a, b) is theTrotterized evolution from time a to time b andZ tkC(v, tk−1 )F (v, tk−1 )dv(4.43)

( Z
)† Z
( Z
)†
tk
u
tk
G(s)ds)A(s)ds)C(tk , tk−1 ) = exp (idu exp (i(4.44)F (tk , tk−1 ) =
tk−1

tk−1

tk−1

tk−1

(
×[A(u), ℏG(tk )] exp (−i

)

Z u

(

Z tk

)
G(s)ds) ,A(s)ds) exp (−i(4.45)

tk−1

tk−1

where A(s) = µB
ℏ cos (ωs)Z. Thus, to lowest order in 1/ω,()
Zu

exp (−iA(s)ds)tk−1= exp{(iBZ(cos (ωu) − cos (ωtk−1 ))/ω)}(4.46)µB≈I −i
Z(cos (ωu) − cos (ωtk−1 ))/ωℏ
so that the relevant integrals over u are
Z tk
tk−1

du cos (ωu) = (sin (ωtk ) − sin (ωtk−1 ))/ωZ tk

(4.47)

du cos (ωu)(cos (ωu) − cos (ωtk−1 ))(4.48)= (2δtω + sin (2tk−1 ω) − 4 cos (tk−1 ω) sin ((tk−1 + δt)ω) + sin (2(tk−1 + δt)ω))/(4ω)(4.49)tk−1

77
Plugging these in we findµB
C(tk , tk−1 ) =((sin (ωtk )
ℏ
)
(
Z

)†
( Z
tk
ℏG(s)ds) [Z, ℏG(tk )] exp (−i− sin (ωtk−1 ))/ω)(exp (i
tk−1

+

tk

ℏG(s)ds) )(4.50)(4.51)

tk−1

i2ζ 2 B 2((2δtω + sin (2tk−1 ω) − 4 cos (tk−1 ω) sin ((tk−1 + δt)ω) + sin (2(tk−1 + δt)ω))
4ω 2
)
)†
(
( Z
Z tk
tk
ℏG(s)ds) )ℏG(s)ds) Z[Z, ℏG(tk )] exp (−i×(exp (i(4.52)(4.53)

tk−1

tk−1

( Z
)†
(
)
Z tk
tk
+(exp (iℏG(s)ds) [Z, ℏG(tk )]Z exp (−iℏG(s)ds) ))))),
tk−1(4.54)

tk−1

where we have only kept the lowest order correction terms. By inspection, we see that there is
no way for a T dependence to enter for higher order terms, and moreover we see that for ω large
enough (Ω >> µB/ℏ) the terms are both integrable over ω and arbitrarily small. We are interested
in N times the B-derivative of this error, where the extra factor of N is because the total error
accumulates at worst as N times the step-wise error. But now we are done, since this shows thatfor all ϵ > 0 there is an Ω(||G||, B, δt) such that in total we have a bound ofZdωJc (ω) ⩽2πζ 2 T 2
2πζ 2 T 2
+ (c(ℏ||G||, µB, δt) + ϵ(ℏ||G||, µB, δt))T 2 =+ α(ℏ||G||, µB, δt)T 2 ,δtδt(4.55)

where we’ve denoted two parts in the coefficient of the second term - one coming from the lowest
order contribution to the right tail of the QFI, and one coming from the inaccuracy in this apR
proximation. So in total we see that dωK(T ) ∈ O(T 2 ). We see furthermore that this argument
generalizes to arbitrary entangled probes. The B−derivative of the time evolution is maximized on
entangled inputs and contributes a factor of n, the number of qubits. To see this, note Eq. (4.34)
for n qubits is instead′∂B U (T ) = −iζN−1 k=n−1
XX
j=0

Θj P0 U0 ...Pj Z (k) Uj ...PN −1 UN −1 ,(4.56)

k=0

where the new index (k) denotes the Z operation on the k th qubit, and the other operators are
generalized to their natural multi-qubit counterparts. Similarly, the Trotter error has a factor of n

78
because it is proportional to the sum of Z (k) , and so the B−derivative of the Trotter error also is
proportional to n. Because both terms in the QFI are quadratic in |ϕ⟩ we find a factor of n2 as inR
the GHZ state example, giving a total bound of dωK(T ) ∈ O(n2 T 2 ) for n qubits.

Chapter 5

Noisy Reservoir Computation [144]5.1

Abstract
In this note we extend the definition of the Information Processing Capacity (IPC) by

Dambre et al [145] to include the effects of stochastic reservoir dynamics.

We quantify thedegradation of the IPC in the presence of this noise.
[1] Dambre et al. Scientific Reports 2, 514, (2012).

5.2

Reservoir Computation
In the traditional framework of a classical reservoir computer, [146] one considers a dynamicalsystem observed at discrete time-steps t = 0, 1, 2, . . ., with internal states s(t) ∈ Rn , driven by inputs
U (t) ∈ Rd . The reservoir dynamics are then encompassed by the dynamical update rules(t) = F s(t − 1), U (t) ,(5.1)

from which m outputs xk (t) ∈ R with k ⩽ m ⩽ n, can be constructed from the reservoir as
xk (t) = G s(t), U (t) .

(5.2)

These outputs can then be utilized to approximate a target function y(t) ∈ R, via a supervised
learning scheme.
Specifically, we first define the concatenated h-step sequence of recent inputs U −h (t) = [U (t−h + 1), U (t − h + 2), . . . , U (t)]. While we may use the reservoir to learn a function of time, the80reservoir outputs themselves can be approximated by maps xhk : U −h (t) 7→ R. In particular, thisis because we require that the reservoir satisfies the fading memory property [145]. A dynamicalsystem has fading memory if, for all ϵ > 0, there exists a positive integer h0 ∈ N, such that for allh > h0 , for all initial conditions, and for all sufficiently long initialization times T ′ > h, the xk (t)at any time t ⩾ 0 are well-approximated by functions xhk :E(xk (t) − xhk [U −h (t)])2 < ϵ(5.3)

where the expectation is taken over the t + T ′ previous inputs.
The signals xk (t) can used to construct estimators (denoted with hats) of y(t)ŷw (t) = wT X(t),(5.4)

where w is a vector of weights, and X(t) is the vector whose entries are xk (t).
Characterizing the learning capabilities of a dynamical system requires analyzing it in alearning-task independent manner. We do so by utilizing a quantity known as the information
processing capacity (IPC) [145]. We first define the capacity to reconstruct some arbitrary signal yasCT [y] = 1 − minwwhere ⟨χ⟩T = T1⟨(ŷw − y)2 ⟩T,⟨y 2 ⟩T(5.5)

PT

t=1 χ(t) denotes a time average. Then for a complete and countably infinite setof basis functions {y1 , y2 , . . .} for the Hilbert space of functions with fading memory [145], the IPC
of a dynamical system is given as
IPC = lim lim

D→∞ T →∞DX

CT [yℓ ] ⩽ n.

(5.6)

ℓ

Intuitively, the IPC can be thought of as a normalized measure of the size of the subspace offunctions learnable by the dynamical system. Here the basis set should be ordered in some senseby the complexity (e.g., the polynomial degree and the memory time h) of the basis functions. The
numerical estimate of the IPC requires defining a probability measure over all possible sequences of
inputs (thus promoting U (t) to a random process with measure ξ) and truncating the basis set atsome sufficiently high, but finite cutoff.

81

5.3

Noisy Reservoir Computation
In this section we extend the ideas from Sec. 5.2 and the formula for the IPC in [145] todescribe the performance of a reservoir in the presence of noise. The internal state and readouts(s(t), X(t)), thus evolve stochastically, even for a fixed input sequence. Formally, what this meansis that in the following, X(t) and U (t) are random variables with probability measures η and ξ.
Furthermore, because the state of the reservoir is a function of the input to the reservoir, we havethat in general X(t) is conditioned on all past U (t).
In what follows, we use the overline A to denote the expectation value of some variable A overthe input signal distribution η. We use ⟨A⟩ to denote expectation values of A over the reservoir noise
distribution ξ. We have otherwise suppressed the notational dependence on U (t) in this analysis.
We find that the IPC is
IPC ≤ Tr (I + Q̃ξ )−1

(5.7)

1
≤ n,
1 + σ̃k2

(5.8)

=

n
X
k=1

where I is the identity matrix and Q̃ξ measures the normalized reservoir output noise covariance.
This is by design a positive (semi-)definite matrix whose eigenvalues we denote as {σ̃k2 : k = 1, . . . , n}
and which are the generalized reservoir noise variances. Thus IPC = n only occurs when Q̃ξ = 0
is the trivial matrix, demonstrating that noise strictly decreases the achievable IPC of a reservoir
computer of given (output) size n. In the following derivation of the inequality (5.7) we assume
that the average reservoir outputs ⟨X⟩ (i.e, averaged over the reservoir noise) satisfy
Qη := ⟨X⟩⟨X⟩T > 0.

(5.9)

We will designate this assumption as the full signal rank condition. We note, however, that this
only simplifies the analysis, and that the result is correct even when the condition does not hold.
We revisit this case at the very end of our discussion.

82
5.3.1

Proof

We limit our proof to the finite case of a truncated basis set (of size D). A common quantity
to consider in any regression analysis is the expected squared error, J, between the predictions madeby the model and the data. Thus we start by defining the expected squared reconstruction error

*D n
+
XX
J=
(yℓ (t) − xk (t)wkℓ )2ℓ=1 k=1

*D n
+
XX
=
(yℓ (t) − xk (t)wkℓ )(yℓ (t) − wkℓ xk (t))ℓ=1 k=1= ⟨(Y T − X T W)(Y − WT X)⟩= Tr ⟨(Y − WT X)(Y T − X T W)⟩,

(5.10)

where here vectors are column vectors, and transposes are row vectors. In particular, Y ∈ RD×1 ,W ∈ Rm×D are the vector of estimated signals and weight matrix, respectively, and X ∈ Rm×1
is defined as before. We have chosen the convention for WT so that WT X is an estimator of thetarget functions.
In this notation we have suppressed and will continue to suppress the dependence on time,so that Y := Y (U −h (t)) are target functions of the finite-sequence inputs U −h (t), and will beapproximated in this scheme by linear functions of the measured outputs X := X(t). That is, thegoal is to find a weight matrix W such that WT X ≈ Y . For computing the IPC, we will take
Y (U −h (t)) to constitute a complete basis of functions on the finite input sequences, as in Eq. (5.6).
Differentiating Eq. (5.10) with respect to W then gives∇W J = ∇W ⟨X T WWT X − Y T WT X − X T WY ⟩= ∇W Tr ⟨X T WWT X − 2XY T WT ⟩= 2⟨XX T ⟩W − 2⟨XY T ⟩,(5.11)

83
allowing us to solve for W via the first order optimality condition ∇W J = 0 as
(5.12)

W∗ = ⟨XX T ⟩+ ⟨XY T ⟩where A+ denotes the Moore-Penrose pseudo-inverse of A. Under the full signal rank condition(5.9) this is a proper inverse.
Defining the reservoir output noise as(5.13)∆X := X − ⟨X⟩,

allows decomposing the output into its deterministic and its noise part X = ⟨X⟩ + ∆X. With this,
we find⟨XX T ⟩ = ⟨X⟩⟨X⟩T + ⟨∆X∆X T ⟩| {z } |{z}Qη(5.14)Qξ+ ⟨X⟩ ⟨∆X T ⟩ + ⟨∆X⟩⟨X⟩T| {z } | {z }00(5.15)= Qη + Qξ ,i.e., we can decompose this second moment matrix into the second moment matrix of the deterministic output signal Qη and the noise covariance Qξ .
Under the full signal rank condition (5.9), we can perform a spectral decomposition⟨X⟩⟨X⟩T = V DV T with positive definite, diagonal matrix D and an orthogonal matrix V andfactor the second moment matrix as 1
1⟨XX T ⟩ = V D 2 I + Q̃ξ D 2 V T(5.16)

where we have defined
1

1Q̃ξ := D − 2 V T Qξ V D − 2 = ⟨∆X̃∆X̃ T ⟩.

(5.17)

X̃ in turn is related to the above diagonalization of the deterministic second moment matrix⟨X⟩⟨X⟩T , corresponding to a basis change for the outputs
1

1X̃ = D − 2 V T X ⇔ X = V D 2 X̃.

(5.18)

84
We can use this same transform to re-express the overlap matrix(5.19)CXY = ⟨XY T ⟩ = ⟨X⟩Y T
1

1= V D 2 ⟨X̃⟩Y T = V D 2 CX̃Y .
Substituting W∗ back into J and transforming to the normalized outputs gives
J(W∗ ) = Tr ⟨Y Y T − 2Y X T W∗ + W∗T XX T W∗ ⟩



−1
T
T
= Tr ⟨Y Y ⟩ − CX̃Y I + Q̃ξ
CX̃Y



−1


= D − Tr I + Q̃ξCX̃Y CTX̃Y  ≥ D − n| {z }(5.20)≤I

We note that the trace in the last line is over the reservoir output indices k = 1, 2, . . . , n whereas
the previous lines have traces over the basis set l = 1, 2, . . . , D. Finally, our claim follows from the
Cauchy-Schwarz inequality and the fact that Tr AB ≥ 0 for positive semi-definite A, B ≥ 0. Also
note that under the full signal rank condition we have limD,T →∞ CX̃Y CTX̃Y → I, where in typicalphysicist fashion we ignore the subtleties around how to carefully take the infinite limits.
For a simple example, consider a deterministic reservoir with orthonormal outputs, i.e. Qη =I. By introducing Gaussian noise on the outputs with covariance matrix Σ, the IPC is boundedexactly as in Eq. (5.7), giving
IPC ⩽

n
X
k=1

1
,
1 + σk2(5.21)

where σk2 are the eigenvalues of Σ, since Qη = I implies Q̃ξ = Qξ = Σ. While we have specified
that the noise is on the outputs of the reservoir, our derivation shows that the result is the same
independent of whether the noise is added to the outputs or is internal to the reservoir.
Finally, note that this proof can easily be extended to the case without the full signal rank
condition (5.9). E.g., if the second signal moment has only rank ñ < n, then it requires replacing1

1

D − 2 with its pseudo-inverse version (D 2 )+ and some special care needs to be taken when canceling1

1(D 2 )+ D 2 = Iñ where Iñ is a diagonal matrix with ñ ones on its diagonal (and otherwise only zeros).
Also, note that in this case the overlap matrix CXY has rank ≤ min (ñ, D), which implies that for85
our optimal weights W∗ we are only sensitive to those output noise contributions that are in thelinear span of the expected output signals. In particular, the noise covariance then generalizes to
1

1Q̃ξ := (D 2 )+ V T Qξ V (D 2 )+ = ⟨∆X̃∆X̃ T ⟩which may have lower rank than Qξ .

(5.22)

Chapter 6

A Theory of Direct Randomized Benchmarking [147]

6.1

Abstract
Randomized benchmarking (RB) protocols are widely used to measure an average error rate

for a set of quantum logic gates. However, the standard version of RB is limited because it only
benchmarks a processor’s native gates indirectly, by using them in composite n-qubit Clifford gates.
Standard RB’s reliance on n-qubit Clifford gates restricts it to the few-qubit regime, because thefidelity of a typical composite n-qubit Clifford gate decreases rapidly with increasing n. Furthermore,
although standard RB is often used to infer the error rate of native gates, by rescaling standard
RB’s error per Clifford to an error per native gate, this is an unreliable extrapolation. Direct RB is amethod that addresses these limitations of standard RB, by directly benchmarking a customizable
gate set, such as a processor’s native gates. Here we provide a detailed introduction to direct
RB, we discuss how to design direct RB experiments, and we present two complementary theories
for direct RB. The first of these theories uses the concept of error propagation or scrambling in
random circuits to show that direct RB is reliable for gates that experience stochastic Pauli errors.
We prove that the direct RB decay is a single exponential, and that the decay rate is equal tothe average infidelity of the benchmarked gates, under broad circumstances. This theory showsthat group twirling is not required for reliable RB. Our second theory proves that direct RB isreliable for gates that experience general gate-dependent Markovian errors, using similar techniques
to contemporary theories for standard RB. Our two theories for direct RB have complementary
regimes of applicability, and they provide complementary perspectives on why direct RB works.

87
Together these theories provide comprehensive guarantees on the reliability of direct RB.

6.2

Introduction
Reliable, efficient, and flexible methods for benchmarking quantum computers are becoming

increasingly important as 5-50+ qubit processors become commonplace. Isolated qubits or coupledpairs can be studied in detail with tomographic methods [148–150], but tomography of general nqubit processes requires resources that scale exponentially with n. Randomized benchmarking (RB)methods [151–165] were introduced partly to avoid the scaling problems that afflict tomography.
RB avoids these scaling problems because, in the most commonly used RB protocols [154, 155, 160],
both the necessary number of experiments [166] and the complexity of the data analysis [155] areindependent of n. RB methods efficiently estimate a single figure of merit—the average infidelity ofa gate set—by (i) running random circuits of varied depths (d) that should implement an identity
operation, (ii) observing the probability that the input is successfully returned (Sd ) versus d, andthen (iii) fitting this data to an exponential. The decay rate of this exponential (r) is an estimateof the gates’ average infidelity.
However, many RB methods have a different scaling problem, introduced by a gate compilation step. Most RB protocols benchmark an n-qubit gate set that forms a group [154–164], sothat they can use group twirls as their theoretical foundations. But a quantum processor’s native
operations do not normally form a group. Instead, subsets of a processor’s native gates can beused to generate a variety of different groups (e.g., the unitary group, the Clifford group, thePauli group, etc). This is a problem for implementing group-based RB methods because, for manyn-qubit groups, the number of basic gates required to implement a typical group element is verylarge, meaning that many group elements can only be implemented with low fidelity.
The standard RB protocol—which we will call “Clifford group RB”—benchmarks the n-qubit
Clifford group [154, 155], estimating an error rate (r) that corresponds closely to the average infidelity of a gate from the n-qubit Clifford group [168, 169]. This protocol runs random sequences ofd + 2 composite gates from that group (see the upper circuit in Fig. 6.1), with a variable benchmark

88

Figure 6.1: An illustration (adapted from [167]) comparing the varied-length random circuits usedin the standard RB protocol of Magesan et al. [154, 155] (“Clifford group RB”) and the streamlined
direct RB protocol that was introduced in [167] and that we investigate further here. Direct RB canbe implemented on almost any n-qubit gate set (a set of circuit “layers” or “cycles”) that generates
a unitary 2-design. The case shown here is when that gate set is a set of Clifford gates (cnot,
Hadamard, and the phase gate) that generate the n-qubit Clifford group. The circuits of direct RB
can be shallower than those of Clifford group RB (e.g., consider the case of d = 0 in each circuit),
allowing for RB on more qubits.

depth (d, with d ⩾ 0). The first d + 1 gates are sampled uniformly from the Clifford group, and thelast gate is selected to invert the proceeding d + 1 gates. For a single qubit there are only 24 Cliffordgates, so each Clifford gate can typically be compiled into a short sequence of native gates. But the2

Clifford group grows quickly with the number of qubits n: there are 2O(n ) n-qubit Clifford gates2n[170, 171], and a single typical n-qubit Clifford gate can be compiled into no fewer than O( logn)one- and two-qubit gates [172–174]. This means that the fidelity of a random Clifford gate degrades
rapidly with n for a fixed quality of one- and two-qubit gates, i.e., r → 1 quickly with increasing n.
This makes it infeasible to reliably estimate Clifford group RB’s error rate r on many qubits—even
with very high fidelity one- and two-qubit gates—because as n increases even the shortest Cliffordgroup RB circuits (d = 0, corresponding to a random n-qubit Clifford gate and its inverse) typicallyhave such small success probabilities that they cannot be distinguished from 21n with a reasonable
number of circuit repetitions (see Fig. 6.2). Indeed, one- and two-qubit Clifford group RB has
been widely implemented [175–188] but we are aware of only two publications presenting Cliffordgroup RB experiments on more than two qubits [189, 190], and none presenting Clifford group RB
experiments on six or more qubits.

89
An alternative to Clifford group RB is to benchmark a gate set that is a smaller group [156–159, 161, 163], enabling group elements to be implemented with fewer native gates. The gatecompilation overhead can then be reduced from Clifford group RB, but often at a cost—the data
analysis and the experiments become more complex. The simple behavior of Clifford group RB,
i.e., the exponential decay of its success probability data, is guaranteed by a particular property of
the Clifford group: conjugation of an error channel by a uniformly random Clifford group element“twirls” that channel into an n-qubit depolarizing channel. This follows because the Clifford group isa unitary 2-design [191–193] (or, equivalently, because the superoperator representation of Cliffordgates consists of the direct sum of two irreducible representations of the Clifford group). The
less like a unitary 2-design a group is (i.e., the more irreducible representations of the group its
superoperator representation decomposes into) the weaker the effect of twirling over it is, and so
the more complex the RB decay curve can be [156–159, 161, 163] (see [163] for a theory of RB overgeneral groups). For general groups, the RB decay curve is a sum of many exponentials, rather than
the single exponential of Clifford group RB.
Another consequence of the gate compilation required in group-based RB protocols is thatthey measure an error per compiled group element. There are infinitely many ways to compile
each group element into native gates, and the observed error rate depends strongly on which groupcompilations is used. This is desirable if we aim to estimate the fidelity with which gates in thatgroup can be implemented. But normally group-based RB error rates are used as a proxy for nativegate performance [184–186, 189]. In particular, it is common to estimate a native gate error ratefrom a Clifford group RB error rate, e.g., by dividing Clifford group RB’s r by the average circuit
size of a compiled gate [184–186] or by the average number of two-qubit gates in a compiled gate
[189] (possibly after correcting for the estimated contribution of one-qubit gates). However, theserescaling methods have little theoretical justification [167, 194]: in general, the resultant error rate
is not a reliable estimate of the average infidelity of the native gates.
One solution to the limitations of Clifford group RB is direct randomized benchmarking[167]. Direct RB is a technique that is designed to retain the conceptual and experimental simplicity

90
and efficiency of Clifford group RB while enabling direct benchmarking of a processor’s native
gates. Like all RB protocols, direct RB utilizes random circuits of variable length. Furthermore,
like Clifford group RB, direct RB’s data analysis simply consists of (1) fitting success probability
data to a single exponential decay, and (2) rescaling the fit decay rate to obtain an estimate of
the benchmarked gates’ average infidelity. But, unlike Clifford group RB, the varied-length portionof direct RB circuits can consist of randomly sampled layers of a processor’s native gates, rather
than compiled group operations (see the lower circuit in Fig. 6.1). Direct RB therefore enables
estimating the average infidelity of native gate layers. The only restriction on the gate set that canbe benchmarked with direct RB is that it must generate a group that is a unitary 2-design (e.g.,
the n-qubit Clifford group).
Direct RB addresses the main limitations of Clifford group RB, but it has been lacking atheory that proves that it is reliable. Those existing theories for RB that are applicable to direct
RB [167, 195, 196] provide few guarantees on direct RB’s behavior. In this paper we provide a
detailed introduction to direct RB and two complementary theories of direct RB. These theories
show that direct RB is reliable—the direct RB success probability follows an exponential decay andthe direct RB error rate is the infidelity of the benchmarked gates—and explain why direct RBworks.
This paper is structured as follows. In Section 6.3 we introduce our notation and review thenecessary background material. In Section 6.4 we provide a detailed definition and introduction to
the direct RB protocol, and we compare it to other RB protocols for benchmarking native gate sets
[153, 196, 197]. The definition for direct RB presented here generalizes that in [167]. This section
includes examples that illustrate how direct RB can be used to benchmark a variety of physically
relevant gate sets—including universal gate sets on few qubits. In Section 6.5 we present a theory
for direct RB on gates that experience stochastic Pauli errors, which is the relevant error modelfor gates that have undergone Pauli frame randomization [198, 199] or randomized compilation[200, 201]. This theory provides a physically intuitive underpinning for direct RB, by using theintuitive concepts of error propagation and scrambling in random circuits. Interestingly, this theory

91
shows that group twirling is not necessary for reliable RB—direct RB can be reliable even when thebenchmarked gate set is very far from approximating a unitary 2-design. This enables us to showthat direct RB is reliable even in the n ≫ 1 regime, where short sequences of layers of native gatescannot approximate a unitary 2-design.
One of the contributions of this paper is a theory for direct RB with general gate-dependent
Markovian errors that is similar in nature to the group twirling (and Fourier analysis) theories forgroup-based RB methods. This theory uses the concept of a sequence-asymptotic unitary 2design, which we introduce in Section 6.6. A sequence-asymptotic unitary 2-design is any gate set
for which length d random sequences of gates from that set create a distribution over unitaries thatconverges to a unitary 2-design as d → ∞. Sequence-asymptotic unitary 2-design are closely related
to various existing notions of approximate or asymptotic unitary designs and the general theoryof scrambling in random circuits [191, 192, 202–208]. However, to our knowledge, the particularconcept and technical results that we require for our theory of direct RB do not appear in theliterature.
In Section 6.7 we present our theory for direct RB under general gate-dependent Markovian
errors, which is based on twirling superchannels (or Fourier transforms on groups). First, we showthat the depth-d direct RB success probability can be computed exactly by taking the dth power of asuperchannel (R) constructed from the permutation matrix representation of the group generated by
the benchmarked gate set. Then we show how the direct RB decay can be (approximately) rewritten
in terms of a smaller superchannel (L) that is constructed from the superoperator representationof the generated group. These results correspond to equivalent theories for Clifford group RB
presented in [168, 169, 209, 210]. We then use the spectral decomposition of these superchannels,
together with the theory of sequence-asymptotic unitary 2-designs, to show that the direct RB
decay is approximately a singe exponential. Finally, we use the L superchannel to show that thedirect RB error rate is equal to a gauge-invariant version of the mean fidelity of the benchmarked
gates, building on Wallman’s derivation of an equivalent result for Clifford group RB [169]. Our twotheories of direct RB are complementary, providing different perspectives as well as different regimes

92
of applicability. Our superchannel-based theory (Section 6.7) is more mathematically rigorous thanour scrambling-based theory (Section 6.5), and it applies to a more general class of errors. However,
our superchannel-based theory relies on an approximation whose size increases with n, whereas ourscrambling-based theory shows that direct RB is reliable even when n ≫ 1. Therefore, together
these theories provide comprehensive evidence for the reliability of direct RB.

6.3

Definitions
In this section we introduce our notation and review the necessary background material.

6.3.1

Gates, gate sets, and circuits

An n-qubit gate G is a physical operation associated with an instruction to ideally implement
a particular unitary evolution on n qubits (we use the term “gate” for consistency with the RB
literature—an n-qubit gate is also often referred to as a “layer” or a “cycle”). We denote the unitarycorresponding to G by U (G) ∈ SU(2n ). It will also often be convenient to use the superoperator
representation of a unitary, so we define G(G) to be the linear map
G(G)[ρ] := U (G)ρU (G)† ,(6.1)

where ρ is an n-qubit density operator. We consider a gate G to be entirely defined by the unitary
U (G). RB methods are agnostic about how an n-qubit gate is implemented, except that an attempt
must be made to faithfully implement the unitary it defines. We will use G to denote a set of n-qubit
gates, which need not be a finite set, and, slightly abusing notation, we will also use G to denotethe corresponding sets of unitaries (i.e., {U (G) | G ∈ G} and superoperators (i.e., {G(G) | G ∈ G})
with the meaning implied by the context.
A quantum circuit C over a n-qubit gate set G is a sequence of d ⩾ 0 gates from G. We will
write this as
C = Gd Gd−1 · · · G2 G1 ,(6.2)

where each Gi ∈ G, and we use a convention where the circuit is read from right to left. The circuit

93
C is an instruction to apply its constituent gates, G1 , G2 , . . . , in sequence, and it implements theunitary
U (C) = U (Gd )U (Gd−1 ) · · · U (G2 )U (G1 ).

(6.3)

In direct RB, and most other RB methods, multiple gates in a circuit must not be combined orcompiled together by implementing a physical operation that enacts their composite unitary (there
are “barriers” between gates).

6.3.2

Fidelity

In our theory for direct RB we will show how the direct RB error rate is related to the fidelityof the benchmarked gates. There are two commonly used version of a gate’s fidelity: average gate
fidelity and entanglement fidelity, defined below. Throughout this work, we will use the Markovian
model for errors, whereby the imperfect implementation of a gate G is represented by a complete
positive and trace preserving (CPTP) superoperator G̃(G) [so, for low-error gates G̃(G) ≈ G(G)].
The average gate fidelity of G̃(G) to G(G) is defined by
Fa (G̃, G) :=Znodψ Tr G̃[|ψ⟩⟨ψ|] G[|ψ⟩⟨ψ|] ,(6.4)

where dψ is the normalized Haar measure [211], and where here we have dropped the dependence of
G and G̃ on G for brevity (we also do this elsewhere when convenient). The entanglement fidelity
is defined by
Fe (G̃, G) := ⟨φ|(I ⊗ Λ)[|φ⟩⟨φ|]|φ⟩,(6.5)

where φ is any maximally entangled state [211] andΛ(G) = G̃(G)G(G)−1 .

(6.6)

The corresponding infidelities, average gate infidelity (ϵa ) and entanglement infidelity (ϵe ),are defined by
ϵa/e (G̃, G) = 1 − Fa/e (G̃, G).

(6.7)

94
These infidelities are related by [211]ϵe =2n + 1
ϵa .
2n

(6.8)

The average gate [in]fidelity is more commonly used in the RB literature, but in this work we use
the entanglement [in]fidelity. Typically we drop the subscript, letting ϵ ≡ ϵe . Note that a gate’s[in]fidelity is not a “gauge-invariant” property of a gate set [168], meaning that it is not technicallymeasurable: we discuss this subtle point when presenting our theory for direct RB with generalgate-dependent Markovian errors (Section 6.7).
Randomized benchmarking methods are typically designed to measure the mean infidelity ofa set of gates. Direct RB is designed to measure
ϵΩ =

X
G∈G

h
i
Ω(G)ϵ G̃(G), G(G) ,(6.9)

where Ω is a probability distribution over the gate set G. Note that, except where otherwise
stated, G does not need to be a finite set, i.e., it can include gates with continuous parameters.
PTheG∈G Ω(G) notation is a shorthand for integrating and/or summing over G according to
the measure Ω(G). When gates experience only stochastic Pauli errors, then ϵΩ is equal to theprobability that a Pauli error occurs on a gate sampled from Ω.

6.4

Direct randomized benchmarking
In this section we first define the direct RB protocol and explain what its purpose is (Sec-tion 6.4.1). Direct RB has flexible, user-specified components, so we then provide guidance on how
to choose these aspects of direct RB experiments (Sections 6.4.3-6.4.5). We then explain why the
direct RB protocol is defined the way it is, and we discuss how it differs from Clifford group RB (Sections 6.4.6-6.4.8). Finally, we compare direct RB to other RB protocols for directly benchmarking
native gate sets (Section 6.4.10).

95
6.4.1

The direct RB protocol

We now define the n-qubit direct RB protocol, and explain what it aims to measure. The
definition given here is more general than that in [167]. For example, in [167], direct RB is definedfor gate sets containing only Clifford gates. Our definition here permits non-Clifford gates. Notethat our definition for direct RB applies to gates on a set of n ⩾ 1 qubits. Generalizing to n ⩾ 1
qudits of arbitrary dimension is conceptually simple, as is the case with Clifford group RB, but we
do not pursue this here. In addition to parameters that set the number of samples (which exist in
all RB protocols), direct RB on n qubits has two flexible, user-specified inputs:(1) A set of n-qubit gates (a.k.a., layers or cycles) to benchmark (G).
(2) A sampling distribution Ω over the gate set G.
We denote direct RB of the gate set G with the sampling distribution Ω by DRB(G, Ω). We
call G a “gate set” for consistency with common RB terminology, but note that G contains n-qubit
gates—i.e., “circuit layers” or “cycles”—not one- and two-qubit gates. All n-qubit gate sets generatesome n-qubit group, or a dense subset of some group a . This group plays an important role in the
direct RB protocol, and we denote it by C (for brevity, for a gate set G that only generates a dense
subset of a group C we also refer to G as generating C). In most of our examples this group willbe the n-qubit Clifford group, but this is not required (the conditions required of C are stated in
Section 6.4.4, which include that it is a unitary 2-design).
Having introduced G and Ω, we are now ready to define the direct RB protocol. DRB(G, Ω)
is a protocol for estimating ϵΩ [see Eq. (6.9)] and it consists of the following procedure:(1) Sample the circuits. For each d in some set of user-specified benchmark depths all
satisfying d ⩾ 0, and for k = 1, 2, . . . , Kd for some user-specified Kd ⩾ 1:(1)1. Choose a target output n-bit string sd,k for this circuit. This can be set to the all-zerosaThroughout the remainder of this paper we use the phrase “G generates the group C” to include the case where G
only generates a dense subset of C.

96
bit string (as in the conventional implementation of Clifford group RB [154, 155]), or
it can be chosen uniformly at random (our recommendation).
(1)2. Sample Fsp uniformly from C (the group generated by G), and then find a circuit Csp
that creates the state|ψ⟩ = U (Fsp )|0⟩⊗n ,(6.10)

from |0⟩⊗n [U (·) maps a gate or circuit to the unitary it implements—see Section 6.3].
That is, find a circuit Csp that satisfies
U (Csp )|0⟩⊗n = U (Fsp )|0⟩⊗n ,(6.11)

meaning that Csp is only required to faithfully implement U (Fsp )’s action on |0⟩⊗n .
The circuit Csp can contain any gates, including gates that are not in G, and it isideally chosen to maximize the fidelity with which ψ is produced. The circuit Csp is
the first part of the sampled direct RB circuit, and we refer to it as the circuit’s state
preparation subcircuit.
(1)3. Independently sample d gates, G1 , G2 , . . . , Gd , from Ω (the user-specified distribution
over G). The circuit Gd · · · G2 G1 is the next part of the sampled direct RB circuit,and we refer to it as the circuit’s “core”.
(1)4. Find a circuit Cmp that, when applied after the two parts of the circuit sampled so far,maps the qubits to |sd,k ⟩, and append it to the circuit sampled so far. That is, Cmp is
a circuit that satisfies
U (Cmp Gd · · · G2 G1 Csp )|0⟩⊗n = |sd,k ⟩.

(6.12)

The circuit Cmp is the final part of the sampled direct RB circuit. As with the circuit Csp , Cmp can contain any gates, and we refer to it as the circuit’s measurement
preparation subcircuit.
The sampling procedure of 1.1-1.4 creates the circuit
Cd,k = Cmp Gd · · · G2 G1 Csp ,(6.13)

97
that, if run on a perfect quantum computer, always outputs the bit string sd,k . That is,|⟨sd,k |U (Cd,k )|0⟩⊗n |2 = 1.
(2) Run the circuits. Run each of the sampled circuits N times, for some user-specified
N ⩾ 1. Estimate each circuit’s success probability (Sd,k ), as the frequency that the circuit
Cd,k returns its success bit string sd,k (we denote this estimate by Ŝd,k ). Note that the direct
RB protocol does not specify the order that the circuits are run, but this is important if
there could be significant drift in the system over the time period of the entire experiment[212–214]. We recommend “rastering” [215] if possible—meaning looping through all of thecircuits N times, running each circuit once in each loop—as this faciliates detecting thepresence of drift and a time-resolved RB analysis [215] (these analyses are not discussed
further herein). If rastering is not possible then the order that circuits are run should be
randomized, as this will typically reduce the impact of drift on the results.
(3) Analyze the data. Fit the estimated average probability of success, Ŝd =k Ŝd,k /Kd ,Pversus benchmark depth (d) to the exponential decay function
Sd = A + Bpd ,(6.14)

where A, B and p are fit parameters (this is the same as the fit function used in Cliffordgroup RB). If the success bit strings were sampled uniformly then fix A = 21n . The estimateof the DRB(G, Ω) error rate of the gates (r̂Ω ) is then defined to ber̂Ω =(4n − 1)(1 − p̂).
4n(6.15)

where p̂ is the fit value of p.
Our definition for the direct RB procedure specifies fitting the data to the standard exponential
decay function of Eq. (6.14), and so DRB(G, Ω) is only well-behaved if the direct RB data hasapproximately this form. However, it is not clear a priori that the direct RB average successprobability will decay exponentially. Moreover, even if the decay is an exponential it is not obvious

98
what the direct RB error rate (rΩ ) measures. Proving that Ŝd ≈ A + Bpd under broad conditions,providing an intuitive explanation for why Ŝd decays exponentially, and then explaining what rΩ
quantifies are three of the main aims of this paper. In particular, we will show that rΩ ≈ ϵΩ .
6.4.2

A simple numerical demonstration of direct RB

We now demonstrate that direct RB works correctly—the decay is an exponential and rΩ ≈ϵΩ —in two simple scenarios. First, consider the case of gate-independent global depolarizing noise,
and assume perfect state preparation and measurement sub-circuits. In this model, after each nqubit gate the n qubits are mapped to the completely mixed state with some probability 1 − λ.
This is arguably the simplest possible error model, and we would expect the error rate measured
by any well-formed RB protocol to be the infidelity of this global depolarizing channel, which isϵ = (4n − 1)(1 − λ)/4n . An explicit calculation confirms that this is the case here:
Sd =

1
1
+ (1 − n )λd ,n2
2(6.16)

which implies that p = λ and rΩ = (4n − 1)(1 − λ)/4n .
Global depolarizing errors are physically unrealistic, so we also consider another error modelthat is more realistic but that is still simple to understand: gate-independent local depolarization. We simulated n-qubit direct RB for a hypothetical n-qubit processor (with n = 2, 4, 6, . . . , 14)
whereby after each n-qubit gate is applied every qubit experiences independent uniform depolarization with an error rate of 0.1%, i.e., the infidelity of each single-qubit depolarizing channel isϵ = 0.001. This error model has the convenient property that we can easily compute ϵΩ . The(entanglement) fidelity of a tensor product of n error channels E1 , . . . , En is the product of E1 to
En ’s fidelities, i.e.,
F (E1 ⊗ · · · ⊗ En , I ⊗ · · · ⊗ I) = F (E1 , I) · · · F (En , I),(6.17)

so ϵΩ is given by ϵΩ = 1 − (1 − 0.001)n ≈ n × 0.001. Note that, because the error rates are gateindependent, ϵΩ is again independent of the sampling distribution Ω in this example. Figure 6.2(upper plot) shows Sd versus d (black circles) as well as the fit exponential (solid lines) and the99
Direct Randomized Benchmarking
1.0

2 qubits, r = 1.9 ± 0.2 × 10 3

Average success probability (Sd )Average success probability (Sd )1.0

Clifford Group Randomized Benchmarking

4 qubits, r = 4.1 ± 0.2 × 10 3
6 qubits, r = 5.9 ± 0.4 × 10 3

0.8

8 qubits, r = 8.1 ± 0.7 × 10 3
10 qubits, r = 1.5 ± 0.1 × 10

2

12 qubits, r = 1.3 ± 0.2 × 10 2

0.6

14 qubits, r = 1.5 ± 0.3 × 10 2
0.4

0.2

0.0

2 qubits, r = 2.2 ± 0.5 × 10 2
4 qubits, r = 1.0 ± 0.1 × 10 1
6 qubits, r = 2.9 ± 0.2 × 10 1

0.8

8 qubits, r = 4.7 ± 0.5 × 10 1
10 qubits, r = 6.7 ± 1.0 × 10 1
0.6

12 qubits, r = 8.9 ± 2.4 × 10 1
14 qubits, r estimation failed

0.4

0.2

0.0
0

50

100

150

200

250

300

350

400

0

1

Benchmark Depth (d)

2

3

4

5

6

7

8

Benchmark Depth (d)Figure 6.2: Simulated n-qubit direct RB and Clifford group RB for n = 2, 4, 6, . . . , 14. Here, direct
RB is benchmarking n-qubit gates that consist of a single layer of parallel one- and two-qubit gates.
Clifford group RB is benchmarking the n-qubit Clifford group generated out of these layers. In thissimulation, an imperfect n-qubit layer is modelled by the perfect unitary followed by independent
uniform depolarization on each qubit at a rate of 0.1%. The points and violin plots are the means
and the distributions of the estimated circuit success probabilities versus benchmark depth (d),respectively, and the lines are fits of the means to Sd = A + Bpd . The estimated RB error rates(r̂), which are reported in the legends, are obtained from the fit decay rates, using Eq. (6.15). We
observe that the direct RB average circuit success probabilities (Sd ) decay exponentially for all n,
and that the direct RB error rate is given by r̂ ≈ 1 − (1 − 0.001)n ≈ n × 0.001. Direct RB istherefore accurately estimating the error rate of the n-qubit gates it is benchmarking. The Cliffordgroup RB error rate grows very quickly with n—as it is benchmarking the n-qubit Clifford group,
whose elements must be compiled into the available layers of parallel one- and two-qubit gates. The
Clifford group RB d = 0 intercept quickly decays to approximately 21n as n increases, at which
point r is close to 100% and it cannot be estimated to reasonable precision with a practical amountof data. This demonstrates that direct RB is feasible on substantially more qubits than standard
Clifford group RB.

estimates for rΩ (in the legend), for each n. We observe that Sd decays exponentially, and thatr̂Ω ≈ ϵΩ . We specify the gate set and sampling distribution used in this simulation in Example 2 ofSection 6.4.4.

6.4.3

Direct RB’s standard sampling parameters

The direct RB protocol has three user-specified parameters in common with all RB protocols,
which we now briefly discuss. These are the user-specified benchmark depths (the values for d), thenumber of repetitions of each circuit (N ), and the number of randomly sampled circuits at eachbenchmark depth (Kd ). These parameters predominantly control purely statistical aspects of theexperiment—specifically, the precision with which the direct RB experiment estimates the under-

100
lying N, Kd → ∞ direct RB error rate. How these parameters control the statistical uncertainty
in an estimated RB error rate, and how to choose these parameters, has been studied in somedetail for Clifford group RB [155, 166, 193, 194, 216, 217]. Much of this work can be applied todirect RB. We will therefore not discuss these parameters in detail herein. Instead, we will provide
some simple numerical evidence that the direct RB error rate can be estimated from reasonableamounts of data. The simulated direct RB experiments of Fig. 6.2 use a practical amount of data:
we used 9 benchmarking depths, Kd = 30, and N = 40, for a total of approximately 104 samples
for the direct RB instance at each value of n. Furthermore, the estimated direct RB error rates
have low uncertainty (the uncertainties reported in the legend are 2σ and they are estimated using
a standard bootstrap). Throughout the remainder of this paper we are predominantly interested
in studying the behaviour of direct RB in the limit of N, Kd → ∞; we denote the average circuit
success probability and the direct RB error rate in this limit by Sd and rΩ , respectively.
6.4.4

Selecting the gate set to benchmark

One of the defining characteristics of direct RB is that the user specifies the n-qubit gate set(G) that is to be benchmarked. This is in contrast with Clifford group RB, where the benchmarked
gate set is necessarily the n-qubit Clifford group—or another group that is a unitary 2-design.
However there is not total freedom in choosing G in direct RB. The direct RB gate set has to satisfy
the following conditions, some of which depend on the performance of the processor that is to bebenchmarked:• The group C ⊆ SU(2n ) generated by G must be a unitary 2-design over SU(2n ). The requirement that C is a unitary 2-design can likely be relaxed (with appropriate adaptions
to the direct RB experiments and/or data analysis) using techniques from RB for generalgroups [156–159, 161–164]. This is an interesting open area of research, as these generalizations have the potential to enable direct RB on any gate set G. Our formal theory fordirect RB (Section 6.7) relies on a slightly stronger condition on G than simply generating aunitary 2-design—it requires that G induces a sequence-asymptotic unitary 2-design.

101
The concept of a sequence-asymptotic unitary 2-design is introduced in Section 6.6, where
we state necessary and sufficient conditions for a gate set to induce a sequence-asymptotic
unitary 2-design. Here we state a simple condition that is sufficient (but not necessary) fora gate set G to induce a sequence-asymptotic unitary 2-design: the gate set G generates aunitary 2-design and contains the identity operation.
• The n qubits to be benchmarked must have sufficiently high fidelity operations for there
to exist a circuit to prepare any random state from {|ψ⟩ = U (C)|0⟩⊗n | C ∈ C} with
non-negligible fidelity. To actually run direct RB, the user also needs an explicit algorithm
for finding these circuits. This is required so that errors in the state and measurement
preparation subcircuits do not mean that Sd ≈ 21n at d = 0 (how far above 21n it is necessary
for S0 to be depends on how much data the user is willing to collect).
• It must be feasible to sample the direct RB circuits using a classical computer, which
requires, e.g., multiplying together arbitrary elements in C, and sampling uniformly from
C. This condition is also required to implement standard RB over C, and it is satisfied if C
is the Clifford group.
None of the three conditions on G, above, require any efficient scaling with the number of
qubits n. Instead, it suffices that each condition is satisfied at the values for n of interest. For
example, perhaps only benchmarking one- or two-qubit gate sets is of interest (note that most RB
experiments are one- or two-qubit Clifford group RB). The conditions also do not require that G
is finite. This is illustrated by our first example, below, of a practically interesting type of gate set
that can be benchmarked with direct RB.
Example gate set 1 [G1 (n)]: An n-qubit gate set constructed from all possible combinations ofparallel applications of cnot gates on connected qubits and the one-qubit gates X( π2 ) and Z(θ) forθ ∈ (−π, π] [where X(θ) and Z(θ) denote rotations by θ around σx and σz , respectively].
This gate set is universal—i.e., it generates SU(2n )—and so G1 (n) does not satisfy all our102
conditions for n ≫ 1 [generating a uniformly (Haar) random pure state is not efficient in n].
However, this gate set does satisfy our conditions for small n. For example, generating any unitary
in SU(4) requires only three cnot gates [218, 219]. The particular n-qubit universal gate set given
above is just an example, and this reasoning holds for (almost) all few-qubit universal gate sets.
Direct RB can therefore be used to benchmark almost any universal gate set over a few qubits b .
This is arguably substantially simpler than the RB method used by Garion et al. [220] to benchmark
a two-qubit gate set containing a controlled Z( π4 ) gate (which is not a Clifford gate). Note that,recently, direct RB of one- and two-qubit universal gate sets has been experimentally demonstrated
in [221].
Example gate set 2 [G2 (n)]: An n-qubit gate set constructed from cnot gates and any set ofgenerators for the single-qubit Clifford group (such as the Hadamard and phase gates).
This is the type of gate set used in the simulations of Fig. 6.2 (and shown in the schematic ofFig. 6.1). In that simulation the gate set consisted of the Hadamard and phase gates as well as cnot
gates between any pair of qubits. That is, this simulation uses all-to-all connectivity. However, notethat direct RB with this gate set can be applied to a processor with any connectivity, as direct RB
does not have to be applied to a processor’s native gate set (applying direct RB to a standardized
set of n-qubit layers could be useful for comparing different processors). In that case, cnot gates
between distant qubits would be synthesized via swap gate chains.
Example gate set 3 [G3 (n)]: An n-qubit gate set constructed from a maximally entangling
two-qubit Clifford gate (e.g., cnot or cphase) and the full single-qubit Clifford group.
Direct RB of a gate set with this form is particularly robust and simple to understand froma theoretical perspective (this is the type of gate set that was used in the direct RB experiments of[167]). As with G2 (n), G3 (n) specifies a family of gate sets. We now specify a particular convenientb

It is only almost any universal gate set rather than every universal gate set as, e.g., it is possible to construct
universal gate sets that require arbitrarily long sequences of gates to implement some one-qubit gates—but suchgate sets are of no practical relevance103
gate set within this family (which is entirely specified given a processor’s connectivity).
Example gate set 4 [G4 (n)]: An n-qubit gate set consisting of all n-qubit gates composed fromapplying a layer L1 and then a layer L2 where these layers have the following forms: L1 is a layer
containing one of the 24 single-qubit Clifford gates on each qubit, and L2 is a layer containing
non-overlapping cnot gates on connected pairs of qubits.
A random gate from G4 (n) locally randomizes the basis of each qubit (and applies somerandom arrangement of two-qubit gates)—if the marginal distribution over L1 layers is uniform
it implements local 2-design randomization (composed with a more complex multi-qubit randomization step). A gate set of this form is assumed in parts of Section 6.5, in order to simplify thetheory presented there. Note that direct RB circuits for G4 (n) have much in common with therandom circuits of Google’s quantum supremacy experiments [32], which are used in cross-entropy
benchmarking [222]. However, these direct RB circuits contain only Clifford gates (making them efficient to simulate), and they contains additional structure (the state preparation and measurement
preparation subcircuits).

6.4.5

Selecting the sampling distribution

There are two main customizable aspects of a direct RB experiment: the gate set (G) andthe sampling distribution (Ω). We now discuss the role of Ω, and how to choose it. The direct RB
sampling distribution Ω can be any probability distribution over G that has support on a subset G′
of G that also satisfies all the requirements for a direct RB gate set (e.g., G′ also generates C). The
sampling distribution Ω and the gate set G control what direct RB measures—direct RB estimatesthe mean infidelity of an n-qubit gate sampled from Ω. Therefore, the primary consideration whenselecting Ω is to choose a distribution that defines an error rate ϵΩ of interest. There are infinite
valid choices for Ω, and so we do not attempt to discuss all interesting choices for Ω here. One
category of sampling distributions we have found useful in experiments is one-parameter familiesof distributions Ωξ in which ξ sets the expected two-qubit gate density of the sampled gate. There104
are many possible such families of distributions—in Appendix 6.10 we briefly describe a family of
probability distributions that has been used in direct RB experiments (the “edge grab” sampler,
introduced in [223]).
The sampling distribution Ω defines the error rate that direct RB measures, but it also impacts
the reliability of direct RB, i.e., whether Sd decays exponentially and how close rΩ is to ϵΩ . So asampling distribution should be chosen for which direct RB will be reliable. For every sampling
distribution satisfying the above requirements, direct RB will be reliable for sufficiently low error
gates. This is because, for any (G, Ω) satisfying the criteria of direct RB, any Pauli error will berandomized by a depth l Ω-random circuit for sufficiently large l (see the theory in Section 6.5).
But this randomization can be very slow, i.e., the required depth l can be very large, and so theerror randomization can be much slower than the rate of errors—which, in Section 6.5, we explain
can result in unrealiable direct RB (e.g., multi-exponential decays). For example, we could haveΩ(G) = 1 − δ for one gate G and some δ ≪ 1. In this case, a length l ≪ 1δ sequence of gates sampledfrom Ω is almost certainly just l repetitions of G—so sequences of this length do not randomize
errors at all. The theory presented in Section 6.5 will help to explain how to choose a sampling
distribution that guarantees reliable direct RB.

6.4.6

The reason for randomized 2-design state preparation and measurement in
direct RB circuits

Direct RB is built on the simple idea of directly benchmarking a gate set G by running varieddepth circuits whose layers are sampled independently from a distribution over G—a class of circuitsthat have been termed Ω-distributed random circuits [221]. However, the depth d direct RB
circuits do not just consist of a depth d Ω-distributed random circuits. They surround the core direct
RB circuit—the Ω-distributed random circuit—with additional structure (see Fig. 6.1). We now
explain the purpose of this additional structure. The direct RB circuits begin with a randomized
state preparation sub-circuit (Csp ) and end with a measurement preparation sub-circuit (Cmp ).
The purpose of Cmp is simply to return the qubits to the computational basis, maximizing error

105
visibility and simplifying the data analysis. The subcircuit has the same purpose as the groupinversion element in Clifford group RB.
The reasons for beginning a direct RB circuit with a randomized state preparation circuit(Csp ) are now explained. First, together Csp and Cmp implement an approximate (state) 2-design
twirl on the error in the core circuit. This is because the state preparation subcircuit generates asample from a (state) 2-design, which (if implemented perfectly) twirls the overall error channel ofthe core circuit, so that it behaves as though it is a global depolarizing channel. This will be shownin each of our complementary theories for direct RB (Section 6.5 and Section 6.7). An alternativeinterpretation of Csp is that it seeds the random walk over the group C induced by a random
sequence of group generators (gates from G) with an initial uniformly random group element (thathas been compiled into the state preparation for efficiency).
There is also a more practical but mundane reason for including the state preparation subcircuit Csp . If Csp is not included in a direct RB circuit, the contribution of errors in Cmp to a direct
RB circuit’s total failure probability can be strongly dependent on the depth of the core circuit (d).
This would pollute rΩ , i.e., rΩ would not accurately quantify the error per layer in the core of thedirect RB circuit (ϵΩ ). To see this, consider direct RB without the randomized state preparation,
and assume an efficient compiler for creating Cmp . Further, assume that G is small (compared toC), so that the distribution over unitaries produced by a short sequence of gates sampled from Ωmust be far from the uniform distribution over C. When d is small, the average depth of Cmp willgrow approximately linearly with d under many circumstances. This is because Cmp is likely tolook very much like the depth-d core circuit that it inverts, with the order reversed and individual
gates inverted. However, as d gets large, the core circuit converges to a random group element. So
the mean depth (and all other properties) of the Cmp circuits asymptote to a fixed value that does
not depend on d. Critically, this means that the depth of Cmp would have a nontrivial dependence
on d. This variation will pollute the decay rate (and so rΩ ), and may even cause non-exponential
decays—its impact is uncontrolled, as the direct RB protocol is agnostic to how Cmp is implemented(just as the Clifford group RB is agnostic about the Clifford compiler). This effect could instead

106
be mitigated by carefully designing the compiler for Cmp , but this is a complicated undertaking.
Instead, the problem is simply solved by the inclusion of Csp . With Csp included, the average
depth of Cmp is independent of d. The only residual d dependence outside of the core circuit is incorrelations between the state that is prepared by Csp and the state that must be mapped to thecomputational basis by Cmp (they are perfectly correlated at d = 0 and uncorrelated as d → ∞).
These correlations can have an effect on rΩ , in principle, but in realistic settings they appear tohave no observable effect on rΩ (see Section 6.5).

6.4.7

The reason for conditional compilation: improved scalability

The state preparation and measurement sub-circuits within a direct RB circuit are very similar
to the first and last gates in a standard Clifford group RB circuit (see Fig. 6.1)—and they playessentially the same roles (see above). However, they differ in a practically important way. Both
Clifford group RB and direct RB begin with a circuit that implements a group element Fsp sampled
uniformly from C, but whereas Clifford group RB demands that this circuit implements U (Fsp ) on
any input state, direct RB only requires that the circuit generates the same state as U (Fsp ) whenapplied to |0⟩⊗n . We call this unconditional and conditional compilation, respectively. The same
distinction holds for the final inversion step, used in direct and Clifford group RB.
We choose to use conditional compilation in direct RB circuits because it substantially increases the number of qubits on which direct RB is feasible—as illustrated by the Clifford and direct
RB simulations shown in Fig. 6.2 (this difference between direct and Clifford group RB is the only
reason for the difference in their S0 values, which is the primary factor setting the number of qubitsthat it is feasible to benchmark). This is because conditional compilation results in circuits that are
shallower and contain fewer two-qubit gates [172]. This is demonstrated in Fig. 6.3, which compares
the mean number of cnot gates in the circuits generated by a conditional and unconditional n-qubit
Clifford compiler. To generate this plot we used open-source compilation algorithms [224] (that we
also used for all the simulations herein). These algorithms are unlikely to generate circuits withminimal two-qubit gate counts (see [172–174] for work on optimal Clifford compilation) although

107
we have found that they perform reasonably well c .
Note that the error rate measured by direct RB is (approximately) independent of the detailsof the compiler used to generate these states. The property of the compilation algorithm that is ofimportance to direct RB is the efficiency with which it generates these states—the higher the fidelityof these states the more qubits direct RB will be feasible on. This is convenient, as algorithms forgenerating many-qubit states/unitaries are typically “black-box” and the properties of the circuits
they generate cannot be easily controlled. In contrast, with Clifford group RB the compiler entirely
defines the physical meaning of the RB error rate—a Clifford group RB error rate cannot be used
to quantify native gate performance without a detailed understanding of the compiled circuits.
c

The algorithm we use is based on Gaussian elimination; the compiled circuits contain O(n2 ) two-qubit gates forboth the state and unitary compilations. This scaling is not optimal: their exist algorithms that generate circuits
containing O(n2 / log(n)) two-qubit gates [172]. But in the n = O(10) regime we have found that our compilations
contain many fewer two-qubit gates than the O(n2 / log(n)) algorithms of [172]. We have not compared our

Number of 2-qubit gatescompilation algorithms to more recent work on Clifford compilation [174].

300
Conditional compilation
Unconditional compilation

200
100
0

2

6

10
Number of qubits

14

18

Figure 6.3: A comparison of the mean number of two-qubit gates in circuits for implementing auniformly random n-qubit Clifford gate either unconditionally (green connected points) or conditionally on the input of |0⟩⊗n (blue line). In the latter case, this is a circuit mapping |0⟩⊗n to a
uniformly random n-qubit stabilizer state. This demonstrates the significant increase in the numberof qubits on which direct RB is feasible—for a fixed two-qubit gate error rate—compared to Cliffordgroup RB. This data was generated using the same open-source compilers that we use for all our
direct RB and Clifford group RB simulations [224], and here we compiled into one-qubit gates andcnot gates between any pair of qubits (i.e., all-to-all connectivity).

108
6.4.8

The reason for randomizing the success outcome

Direct RB specifically allows for uniform randomization of the success bit string (this is also
possible with Clifford group RB [225], but it is not standard practice). This bit string randomization
is not essential, but in our view it is preferable. This is because it guarantees that the d → ∞asymptote of the average RB success probability Sd is 21n (for all Markovian errors), so we can fix
A = 21n in Sd = A + Bpd . As Harper et al. [225] discuss the motivation and statistical impact of
this bit string randomization (in the context of Clifford group RB), we do not do so further here.

6.4.9

The error rate convention: choosing the decay rate scaling factor

For both direct and Clifford group RB, data are analyzed by fitting the average successprobabilities to Sd = A + Bpd and then mapping p to an error rate r (or a fidelity 1 − r). In
Clifford group RB (and other RB protocols), p is conventionally mapped to an error rate defined byr′ = (2n − 1)(1 − p)/2n [154, 155]. This is not the definition that we use. As specified in Eq. (6.15),
we define the direct RB error rate to be r = (4n − 1)(1 − p)/4n . For n ≫ 1 the difference betweenr and r′ is negligible, but it is substantial for n ∼ 1. Our decision to use Eq. (6.15) to define r is
not specific to direct RB. The RB error rate r′ corresponds to the gate set’s mean average gate
infidelity, whereas r corresponds to the mean entanglement infidelity. We choose to use adefinition for the RB error rate that correspond to entanglement infidelity because of entanglement
infidelities convenient properties. For example, for Pauli stochastic errors the entanglement infidelity
corresponds to the probability of any Pauli error occurring (see Section 6.3) and the entanglement
fidelity of multiple gates used in parallel is equal to the product of their entanglement fidelities(assuming no additional errors when gates are parallelized)—see Eq. (6.17). Due to these convenient
properties, this choice has also been made with other scalable benchmarking techniques (e.g., cycle
benchmarking [226]).

109
6.4.10Comparison to other methods for native gate randomized benchmarking

Direct RB is not the first or only proposal for RB directly on a gate set that generates agroup. Below we explain the relationship between RB and each of these protocols:• The method of Knill et al. [153] benchmarks a set of gates that generate the one-qubit
Clifford group, and this has been widely used as an alternative to Clifford group RB (e.g.,
see the references within [227]). That method consists of uniform sampling over a specific
set of one-qubit Clifford group generators. So it is a particular example of direct RB (aspointed out by Boone et al. [227]) except that it does not include the initial stabilizer state
preparation step (which is of little importance in the single-qubit setting). So our theory
for direct RB is further evidence that the protocol of [153] is just as reliable and arguably
as well-motivated as Clifford group RB, although it measures a different error rate.
• The extensions of the method of Knill et al. up to three qubits [197] also fit within theframework of direct RB except that, again, in that method there is no stabilizer state
initialization.
• Independently of the development of direct RB, França and Hashagen proposed “generator
RB” [196], which is also direct RB without the stabilizer state preparation step and withoutuser-configurable sampling (but note that some of the theory within [196] also applies todirect RB).
• Since the development of direct RB, a protocol called mirror RB [190, 221] has been introduced that adapts direct RB to improve its scalability. Mirror RB methods replace thestabilizer state preparation and measurement sub-circuits from direct RB with a mirror
circuit reflection structure, meaning that they contain no large subroutines. Mirror RB
techniques are more scalable than direct RB, and they are designed to measure the sameerror rate (ϵΩ ). However, the error rate measured by existing mirror RB methods is a lessreliable estimate of ϵΩ (it is typically a slight underestimate—see [190, 221] for further de-

110
tails), so mirror RB is not a strict improvement on direct RB. Much of the theory presented
in this paper is also applicable, or adaptable, to mirror RB.

6.5

Understanding direct RB using error scrambling
Elided.

6.6

Sequence-asymptotic unitary designs
Our theory for direct RB with general Markovian errors (Section 6.7) relies on the concept ofsequence-asymptotic unitary 2-designs. In this section we define sequence-asymptotic unitary
2-designs and we prove some results about their properties. Informally, a sequence-asymptotic
unitary 2-design is a gate set G and a sampling distribution Ω over G such that the set of unitaries
created by length l sequences of gates sampled from Ω converges to a unitary 2-design as l → ∞. The
concept of sequence-asymptotic unitary 2-designs is related to approximate unitary 2-designs [191,192], and the broader theory of convergence to unitary t-designs and other scrambling propertiesof random circuits [202–208]. But, because we only use sequence-asymptotic unitary 2-designs as atool for studying direct RB, we do not discuss the relationship with this prior work.
A plausibly sufficient condition for (G, Ω) to constitute a sequence-asymptotic unitary 2design is that G generates a unitary 2-design (assuming that Ω has support on all of G). Thisis because, for a wide class of generating sets and probability distributions, a random sequence ofgroup generators converges to a random group element with increasing length sequences [228–230].
However, this convergence does not happen for all probability distributions over all generating sets:
the long-length distribution can oscillate between the uniform distribution on different parts ofthe group (an example is given towards the end of this section, and this is only possible if G iscontained within a coset of a proper subgroup of C). In this section, we will show the following:
given a set of generators for any group that is a unitary 2-design, length-l sequences of Ω-distributed
and independently sampled generators converge to a unitary 2-design as l → ∞, if and only if theset of length-l sequences also generate unitary 2-designs for all l.

111
6.6.1

NotationFirst we need to define some notation. In this section we consider a quantum system ofdimension d, without the restriction to d = 2n used throughout the rest of this paper. Let L (Hd )denote the space of linear operators on the d-dimensional Hilbert space Hd . Let T (Hd ) denote thespace of superoperators on Hd , which is the set of all linear operators S whereby S : L (Hd ) →L (Hd ). In the following proofs, we will use Hilbert-Schmidt space, and superoperator “stacking”.
Let
B = {B0 , B1 , . . . , Bd2 −1 }(6.18)

be an orthonormal basis for L (Hd ), with respect to the Hilbert-Schmidt inner product⟨X, Y ⟩ = Tr X † Y .

(6.19)√
We can always choose this basis such that B0 = I/ d and Tr(Bj ) = 0 for j > 0, and this will
be assumed herein. For example, with d = 2n the elements of this basis can be the n-qubit Pauli√
operators multiplied by 1/ 2n .
Any ρ ∈ L (Hd ) can be expanded with respect to the basis B as
ρ=X
⟨Bj , ρ⟩Bj .

(6.20)j

As such, ρ may be represented by the column vector| ρ⟩⟩ = (⟨B0 , ρ⟩ , ⟨B1 , ρ⟩ , . . . , ⟨Bd2 −1 , ρ⟩)T .

(6.21)

Similarly, a superoperator X may be represented as the matrix X HS with elements XijHS = ⟨Bi , X Bj ⟩.
In the following we will drop the superscript; we will not use a distinct notation for a superoperator
and the matrix representation of the superoperator acting on Hilbert-Schmidt space.
In this and the next section we will be considering linear maps from superoperators to superoperators, which we will refer to as “superchannels” and denote using the script font (e.g., L). When considering superchannels it is often convenient to represent matrices acting on HilbertSchmidt space (i.e., superoperators) as vectors in a vector space, and superchannels as matrices

112
acting on that vector space. To achieve this we use the invertible linear “stacking” mapvec (|Bj Bk |) = |Bk ⊗ |Bj ,(6.22)

which stacks the columns of a matrix. We denote the inverse “unstacking” operation by
unvec (|Bk ⊗ |Bj ) = |Bj Bk |.

(6.23)

vec(ABC) = C T ⊗ A vec(B)(6.24)

From this definition, it follows thatfor Hilbert-Schmidt matrices A, B and C. Therefore, any superchannel K with the action K(B) =P
i Ai BC i may be rewritten as a matrix
mat[K] =Xi


CiT ⊗ Ai ,(6.25)

that acts on vectors |B) ≡ vec(B), where we will use the |·) notation to succinctly denote stacked
superoperators while distinguishing them from pure states in Hilbert space (denoted |·⟩) and HilbertSchmidt vectors (denoted | ·⟩⟩).
6.6.2

Ordinary unitary designs

Before introducing sequence-asymptotic unitary designs, we first review the definition of an“ordinary” unitary t-design [191, 192]. Although not the original definition [192], an equivalentdefinition for a unitary t-design is [191]:
Definition 1. The gate set G ⊆ U is a unitary t-design if
1 X ⊗t
G =
|G|
G∈G

Z
U ∈U

U ⊗t dµ(6.26)

where dµ is the Haar measure on U.
The class of designs important herein are the unitary 2-designs. The definition of a unitary
t-design implies that an equivalent characterization of a unitary 2-design is the following: a gate set

113
is a unitary 2-design if and only if, when it “twirls” any channel, it projects this channel onto thespace of depolarizing channels. Specifically, define the G-twirl superchannel TG : T (Hd ) → T (Hd )by
TG (E) =1 X
GEG † .
|G|

(6.27)G∈G

The twirl superchannel is a linear operator on superoperators. If G is a unitary 2-design thenTG (E) = Dλ(E) ,λ(E) =

Tr[E] − 1,d2 − 1(6.28)

for any trace-preserving superoperator E ∈ T (Hd ) [156], where Dλ(E) is a “global” depolarizing
channel (i.e, uniform depolarization on the d-dimensional space).
Eq. (6.28) implies that the twirl superchannel for any unitary 2-design is a rank 2 projector,
with support on the depolarizing subspace Sdep spanned by the identity channel and any non-identity
depolarizing channel. Explicitly,
Sdep = span {D0 , I} ,(6.29)

where I denotes the identity superoperator and D0 is the completely depolarizing channel.
6.6.3

Sequence-asymptotic unitary designs

We now introduce the notion of a sequence-asymptotic unitary 2-design. This concept isdistinct from the “asymptotic unitary 2-designs” of Gross et al. [191], which are asymptoticallyunitary 2-designs with respect to scaling of the dimension d. This definition will be in terms of a
set G of unitaries and a probability distribution Ω over G. Note that we do not assume that G isfinite. Instead, we assume the weaker condition that G contains a finite number of “gates” that areeach either fixed (i.e., a constant matrix) or continuously parameterized. For example, a one-qubit
gate set of this sort G is the Hadamard gate and a continuous family of rotations around σz . The
probability distribution Ω should be interpreted as an arbitrary weighted mixture of probability
distributions over these fixed or parametrized gates. That is, sampling from Ω consists of first
selecting one of the gates, and if this selected gate is continuously parameterized we then further
sample a value for the continuous parameter(s) according to some distribution. We also assume that

114
all of the probability density functions over any continuous parameter(s) are continuous function.
P
So when we use the notation G∈G this should be interpreted as a sum over the finite number of(perhaps parameterized) gates and an integral over any continuous parameters in each gate.
Definition 2. Let G ⊆ U be a unitary gate set and Ω : G → [0, 1] be a probability distribution over
G. The tuple (G, Ω) is a sequence-asymptotic unitary t-design if
!m Z
X
⊗t
lim
Ω(G)G
=
U ⊗t dµ.
m→∞(6.30)U ∈U

G∈G

If G is a unitary t-design then (G, Π) is also a sequence-asymptotic unitary t-design, where Πis the uniform distribution (this can be shown by induction).
Definition 3. The (G, Ω)-twirl superchannel is the linear map TG,Ω : T (Hd ) → T (Hd ) with theactionTG,Ω (E) =XΩ(G)GEG † .

(6.31)G∈G

If (G, Ω) is a sequence-asymptotic unitary 2-design, Definition 2 implies that the mth power ofthe (G, Ω) twirl superchannel TG,Ω converges to a rank 2 projector as m → ∞. As such, the absolutevalues of the eigenvalues of TG,Ω outside of this subspace must be strictly bounded above by 1. The
converse is also true: this condition on the eigenvalues of TG,Ω is sufficient for any (G, Ω) to be asequence-asymptotic unitary 2-design. We state this more formally in the following proposition, forwhich we will require the notion of the fixed points of a linear map. The fixed points of a linear
map L : V → V , for some vector space V , are those v ∈ V such that L(v) = v. That is, they are
the eigenvectors with eigenvalue 1. For example, the space of all fixed points of the twirling map
for any unitary 2-design is Sdep [as defined in Eq. (6.29)]. The following are formal requirements onproperties of the map TG,Ω , that will allow us to characterize the sequence asymptotic 2-designs.
Proposition 1. Let G ⊆ U be a unitary gate set and Ω : G → [0, 1] be a probability distribution
over G. The tuple (G, Ω) is a sequence-asymptotic unitary 2-design if and only if the following two
conditions hold:(1) The space of all fixed points of TG,Ω is the subspace of depolarizing channels Sdep .

115(2) If TG,Ω (E) = λE with E ∈/ Sdep then |λ| < 1.
Proof. We first prove that if (G, Ω) is a sequence-asymptotic unitary 2-design then (1) and (2) arem converge
implied. If (G, Ω) is a sequence-asymptotic unitary 2-design then the fixed points of TG,Ω

to Sdep as m → ∞. The superchannel TG,Ω has complex eigenvalues of modulus at most 1. OneP
way to see this is by observing that mat [TG,Ω ] := G∈G Ω(G)G ∗ ⊗ G, which has the same spectrummas TG,Ω , is a convex sum of unitaries, and so it is a CPTP map. Hence, any fixed point of TG,Ω

is an eigenchannel of TG,Ω with an eigenvalue of unit modulus, and so it is also a fixed point ofTG,Ω if this eigenvalue is 1. Moreover, all of the unit modulus eigenvalues of TG,Ω must be equal tom would not have a convergent spectrum as m → ∞,1, because if this was not the case then TG,Ω

which would imply that (G, Ω) was not a sequence-asymptotic unitary 2-design. Hence, the spaceof fixed points of TG,Ω is Sdep , and the eigenvalues for all eigenchannels outside of this subspacehave absolute value strictly less than 1.
It remains to prove that if (1) and (2) hold then (G, Ω) is a sequence-asymptotic unitary 2m converges to a rank-2 projector onto
design. If (1) and (2) hold this immediately implies that TG,Ω

Sdep as m → ∞. Projection onto this subspace is the action of the right hand side of Definition 2
for t = 2 [156], so (1) and (2) imply that Definition 2 holds for t = 2.
We now prove a lemma that will be of use later. Here and later, we let Gk = {gk . . . g1 | gi ∈ G}
be the gate set consisting of the set of all length k sequences of gates from G and Ωk be the
distribution over Gk obtained by k convolutions of Ω, i.e., Ωk (gk . . . g1 ) = Ω(gk ) · · · Ω(g1 ).
Lemma 1. The tuple (G, Ω) is a sequence-asymptotic unitary 2-design if and only if, for all k > 0,
(Gk , Ωk ) is a sequence-asymptotic unitary 2-design.
Proof. Suppose (G, Ω) is a sequence-asymptotic unitary 2-design. The composite of twirling maps
is a twirling map. Thus by taking the composite map, (1) and (2) of Proposition 1 hold. The reverse
direction is trivial.
Proposition 1 allows us to test whether or not a given (G, Ω) tuple is a sequence-asymptotic

116
unitary 2-design, by explicitly calculating the spectrum of TG,Ω . However, for an n-qubit gate set,
this brute force calculation becomes quickly intractable as n increases—using the stacked HilbertSchmidt representation, TG,Ω is a 16n × 16n dimensional matrix. Moreover, this proposition is
not particularly instructive for the task of finding asymptotic unitary 2-designs. In the following
proposition and theorem, we give an alternative characterization of sequence-asymptotic unitary
2-designs.
Proposition 2. Let G ⊆ U be a unitary gate set and Ω : G → [0, 1] be a probability distribution
over G with support on all of G. Then (G, Ω) is a sequence-asymptotic unitary 2-design if andonly if the subspace of T (Hd ) containing all superoperators that are preserved up to a (possiblysuperoperator-dependent) phase under conjugation by all elements of G is the space of depolarizing
channels Sdep .
Proof. Assume that (G, Ω) is a sequence-asymptotic unitary 2-design. Let S denote the set of all
superoperators preserved under conjugation by all elements of G up to a phase, meaning the setof all S ∈ T (Hd ) satisfying GSG −1 = eiθS S for all G ∈ G, where θS is a phase that depends on
the superoperator being conjugated. If the superoperator S is preserved under conjugation up toa phase by all elements of G then it is a unit-modulus eigenvector of the (G, Ω)-twirl superchannel
TG,Ω . Thus, by (2) in Proposition 1, it must have θS = 0 because (G, Ω) is a sequence-asymptotic
unitary 2-design by assumption. Hence, S is a subspace of the set of all fixed points of TG,Ω . If
GΩ is a sequence-asymptotic unitary 2-design then Proposition 1 states that the space of all fixed
points of TG,Ω is Sdep . Therefore, as all depolarizing maps are preserved under conjugation by any
unitary gate, if (G, Ω) is a sequence-asymptotic unitary 2-design then S = Sdep .
It remains to show that S = Sdep implies that (G, Ω) is a sequence-asymptotic unitary 2design. Assume that S = Sdep , let R be any element of G, and consider the gate set R−1 G = −1
R G | G ∈ G . Now consider any superoperator V satisfying GVG −1 = eiθV V for all G ∈ R−1 G
and some θV . The identity superoperator I is in R−1 G, and hence eiθV = 1 for any such V. As
such, the only superoperators preserved up to phase under conjugation by all elements of R−1 G are117
preserved with a phase of 1. Thus, because S = Sdep by assumption, the space of all superoperatorspreserved under conjugation by all elements of R−1 G contains Sdep since we see, for S ∈ Sdep ,(R−1 G)S(R−1 G)−1 = e−iθS RSR−1 = e−iθS S. The space of superoperators preserved up to a given
phase under conjugation by all elements of GR−1 is related via unitary conjugation to the space ofsuperoperators preserved up to the same phase under conjugation by all elements of G. Hence, if
V is preserved up to phase under conjugation by all elements of G, it is preserved with a phase of
1 and V ∈ Sdep .
Now, consider the (G, Ω)-twirl superchannel TG,Ω , and consider any V satisfying TG,Ω (V) =eiθ V for some eiθ . For clarity, let us express this as a matrix acting on a vector, by using the “stack”
operation. We then haveX
G∈GΩ(G)[G ∗ ⊗ G] vec(V) = eiθ vec(V)(6.32)

where the matrix acting on vec(V) is a convex sum of unitaries, or a weighted sum and integral
over unitaries with the total “weighting” integrating to 1. Because unitaries preserve the length ofvectors, this can only hold if [G ⊗ G] vec(V) = eiθ vec(V) for all G ∈ G for which Ω(G) > 0, and thisis all G ∈ G by the conditions of the proposition. Hence, returning to the unstacked representation,
we have that
GVG −1 = eiθ V(6.33)

for all G ∈ G. But we have shown above that, if S = Sdep , this holds if and only if V ∈ Sdep andeiθ = 1. As such, any eigen-operator of TG,Ω outside of the depolarizing subspace has an eigenvalue
with absolute value strictly less than 1 in modulus. Proposition 1 then implies that (G, Ω) is asequence-asymptotic unitary 2-design.
Proposition 2 implies the following theorem:
Theorem 1. Let G ⊆ U be a unitary gate set that either generates a group or generates a dense
subset of a group, C, and let Ω : G → [0, 1] be a probability distribution over G with support on all of
G. Let Gk = {g1 . . . gk | gi ∈ G}, which generates a group, or a dense subset of a group, Ck . Then
(G, Ω) is a sequence-asymptotic unitary 2-design if and only if, for all k, Ck is a unitary 2-design.

118
Proof. First we show that if each Ck is a unitary 2-design, then G is a sequence asymptotic 2design. If G generates a unitary 2-design, then the depolarizing subspace is a subspace of the spacepreserved up to a phase by G. Suppose there is some element in the preserved space that is not
depolarizing. There is a k-string (a length k sequence of G ∈ G) that brings this phase arbitrarilyclose to one. Then, in the group generated by these k-strings, this element is also preserved by Ck
with a phase arbitrarily close to 1. However, we also have that Ck is a unitary 2-design and thus
element must also belong to the depolarizing subspace, and thus we have a contradiction.
Next, we prove that if there is some Ck that is not a unitary 2-design, then G is not asequence asymptotic 2-design. Let SGk and SCk denote the space of all superoperators preserved
under conjugation up to a phase by all elements of Gk and Ck , respectively. Because Gk generates
Ck , or generates a dense subset of Ck , then SGk = SCk . If Ck is not a unitary 2-design then since
SCk = SGk , by Proposition 2 the elements of SGk preserve some superoperator up to a phase thatis not a depolarizing channel and thus SGk ̸= Sdep . But if SGk ̸= Sdep , then SCk ̸= Sdep . Therefore,(Gk , Ωk ) is not a sequence-asymptotic unitary 2-design, and hence by Lemma 1, (G, Ω) is not asequence-asymptotic unitary 2-design.
Theorem 1 characterizes the class of all gate sets, and probability distributions over thosegate sets, that form sequence-asymptotic unitary 2-designs. Perhaps surprisingly, a necessary andsufficient condition is that the generated group by Gk is a unitary 2-design for every k. As an√√
example of a consequence of this theorem, consider the gate set G = { ZH, Z ZH} for which
both elements have a period of 3. G generates a subgroup of the Clifford group that is a unitary
2-design. However, G3 = {I, X, Y, Z} is the Pauli group, which does not form a 2-design. Hence, G
does not induce a sequence asymptotic 2-design.
In any instance where a gate set G does generate a unitary 2-design but does not induce
a sequence-asymptotic unitary 2-design (for any Ω), G can be altered so that it can induce asequence-asymptotic unitary 2-design by adding a single element to G. In particular, any gate set
G that generates a unitary 2-design and contains the identity gate is a sequence-asymptotic unitary119
2-design. This is a practically reasonable and easy-to-verify condition on a gate set.

6.7

A theory of Direct RB with gate-dependent errors
In this section we present two closely-related theories for direct RB under general Markovian

errors that build on (and are of similar rigor to) the most accurate theories of Clifford group RB
under general gate-dependent errors [168, 169, 209, 210]. These theories show that the direct RB
decay is an exponential (Sd ≈ A+Bpd ) for sufficiently small gate errors, and that the direct RB error
rate (rΩ ) is equal to a gauge-invariant definition for the Ω-weighted average infidelity (ϵΩ ) of thebenchmarked gate set. This section begins with some preliminaries: in Section 6.7.1 we introduce
some assumptions and notation that are used in this section but not elsewhere in the paper; andin Section 6.7.2 we review the concept of gauge, and gauge-invariant metrics. In Section 6.7.3 weintroduce the first of the two closely-related theories for direct RB that we present in this section.
This theory—which we refer to as the R-matrix theory for direct RB—combines the regular
representation of the group C with the superoperators for the imperfect gates to construct a matrix
R that can be used to exactly compute the direct RB decay (Sd ).
In Sections 6.7.4-6.7.7 we then present our second theory for direct RB with general gatedependent errors—which we refer to as the L-matrix theory for direct RB—that enables us toshow that rΩ ≈ ϵΩ . This theory combines the superoperator representation of the group C withthe superoperators for the imperfect gates to construct a matrix L that is an imperfect versionof the twirling superchannel that appears in our theory of sequence-asymptotic unitary 2-designs(Section 6.6). In Section 6.7.4 we show how Sd can be approximated by a function of the dth powerof this L matrix. In Section 6.7.5 we then show how Sd can be expanded into a linear combinationof many exponentials, using the spectral decomposition of L. In Section 6.7.6 we use the propertiesof sequence-asymptotic unitary 2-designs (Section 6.6) to show that all but two terms in this linear
combination are negligable, i.e., Sd ≈ A + Bγ d where γ is an eigenvalue of L that is close to 1.
This implies that direct RB approximately measures rγ = (4n − 1)(1 − γ)/4n , i.e., rΩ ≈ rγ . In
Section 6.7.7 we conclude our theory, by building on a proof of Wallman’s [169] to show that rγ is120equal to a gauge-invariant definition for the Ω-weighted average infidelity (ϵΩ ) of the benchmarked
gate set. Finally, in Section 6.7.8 we use the exact R-matrix theory for direct RB, and numericalsimulations, to validate our approximate L-matrix theory for direct RB—by demonstrating that|(rΩ − rγ )/rγ | is small for a selection of randomly generated error models on a single-qubit gate
set. The theories presented in this section have a close relationship to the Fourier transforms overa group, as is the case for Clifford group RB [209].

6.7.1

Assumptions and notation

Here we state assumptions and notation that are used throughout this section but that do
not necessarily apply elsewhere in the paper. We will assume that the state preparation and measurement components of direct RB circuits contain only gates from the benchmarked gate set (G).
This is without loss of generality, as the sampling distribution Ω need not have support on all of G.
We also assume (1) unconditional compilation of the initial group element and final inversion groupelement, and (2) no randomization of the output bit string. These two assumptions mean thatthe state preparation circuit implements a random element of C, and the measurement preparation subcircuit implements the unique element in C that inverts the preceding circuit (as in Cliffordgroup RB). These assumptions are not necessary for much of theory in this section, but they greatly
simplify notation.
Unlike elsewhere in this paper, we will use G ∈ G and C ∈ C to represent group elements[elsewhere we use, e.g., U (G)] rather than instructions to implement logic operations. Throughout
this paper, we denote the superoperator representing the imperfect [perfect] implementation ofG ∈ G by G̃(G) [G(G)]. Similarly, in this section we will denote the superoperator representing
the imperfect [perfect] implementation of C ∈ C (which are used in the state preparation and˜measurement stages of direct RB circuits) by C(G)[C(G)]. Note that C(·) is a representation of thegroup C. We will assume a low-error gate set, so that each G̃(G) is a small deviation from G(G)˜(i.e., G̃(G) ≈ G(G)) for all G ∈ G and similarly C(C)
≈ C(C) for all C ∈ C. We will quantify thisassumption when used.

121
6.7.2

Background: gauge-invariant metrics

In this section we will show that the direct RB error rate (rΩ ) is related to a gauge-invariant
version of the average infidelity of an n-qubit gate sampled from Ω (ϵΩ ). We will therefore firstreview the concept of gauge in gate sets, as well as gauge-variant and gauge-invariant propertiesof a gate set. The sets of all as-implemented and ideal operations in direct RB circuits can be
represented by
Gall = {⟨⟨x| }x∈Bn ∪ {G(G)}G∈G ∪ {C(C)}C∈C ∪ {| 0n ⟩⟩} ,(6.34)˜
G̃all = {⟨⟨Ex | }x∈Bn ∪ {G̃(G)}G∈G ∪ {C(C)}C∈C ∪ {| ρ⟩⟩} ,(6.35)

respectively, where Bn is the set of all length n bit strings, Ex is a measurement effect representing
the x measurement outcome (so ⟨⟨Ex | ≈ ⟨⟨x| ), and ρ represents imperfect state initialization (so| ρ⟩⟩ ≈ | 0n ⟩⟩). (Furthermore, due to our assumption in this section that the target bit string isalways the all-zeros bit string, we only need the x = 0n effect).
All ideal and actual outcomes of circuits in direct RB can be computed from Gall and G̃all , respectively. However, the predictions of these models are unchanged under a gauge transformation—
all operation sets of the formonG̃all (M) = ⟨⟨Ex | M−1 x∈Bn ∪ MG̃i M−1 ∪ {M| ρ⟩⟩} ,(6.36)

make identical predictions for the outcome distributions of all circuits, where M is any invertible
linear map, called a gauge transformation [148, 149, 231, 232]. If f is a property of G̃all , it is only
experimentally observable iff G̃all = f G̃all (M) ,(6.37)

for all M, i.e., f must be gauge-invariant. Many metrics of error for a gate or set of gates thatare defined as functions of process matrices are not gauge-invariant [148, 149, 168, 231, 232]. This
includes the [in]fidelity of a superoperator, so ϵΩ is not gauge-invariant. This implies that an RB
error rate (which is observable) cannot be equal to the standard, gauge-variant definition of the122average infidelity of a gate set [168], i.e., to connect RB error rates to infidelities we need a gaugeinvariant definition for infidelity [168, 169, 210].

6.7.3

An exact theory of direct randomized benchmarking

Here we present a theory for direct RB that is exact, generalizing a theory for Clifford groupRB presented in [168]. This theory, which we call the R-matrix theory for direct RB, provides anexact formula for the average success probability (Sd ), as a function of an imperfect gate set G̃
and a sampling distribution Ω. This formula is efficient in d (but not n) to compute. We will findthis theory useful for verifying the accuracy of the approximate theory (that shows that rΩ ≈ ϵΩ )introduced later in this section. Our exact theory requires one additional assumptions: we requirethat G generates a finite group C.
Our theory starts from the formula for Sd that follows directly from the definition of Sd andthe direct RB circuits. Specifically,
Sd = ⟨⟨E0 | S˜d | ρ⟩⟩,(6.38)

where
S̃d =

X
Gd ∈G

···

X X

[Ω(Gd ) · · · Ω(G1 )Π(C) ×

G1 ∈G C∈C˜
˜ −1 G−1 · · · G−1 )G̃(Gd ) · · · G̃(G1 )C(C)],(6.39)C(C1dwhere (as above) Π is the uniform distribution. To expresses S̃d (and so Sd ) in an efficient-in-dform, we use the regular representation [R(·)] of the group C. To specify these matrices explicitly,
we index the elements of C by i = 1, 2, . . . , |C|, with i = 1 corresponding to the identity groupelement. Then R(G) is the C × C matrix with elements given by R(G)jk = 1 if GCk = Cj andR(G)jk = 0 otherwise. So, R(G) is a permutation matrix in which the ith row encodes the productof G and the ith group element. We now use the regular representation to define a matrix [RG,G̃,Ω ]that can be used to model sequences of gates from G̃ that are independent samples from Ω. We

123
define RG,G̃,Ω by
RG,G̃,Ω =X
G∈GΩ(G)R(G) ⊗ G̃(G).

(6.40)

We now express S̃d in terms of R matrices, and to do so we will need the vectorv = (1, 0, 0, . . . )T ⊗ I,

(6.41)

where I is the 4n × 4n dimensional identity superoperator (and the first entry of v corresponds tothe identity group element). We can express S̃d in terms of R matrices and v as follows:dS̃d = v T RC,C̃,Π RG,
Rv,
G̃,Ω C,C̃,Π

(6.42)

noting that RC,C̃,Π represents the application of a uniformly random element from C̃. To understand
why this equation holds, considerdw = RC,C̃,Π RG,
Rv.
G̃,Ω C,C̃,Π

(6.43)

This is a vector of superoperators where the ith element of w consists of an average over all sequencesof superoperators consisting of (1) a uniformly random element of C̃, (2) d elements of G̃ sampledfrom Ω, and (3) a uniformly random element of C̃, conditioned on the overall ideal (error-free)
action of the sequence being equal to the group element C(Ci ). To average over all direct RB
sequences of length d we simply need to condition on the overall action of the sequence being theidentity—which is v T w.
To obtain our final result—a formula for Sd —we substitute Eq. (6.42) into Eq. (6.38) to obtaind
Sd = ⟨⟨E0 | v T RC,C̃,Π RG,
Rv | ρ⟩⟩.
G̃,Ω C,C̃,Π

(6.44)

This formula for Sd is efficient in d to calculate (simply via multiplication of R matrices, or via thespectral decomposition of the two R matrices). The R matrices grow quickly with both the size ofthe generated group (i.e., with |C|) and n. However, the R matrices are sufficiently small when C is
the single-qubit Clifford group (they are 96 × 96 matrices) to use this theory to numerical evaluate
Sd (without resorting to sampling) for any single-qubit gate set G that generates the Clifford group.

124
As such, this theory is useful for numerically studying direct RB in the single-qubit setting—and
this will allow us to validate our theory that shows that rΩ ≈ ϵΩ (see Section 6.7.8).
Equation (6.44) can be rewritten in terms of the dth power of the eigenvalues of RG,G̃,Ω . However, this does not immediately imply that Sd ≈ A + Bpd (each R matrix has 4n × |C| eigenvalues).
6.7.4

Modelling the direct RB decay with twirling superchannels

In this and the subsequent subsections, we present our L-matrix theory for direct RB. In
this theory, we will show how the direct RB decay, and error rate, can be approximately expressedin terms of the following matrix
LG,G̃,Ω =X
G∈GΩ(G)G(G) ⊗ G̃(G).

(6.45)

Like the R matrix [Eq. (6.40)], this L matrix consists of summing over a tensor product of (1) arepresentation of the group C generated by C and (2) the as-implemented superoperators. In thecase of the R matrix, this representation of the group was the regular representation [R(·)], and in
the case of the L matrix it is the standard superoperator representation [G(·)] of the group.
To derive our theory, we first show how the direct RB success probability (Sd ) can be approximately expressed as a function of twirling superchannels Q whereby mat[Q] = L. In subsequent
subsections, this will enable us to show that the direct RB decay is approximately exponential, andthat rΩ ≈ ϵΩ . Our theory will use error maps for the as-implemented group elements C̃. Let
˜
Λ(C) = C(C)† C(C),(6.46)˜
for C ∈ C, which is the “pre-gate” error map for the superoperator C(C).
We also use the average
error map (Λ̄), defined byΛ̄ =X

Π(C)Λ(C).

(6.47)C∈C

The first step to deriving our L-matrix theory for direct RB is to approximate the imperfectinitial group element using a gate-independent error channel. To do so, we first note that Eq. (6.39)

125
can be rewritten as
S̃d =

X X
C∈C Gd ∈G···X[Π(C)Ω(Gd ) · · · Ω(G1 ) ×

G1 ∈G

−1
˜ G̃(Gd ) · · · G̃(G1 )C({CG˜C(C)
d · · · G1 } )], (6.48)

i.e., we have rewritten the first group element in terms of the Gi and the inversion group element(which is now denoted by C). Now we have that
−1
˜
C({CG
d · · · G1 } ) =G † (G1 ) · · · G † (Gd )C † (C)Λ({CGd · · · G1 }−1 ), (6.49)
and by substituting this into Eq. (6.48), we obtain
S̃d =

X X
C∈C Gd ∈G···X[Π(C)Ω(Gd ) · · · Ω(G1 ) ×

G1 ∈G˜ G̃(Gd ) · · · G̃(G1 ) ×C(C)G † (G1 ) · · · G † (Gd )C † (C)Λ(({Gd · · · G1 }−1 )]. (6.50)
By replacing the error maps in this equation (i.e., the error maps for each initial group element)with their average, we obtain
S̃d =

X X
C∈C Gd ∈G···X[Π(C)Ω(Gd ) · · · Ω(G1 ) ×

G1 ∈G˜ G̃(Gd ) · · · G̃(G1 ) ×C(C)G † (G1 ) · · · G † (Gd )C † (C)]Λ̄ + ∆d , (6.51)
where ∆d absorbs the approximation error. This approximation breaks the dependencies, enabling
independent averaging, as we can rearrange Eq. (6.51) to
(
S̃d =

X
C∈C

 X
˜
Π(C)C(C)Ω(Gd )G̃(Gd ) · · ·
Gd ∈G

n X
G1 ∈G

)

o
†
†
Ω(G1 )G̃(G1 )G (G1 ) · · · G (Gd ) C (C) Λ̄ + ∆d . (6.52)†

126This can be more concisely expressed using two superchannels:S̃d = QC,C̃,Π ◦ QdG,G̃,Ω [I]Λ̄ + ∆d ,(6.53)

where I is the identity superoperator, and QG,G̃,Ω and QC,C̃,Π are imperfect twirling superchannels,
defined by
X

QG,G̃,Ω [E] =G∈GX

QC,C̃,Π [E] =C∈CΩ(G)G̃(G)EG † (G),(6.54)†˜
Π(C)C(C)EC(C).

(6.55)

By substituting Eq. (6.53) into Eq. (6.38) we obtain
Sd = ⟨⟨E0 | QC,C̃,Π ◦ QdG,G̃,Ω [I] ρ′+ δd ,(6.56)

where | ρ′ ⟩⟩ = Λ̄| ρ⟩⟩ and(6.57)

δd = ⟨⟨E0 | ∆d | ρ⟩⟩.

Equation (6.56) represents the average direct RB success probability in terms of imperfect twirlingsuperchannels, which builds on similar results for Clifford group RB [168, 169, 233].
Later we will apply the theory of sequence-asymptotic unitary 2-designs (Section 6.6) tounderstand Sd , and to do so we will find it useful to replace the Q superchannels in Eq. (6.56) withthe closely related L matrices [see Eq. (7.14)]. Consider
vec(QC,C̃,Π ◦ QdG,G̃,Ω [I]) = mat(QC,C̃,Π )mat(QG,G̃,Ω )d |I),(6.58)

where, as introduced in Section 6.6, vec[·] denotes the “stacking” map, mat[·] turns a superchannel
into a matrix and |·) turns a superoperator into a vector. So, e.g., we find thatmat[QG,G̃,Ω ] =X
G∈GΩ(G)G(G) ⊗ G̃(G) = LG,G̃,Ω ,(6.59)

where LG,G̃,Ω is the matrix introduced in Eq. (7.14) [here we have assumed a Hermitian basis forsuperoperators, so that G(G)’s element are real numbers]. We can therefore rewrite Eq. (6.56) interms of two L matrices:hi
Sd = ⟨⟨E0 | unvec LC,C̃,Π LdG,G̃,Ω |I) ρ′+ δd .

(6.60)

127
Equation (7.12) contains an approximation error term (δd ) and we now bound this term.
Using basic properties of CPTP maps and the diamond norm it may be shown that|δd | ⩽ ∥∆d ∥⋄ ⩽X
C∈C

Π(C) Λ̄ − Λ(C) ⋄ ≡ ∆,(6.61)

where ∥·∥⋄ is the diamond norm d . Error maps are not gauge-invariant, so the size of δd (and the sizeof our upper-bound on |δd |) depends on the representation of G̃all . Even for low-error gates—here
meaning that every Λ(C) is close to the identity in some representation—∆ and δd can always be
made large by choosing a “bad” gauge in which to express G̃all . But our bound holds for any CPTP
representation, and so |δd | ⩽ ∆min with ∆min the minimum of ∆ over all CPTP representationsof G̃all . In the following, we will assume we are using a representation in which δd is small, which
always exists with sufficiently low-error gates. Finally, we note that it is possible to improve ourbound on δd , using an analysis similar to Wallman’s [169] treatment of Clifford group RB, whichbounds a quantity that is closely related to δd from above by a function that decays exponentially
in d.

6.7.5

Modelling the direct RB decay using the spectrum of twirling superchannels

We have shown that the direct RB average success probability (Sd ) can be approximated bya function of the dth power of LG,G̃,Ω [see Eq. (7.12)]. This implies that Sd can be approximatedusing the spectral decomposition of LG,G̃,Ω . We now derive the functional form of this spectral
decomposition, which in the subsequent subsection we will use to show that Sd ≈ A + Bpd . Letγ̃i for i = 1, 2, . . . , 16n denote the eigenvalues of LG,G̃,Ω ordered by descending absolute value, i.e.,|γ̃i | ⩾ |γ̃i+1 | for all i. Then Eq. (7.12) implies that Sd may be written asd
Sd = ω̃1 γ̃1d + ω̃2 γ̃2d + · · · + ω̃16n γ̃16n + δd ,d(6.62)An explicit derivation that immediately implies this relation is given in [168] in the context of an equivalent relationin Clifford group RB (see the supplemental material in [168], and also see [155, 233] for similar work). Note that
this upper-bound is d independent, unlike the similar bound for approximating Clifford group RB decay curves in[155] that grow with d and which is therefore not able to provide useful guarantees on Sd [168, 169].

128
where the ω̃k are d-independent parameters. Note that the eigenvalues of an L matrix are gaugeinvariant—because gauge transformations act on a L matrix by matrix conjugation, with a matrixof the form I ⊗ M—so the γ̃i are gauge-invariant.
We now derive a formula for ω̃k . It is convenient to rewrite Eq. (7.12) as
Sd = ⟨⟨E0 | S̃d′ ρ′+ δd ,(6.63)wherehiS̃d′ = unvec LC,C̃,Π LdG,G̃,Ω |I) .

(6.64)

We now write S̃d′ in terms of the eigenvectors and eigenvalues of the two L matrices, LG,G̃,Ω andLC,C̃,Π . Let η̃i for i = 1, 2, . . . , 16n denote the eigenvalues of LC,C̃,Π ordered by descending absolutevalue, i.e., |η̃i | ⩾ |η̃i+1 | for all i (noting that γ̃i denote the ordered eigenvalues of LGG,Ω
˜ ). Moreover, let LC,C̃,i and RC,C̃,i (LG,G̃,i and RG,G̃,i ) be superoperators that, when stacked, are mutuallyorthonormal left and right eigenvectors of the matrix LC,C̃,Π (the matrix LG,G̃,Ω ) with eigenvalueη̃i (eigenvalue γ̃i ). That isLG,G̃,Ω RG,G̃,i = γ̃i RG,G̃,i ,(6.65)(LG,G̃,i LG,G̃,Ω = (LG,G̃,i γ̃i ,LC,C̃,Π RC,C̃,i = η̃i RC,C̃,i ,(6.66)(LC,C̃,i LC,C̃,Π = (LC,C̃,i η̃i ,(6.67)(6.68)with
LG,G̃,i | RG,G̃,j = δij ,LC,C̃,i | RC,C̃,j = δij .

(6.69)

Writing the two L matrices in terms of their eigenvalues and their left and right eigenvectors, andsubstituting this decomposition of the L matrices into Eq. (6.64), we obtain
S̃d′ =Xjkhiη̃j γ̃kd unvec |RC,C̃,j )(LC̃,C,j |RG,G̃,k )(LG,G̃,k ||I) .

(6.70)

By substituting Eq. (6.70) into Eq. (6.63) and comparing the resultant equation to Eq. (6.62), it
follows thatω̃k =′E0′ Sd,k| ρ⟩⟩(6.71)

129
where
′
Sd,k
=

X
j

i
h
η̃j unvec |RC,C̃,j )(LC,C̃,j |RG,G̃,k )(LG,G̃,k ||I) .

(6.72)

This formula shows that we can quantify ω̃k by studying the overlap in the eigenchannels of the two
L matrices, as well as the spectrum of LC,C̃,Π .
6.7.6

The direct RB decay is approximately exponential

We have now shown that the average success probability (Sd ) in direct RB can be written asa linear combination of the dth powers of the 16n eigenvalues of the L matrix (γ̃k ), and we havederived formula for the coefficients in this sum (ω̃k ). We now use the theory of sequence-asymptotic
unitary 2-designs (Section 6.6), as well as the theory of ordinary unitary 2-designs, to show thatSd ≈ A + Bγ d where γ ≡ γ̃2 is the second largest eigenvalue of LG,G̃,Ω . Our theory relies on two
properties of G, Ω, and G̃: (1) (G, Ω) is a sequence-asymptotic unitary 2-design, and (2) the gatesare low error, i.e., G̃ is close to G and C̃ is close to C.
This second condition, above, implies that both the L matrices that appear in Eq. (6.64)[LG,G̃,Ω and LC,C̃,Π ] will be small perturbations on the L matrices for perfect, error-free gate sets
[i.e., LG,G,Ω and LC,C,Π ]. The group generated by G, i.e, C, is a unitary 2-design. Therefore
LC,C,Π =X
C∈C

Π(C)C(C) ⊗ C(C),(6.73)

is a projection onto the space spanned by a completely depolarizing channel [D0 ] and the identity
superoperator [I] (see Section 6.6). Letting ηi denote the ordered eigenvalues of LC,C,Π , we have that(1) η1 = η2 = 1, with the two-dimensional eigen-space for this eigenvalue spanned by D0 and I, and(2) ηi = 0 for i ⩾ 3. Similarly, Propopsition 1 implies that, because (G, Ω) is a sequence-asymptotic
unitary 2-design,
LG,G,Ω =X
G∈GΩ(G)G(G) ⊗ G(G),(6.74)

has two unit eigenvalues, which also correspond to the two-dimensional depolarizing subspace, andall other eigenvalues have absolute value strictly less than one. So, letting γi denote the orderedeigenvalues of LG,G,Ω , we have that (1) γi γ1 = γ2 = 1, with the two-dimensional eigen-space for130
this eigenvalue spanned by D0 and I, and (2) |γi | ⩽ 1 for all i ⩾ 3. Using Eqs. (6.71)-(6.72), thisthen implies that if G̃ = G and C̃ = C, i.e., the gates are perfect, then ω̃k = 0 for k ⩾ 3.
Consider now the case of imperfect but low-error gates, i.e., G̃ ≈ G and C̃ ≈ C. As long as
the errors are small, the eigenvalues of LG,G,Ω will be close to those of LG,G̃,Ω . This is demonstrated
with an example in Fig. 6.4. Furthermore, as long as the errors are small enough so that the gap
between the 2nd largest eigenvalue and the smaller eigenvalues of LG,G,Ω does not close, the spanof the eigenvectors of γ̃1 and γ̃2 is a small perturbation on the depolarizing subspace. Similarly, as
long as the errors are small enough so that the gap between the 2nd largest eigenvalue and the zero
eigenvalues of LC,C,Π does not close, the span of the eigenvectors of η̃1 and η̃2 is a small perturbationon the depolarizing subspace. This then implies that|ω̃k | ≪ 1 ∀k ⩾ 3.

(6.75)

Substituting this into Eq. (6.62), and by noting that 1 is always an eigenvalue of an L matrix for
trace-preserving superoperators (we are assuming CPTP superoperators throughout), we find that
Sd = A + Bγ d + δd + δd′where A = ω1 , B = ω2 , γ = γ̃2 and δd′ =(6.76)

P16nndk⩾3 γ̃k ωk consists of a sum of 16 − 2 small terms.

Therefore, for small n, we have thatSd ≈ A + Bγ d ,(6.77)

i.e., the direct RB average success probability decay is approximately an exponential (plus a constant).
Interestingly, note that the above theory does not show that |δd′ | is small for n ≫ 1. Thisis because we have only shown that δd′ is the sum of 16n − 1 small terms. Furthermore, the above
theory has limited applicability in the n ≫ 1 regime for another reason: for any G consisting ofgates constructed from parallel one- and two-qubit gates, the upper spectral gap in the LG,G,Ω
matrix must decrease in size as n increases. This is because the spectral gap is closely related toconvergence to a unitary 2-design, and the number of layers of one- and two-qubit gates needed to131

Eigenvalues of

1.00
0.75

Imaginary

0.50
0.25
0.00
0.25
0.50
0.75
1.00

1.0

0.5

0.0
Real

0.5

1.0

Figure 6.4: The eigenvalues of the LG̃,G,Ω matrix for the single qubit gate set G = {X( π2 ), Y ( π2 )}without errors [red circles], i.e., G̃ = G, and with a small coherent over-rotation error on each gate
[black crosses], i.e. G̃ = {X( π2 + ϵ), Y ( π2 + ϵ)}, with ϵ = 0.1. There are two unit eigenvalues forthe error-free gate set, and all other eigenvalues have absolutely value strictly less than 1. Herethe upper spectral gap—meaning the difference in the magnitudes of the second and third largest
eigenvalues of L—is 12 for the error-free gate set, and it is still close to 12 for the imperfect gates.

132
implement a good approximation to unitary 2-design increases with n. Yet our less formal theory
for direct RB—presented in Section 6.5—shows that direct RB is reliable in the n ≫ 1 setting, i.e.,
we do have Sd ≈ A + Bpd for some A, B and p, even when n ≫ 1. It is an interesting open question
as to how to obtain this result from the R or L matrix theories presented in this section.
6.7.7

The direct RB measures a version of infidelity

We have now shown that the direct RB success probability (Sd ) is approximately given by
Sd ≈ A + Bγ d where γ is the second largest eigenvalue of the superchannel LG,G̃,Ω . The direct
RB error rate (rΩ ) is defined to be rΩ = (4n − 1)(1 − p)/4n , where p is obtained by fitting Sd to
Sd = A + Bpd . The theory presented so far therefore implies that rΩ ≈ rγ whererγ = (4n − 1)(1 − γ)/4n .

(6.78)

The final part of our theory is to show that rγ is (exactly) equal to a gauge-invariant version of
the mean infidelity of a gate sampled from Ω (ϵΩ ). That is, we now show that direct RB measuresthe Ω-weighted average infidelity of the gate set G̃ it is used to benchmark, in the same sense that
Clifford group RB measures the mean infidelity of an implementation of the Clifford group [168,169].
First we introduce a term for a gate set G̃ that has sufficiently small errors for the upper
spectral gap in the LG,G̃,Ω to not close (which is a condition that we required, above, to show thatSd ≈ A + Bγ d ).
Definition 4. Let (G, Ω) be a sequence-asymptotic unitary 2-design, and let G̃ represent some
implementation of G. This implementation is (G, Ω)-benchmarking compatible if the upper spectral
gap of LG,G̃,Ω does not close.
Next, we present a rather technical proposition, that generalizes a result derived by Wallman[169] (see Theorem 2 therein) for Clifford group RB:
Proposition 3. Let (G, Ω) be a sequence-asymptotic unitary 2-design, and let G̃ represent an implementation of G that is sufficiently low error to be (G, Ω)-benchmarking compatible. Let γ be theestimate from fitting simulated data (r )

133

0.010

= 0.001= 0.002= 0.003= 0.004= 0.005= 0.006= 0.007= 0.008= 0.009= 0.01

0.008
0.006
0.004
0.002
0.000
0.000

0.002
0.004
0.006
0.008
prediction from -matrix theory (r )0.010

Figure 6.5: Verifying the accuracy of our L matrix theory for direct RB, using numerical simulations.
We show the theory’s prediction for the direct RB error rate (rγ ) versus an estimate r̂Ω of the true
direct RB error rate (rΩ ) obtained by simulating direct RB. Each data point shows rγ versus r̂Ωfor a different randomly generated Markovian error models (see main text for error model details).
We find that rγ and r̂Ω are consistent, but note that small systematic deviations between rγ andrΩ would not be visible due to the finite sample error on r̂Ω .

second largest eigenvalue of LG,G̃,Ω by absolute value, and let E1 and Eγ be eigenoperators of LG,G̃,Ω
with eigenvalues 1 and γ, respectively. Then E = E1 + Eγ satisfies
LG,G̃,Ω (E) = Dγ E,(6.79)

where Dγ is the depolarization channel with decay constant γ.
Proof. See Appendix 6.9.
We now use Proposition 3 to prove that rγ = ϵΩ (G̃(E), G). HereϵΩ (G̃(E), G) =X
G∈GΩ(G)ϵ(E G̃(G)E −1 , G(G)),(6.80)

where E is as defined in Proposition 3. Equation (6.80) is the Ω-weighted average infidelity between
the superoperators in G and G̃(E), where G̃(E) expresses the imperfect operations in a particulargauge e :noG̃(E) := E G̃(G)E −1 | G ∈ G .
e(6.81)

Note that the E transformation is also applied to the SPAM whenever any probabilities are to be calculated, but
for simplicity we have dropped the SPAM operations from the notation throughout this subsection.

134
Note that the superoperators in G̃(E) are not necessarily completely positive (this follows because
it is true in the case of Clifford group RB [168, 169], and Clifford group RB is a special case of
direct RB). Furthermore, note that G̃(E) is ill-defined if E is singular, and in any such case we takeG̃(E) to instead be defined using a small perturbation on E. In the following proposition, which is a
generalization of a result for Clifford group RB shown in [168, 169], we prove that rγ = ϵΩ (G̃(E), G).
Proposition 4. Let G, Ω, G̃, γ and E be as defined in Proposition 3. Then rγ = (d2 − 1)(1 − γ)/d2
satisfies(6.82)

rγ = ϵΩ (G̃(E), G).

Proof. From Proposition 3 we have that LG,G̃,Ω (E) = Dγ E, and so LG,G̃,Ω (E)E −1 = Dγ . Taking theentanglement infidelity to the identity superoperator of both sides of this equality, and using thedefinition of LG,G̃,Ω , we have that
!
ϵ

X

Ω(G)G

G∈G

−1

(G)E G̃(G)E−1,I= ϵ(Dγ , I)

(6.83)

Now using the well-known relation [155]ϵ(Dλ , I) =(4n − 1)(1 − λ)4n(6.84)and the definition of rγ , it follows that ϵ(Dγ , I) = rγ . Using this equality, the linearity of ϵ, andthe easily verified relation ϵ(B −1 A, I) = ϵ(A, B) for any superoperators A and B with B invertible,
Eq. (6.83) implies that
X
G∈G



Ω(G)ϵ E G̃(G)E −1 , G(G) = rγ

(6.85)

By noting that G̃(E) = {E G̃(G)E −1 | G ∈ G}, it follows immediately from the definition of ϵΩ ,which is the average infidelity of a gate sampled from Ω, that the left hand side of this equation isthe ϵΩ (G̃(E), G). As such rγ = ϵΩ (G̃(E), G), as required.
We now summarize the results of our theory for direct RB in the following lemma:
Lemma 2. The direct RB error rate (rΩ ) is approximately equal to the Ω-weighted average infidelityof the benchmarked gate set: rΩ ≈ ϵΩ (G̃(E), G).

prediction from -matrix theory (r )

0.010
0.008
0.006
0.004

135
= 0.001= 0.002= 0.003= 0.004= 0.005= 0.006= 0.007= 0.008= 0.009= 0.01(r r )/r
400
200

0.002
0.000
0.000

0

0.002

0.01

0.004
0.006
exact RB rate (r )

0.00

0.008

0.010

Figure 6.6: Verifying the accuracy of our L matrix theory for direct RB, using our exact R matrix
theory. We show the L matrix theory’s prediction for the direct RB error rate (rγ ) versus rΩ
computed from our exact R matrix theory (the R matrix theory enables computing Sd exactly, andwe then fit Sd to Sd = A + Bpd ). Each data point shows rγ versus rΩ for the same randomly
generated Markovian error models as in Figure 6.5. The inset shows a histogram of the relative
error [(rΩ − rγ )/rΩ ], demonstrating that our L matrix theory is extremely accurate. We observe
extremely close agreement between rΩ and rγ , with no more than ±1% relative error (see inset
histogram).

6.7.8

Validating the theory using numerical simulation

In this section we verify the correctness of our L matrix theory for direct RB, using numericalsimulations. We simulated direct RB on a single qubit with a gate set consisting of X( π2 ) and Y ( π2 )gates. We choose an error model consisting of random Markovian errors with rates that contributeno more than ε to the gate’s infidelity, for some small ε > 0, using the error generator formalism in[234]. That is, the superoperator G̃(G) for each gate is modelled as!
12X
G̃(G) = exp
vi (G)Vi G(G)(6.86)

i=1

where the Vi form a basis for the space of Markovian errors. For this example, the 12 error generators
consist of three each of C, A, S and H type generators defined in [234]—where S and H are Paulistochastic and Hamiltonian error generators respectively, while the remaining “Pauli-correlation”
and “active” error generators augment the stochastic errors. For Hamiltonian errors, the leading
order contribution to a gate’s infidelity is at second order, whereas for stochastic errors it is at firstorder. Therefore, the Hamiltonian error rates (the vi ) are chosen uniformly at random from choose√[0, ε], while the rates (the vi ) for all other errors are chosen uniformly from [0, ε]. We do this136
for 785 different error models, varying ε from ε = .001 to ε = .01. The results of the numericalexperiments are shown in Figs. 6.5 and 6.6.
For each error model, Fig. 6.5 shows the prediction for rΩ from the L matrix theory (which isrγ ) versus an estimate r̂Ω for rΩ obtained by simulating direct RB circuits, and fitting the data to
Sd = A + Bpd . Error bars are 1σ, and they are computed using a standard bootstrap. We observe
good agreement between rγ (the theory’s prediction) and rΩ . However, quantifying any small
underlying discrepancies between rΩ and rγ is difficult due to the uncertainties in the r̂Ω —whicharise from simulating a finite set of direct RB circuits. To remove this uncertainty, we computed
the exact Sd values, using our exact R matrix theory for direct RB, and we then fit these exact
values for Sd to Sd = A + Bpd to obtain rΩ . Figure 6.6 shows rΩ versus rγ , and a histogram (inset)of the relative error (rΩ − rγ )/rΩ . We find extremely close agreement between rΩ and rγ , with nomore than ±1% relative error. This therefore demonstrates the accuracy of our L matrix theory fordirect RB.

6.8

Conclusion
Direct RB [167] is a method for benchmarking a set of quantum gates that does not require theinflexible circuit sampling and the complex gate-compilation of standard Clifford group RB [154].
This means that direct RB can benchmark more qubits than Clifford group RB, and direct RB canreliably extract information about the performance of the native gates of a device. In this paper wehave presented a general definition for the direct RB protocol, and theories that explain how andwhy direct RB is broadly reliable. Our theory shows that direct RB is reliable as long as, wheneveran error occurs, it almost certainly spreads on to many qubits before another error occurs. This
proof that direct RB is reliable does not rely on convergence of a sequence of gates to a uniformly
element from a unitary 2-design. This is crucial for proving that direct RB can be reliable on n ≫ 1qubits with realistic error rates, as the number of layers required for such convergence necessarily
increases with n. Furthermore, our theories show that the direct RB error rate can be interpreted
as the average infidelity of the benchmarked gates, building on similar results for Clifford group

137
RB [168, 169].
Clifford group RB has been extended to an entire suite of methods, including methods for
estimating the error rates of individual gates [160, 235, 236], for quantifying the coherent componentof gate errors [237–239], for estimating leakage and loss [233, 240, 241], for calibrating gates [187,188], and for quantify crosstalk [165, 242]. We anticipate that all of these methods can be adapted
to the more flexible direct RB framework, and fully developing these techniques is an interesting
direction for future research. One limitation of direct RB is that it is not indefinitely scalable—
as shown in Fig. 6.3—because initialization and measurement of a random stabilizer state requires
O(n2 / log n) two-qubit gates. Recent work on mirror RB [190, 221, 243] has shown that it is possible
to adapt direct RB to remove these expensive steps, and much of the theory and many of the analysis
techniques used in this paper are also applicable to that more scalable protocol.

6.9

Proof of Proposition 3

Proof. By linearity LG,G̃,Ω (E1 + Eγ ) = E1 + γEγ . If γ = 1 then the proposition holds trivially, so
consider the case of γ < 1. If γ < 1 then, because G̃ is a low-error implementation of G, both 1and γ are non-degenerate eigenvalues of LG,G̃,Ω . Therefore, up to constant scalings, E1 and Eγ are
unique. DefineG̃Ω−avg ≡X
G∈GΩ(G)G̃,GΩ−avg ≡XΩ(G)G,(6.87)

G∈G

where we abuse notation and consider all gates as these as superoperators. Let ⟨⟨V | denote someleft eigenvector of the Hilbert-Schmidt space matrix G̃Ω− avg with eigenvalue 1, i.e., ⟨⟨V | G̃Ω− avg =⟨⟨V | . This eigenvector exists because the elements of G̃ are trace-preserving maps, which implies
that G̃Ω−avg is also a trace-preserving map, and all trace-preserving maps have 1 as an eigenvalue.
Because the the elements of G are the superoperators for unitary gates, G −1 | B0 ⟩⟩ = | B0 ⟩⟩ for all√
G ∈ G, where B0 = Id / d (here we are using the basis for Hilbert-Schmidt space introduced in

138
Section 6.6.1). This is the statement that unitary maps are unital. Hence we have that
LG,G̃,Ω (| B0 ⟩⟩⟨⟨V | ) =X
G∈GΩ(G)[G −1 | B0 ⟩⟩⟨V |G̃]

= | B0 ⟩⟩⟨⟨V | G̃Ω−avg ,(6.88)= | B0 ⟩⟩⟨⟨V |and therefore
E1 = | B0 ⟩⟩⟨⟨V | .

(6.89)

Using the “stacked” representation discussed in Section 6.6.1, we may turn LG,G̃,Ω (Eγ ) = γEγ into a
more standard eigenvector equation.
Let Gu−1 = G −1 − | B0 ⟩⟩⟨⟨B0 | for all G ∈ G, which is known as the unital component of G −1 .
Because the elements in G are unitary gates⟨⟨B0 | Gu−1 = Gu−1 | B0 ⟩⟩ = 0,

(6.90)

for all G ∈ G. Therefore, for any superoperator X ,vec(LG,G̃,Ω (X )) =

X
G∈G

=

X
G∈G

Ω(G)[G̃ T ⊗ G −1 ] vec(X )Ω(G)[G̃ T ⊗ (| B0 ⟩⟩⟨B0 | + Gu−1 )] vec(X )(6.91)= [L0 + L⊥ ] vec(X )T⊗ | B0 ⟩⟩⟨B0 | andwith L0 = G̃Ω−avg

L⊥ =

X
G∈G

Ω(G)(G̃ T ⊗ Gu−1 ).

(6.92)

Because Gu−1 | B0 ⟩⟩⟨B0 | = | B0 ⟩⟩⟨B0 |Gu−1 = 0, we have that the L0 and L⊥ matrices satisfy
L0 L⊥ = L⊥ L0 = 0.

(6.93)

As such, the set of eigenvalues of L is the union of the sets of eigenvalues of L0 and L⊥ . Moreover,
if V is an eigen-operator of L and if the associated eigenvalue is not an eigenvalue of L0 , thenvec(V) =

XX
j

k>0

vjk | Bj ⟩⟩ ⊗ | Bk ⟩⟩,(6.94)

139
for some vjk , and so ⟨⟨B0 | V = 0.
We now show that γ is not an eigenvalue of L0 , which then implies that ⟨⟨B0 | Eγ = 0. The
eigenvalues of L0 are the eigenvalues of G̃Ω−avg , together with 0. GΩ−avg is a sequence-asymptotic
2-design, and thus has 1 as a non-degenerate eigenvalue. We will show this is also true for G̃Ω−avg .
In [169], the author shows that, if the generating set is a 2-design, G̃Ω−avg will generally have aunique largest eigenvalue. We extend this argument to sequence-asymptotic 2-designs. The limiting
map of the generator twirl is a 2-design twirl, and so in the limit the result from [169] holds. Now,
suppose that it did not hold that the generator twirl had a non-degenerate 1-eigenspace. At nopoint in the limiting sequence could it acquire one, thus it has to have one to begin with.
Now, suppose by way of contradiction that γ is an eigenvalue of G̃Ω−avg . Since γ is a perturbation of 1, in the limit that the perturbation goes to 0, G̃Ω−avg would acquire 1 as a degenerate
eigenvalue, thus γ cannot be an eigenvalue of G̃Ω−avg .
Thus, Eγ is not an eigenvector of L0 and hence(6.95)⟨⟨B0 | Eγ = 0.
This equality implies that
Eγ =

XX
i

j>0

αij | Bj ⟩⟩⟨⟨Bi | ,(6.96)

for some αij . A depolarization channel Dλ maps ρ → ρ with probability λ and ρ → I/d withprobability 1 − λ. As such, in Hilbert-Schmidt space with the basis considered here
Dλ = | B0 ⟩⟩⟨⟨B0 | + λXi̸=0| Bi ⟩⟩⟨⟨Bi | .

(6.97)

Hence, from Eqs. (6.96) and (6.89) we have that E1 + γEγ = Dγ (E1 + Eγ ). Finally, using this relationand considering the action of LG,G̃,Ω on E = E1 + Eγ , we have that
LG,G̃,Ω (E) = E1 + γEγ ,= Dγ (E1 + Eγ ),= Dγ E,
which concludes the proof.

(6.98)

140
Note that the E in this proposition is not necessarily a completely positive map, and generically
it is not.

6.10

Sampling distributions
In this appendix we present one possible family of sampling distributions (Ω), called the edge

grab sampler and first introduced in [223]. This sampling distribution is designed for sampling froman n-qubit gate set G containing gates consisting of a single layer of parallel one- and two-qubit
gates, with one-qubit gates from some set G1 and two-qubit gates only between a connected qubits
as specified by an edge list E (that can contain directed edges). The sampling distribution is¯ The two-qubit gate density ofparameterized by the mean two-qubit gate of the sampled gates ξ.
an n-qubit layer is simply ξ = 2αn where α is the number of two-qubit gates in the layer. The edge
grab sampling distribution Ωξ̄ is most easily described by an algorithm for drawing a sample fromΩξ̄ . Drawing a sampling consists of the following procedure(1) Select a candidate set of edges Ec . Initialize Ec to the empty set, and initialize Er tothe set of all edges E. Then, until Er is the empty set:
1.1 Select an edge v uniformly at random from Er .
1.2 Add v to Ec and remove all edges that have a qubit in common with v from Er .
(2) Select a subset of the candidate edges. For each edge in Ec , include it in a final edge¯
set Ef with a probability of nξ/(2|Ec |) where |Ec | is the total number of edges in Ec . If¯
nξ/(2|Ec |) > 1 construct a new edge set sample Ec .
(3) Construct the sampled gate. The sampled gate G ∈ G is then constructed by adding a
two-qubit gate on each edge in Ef and for all remaining qubits independently and uniformly
sample a single-qubit gate from G1 to include in G.
¯
The expected number of two-qubit gates in G is nξ/2,
so this sampler generates an n-qubit¯ One useful property of this sampling is thatlayer with an expected two-qubit gate density of ξ.

141
the probability of sampling any particular n-qubit gate G is non-zero for every G ∈ G (except if¯ξ¯ = 0 or ξ¯ = 1). Finally, note that the edge grab algorithm is invalid if nξ/(2|Ec |) > 1 for any
possible candidate edge set Ec . For an even number of fully-connected qubits, ξ¯ can take any value
between 0 and 1. But for any other connectivity the maximum achievable value of ξ¯ is smaller. In
our experiments, we set ξ¯ = 18 . This is an achievable value of ξ¯ in the edge grab algorithm for all
the processors that we benchmarked.

Chapter 7

The Impact of Markovian Errors on Random Circuits

7.1

Abstract
In addition to the characterization of quantum gate sets, an important task for understanding

the performance of quantum computers is the characterization of quantum circuits. In general, thisis a challenging problem to solve efficiently, given the complexity of arbitrary circuits. In this work,
we make progress towards a solution of this problem by studying the impact of Markovian errors on
random circuits, paying careful attention to Hamiltonian errors. By considering the factorization ofa noisy circuit into a perfect circuit composed with a noise process, we find a perturbative expression
for the survival probabilities of the circuits we study. From this expression we learn the precise way
in which the gate set errors impact the deviations of the circuits from their ideal behavior.

7.2

Introduction
It is relatively unknown how errors, especially coherent errors, affect the success probabilitiesof quantum circuits. Ensuring that circuits have high success probabilities, in turn, is necessary forreliable quantum computation. For these reasons, developing detailed models of how errors affectthe success probability of quantum circuits is one of the most important next steps in quantum
characterization. One simple technique for characterizing quantum circuits is to first assess individual gates in a gate set, and then to use those errors to predict properties of circuits constructedfrom them, such as the success probability. While this idea is simple, even its best answers provide
surprisingly little insight and place a large burden on experimental efforts. Bounds using the dia-

143
mond norm are often overly conservative, and rough heuristics such as multiplying the fidelities canfail to take advantage of circuit structure. As a simple example, coherent errors, if known precisely,
can often be echoed away.
In this paper we further develop this line of research. To do this, we will constrain our modelof errors and circuits. Despite this restriction, we will still be able to observe several interesting
behaviors, and understand general features of the structure and strength of the noise. The first
restriction will be to Markovian errors. In the case of a single gate with an error model thatconsists solely of Markovian errors, there are at least two natural ways to represent errors that areequivalently expressive. These are pre- and post- circuit factorizations [244]. These factorizations
have the advantage of being interpretable, since they model the action of the imperfect operation
as being the perfect operation composed with an error process. For circuits, such a decompositionis also desirable for the same reason. In this work, we will consider the post-circuit representation.
For certain kinds of circuits, finding this decomposition may be easier than for others. Specifically, random circuits have been shown to scramble errors, causing cancellation, and producing error
channels that can be effectively stochastic and even depolarizing. Circuits with repeated elements,
such as those used for amplitude amplification in Grover’s algorithm, may additionally amplify coherent errors resulting in post-circuit error channels that may themselves be coherent with similar
structure to the original gate’s error channel. Other circuits yet, such as error correction circuits,
will in general be more complicated.
To circumvent some of the challenges of analyzing specific circuits, and to take advantage ofmany of the properties of random circuits, many theoretical and experimental works use randomized
compiling techniques [245] to simplify the structure of the noise channels in a circuit. Randomized
compiling refers to the introduction of classical randomness into a quantum circuit in such a way that
non-stochastic errors are converted into nearly stochastic ones. Randomized compiling, however,remains undesirable for two related reasons. First, it fails to take advantage of the coherence ofquantum systems, in the sense that coherent errors can be echoed away if understood precisely.
Second, in ignoring the coherence of the errors and converting them into stochastic errors, the error

144
rate of the circuit is in general increased.
In this paper, we study in detail the propagation of small, Markovian errors through a class ofrandom quantum circuits that are similar to, but more general than, those used in direct randomized
benchmarking (DRB) [246]. After introducing the required technical background, we will prove a
few interesting technical results about the leading order contributions to the effective error rate ofthese circuits. We will conclude by analyzing the family of DRB experiments that draw from thetwo-gate gate set consisting of RX(π/2) and RY (π/2).

7.3

Errors, Gate Sets and Gauge in Quantum Circuits
In this section, we provide the necessary definitions and background for the rest of the paper.

We start by defining a quantum circuit. A quantum circuit is a program for a quantum computerdescribed as a sequence of layers. A layer is a collection of quantum gates on one or two qubits tobe applied simultaneously. A quantum gate is generally described by a unitary operator that acts
on a vector in the Hilbert space spanned by the states of the qubits in the quantum computer, andcomputing the success probabilities is done via Born’s rule, the details of which will be assumed tobe known by the reader.
In general, a quantum computer will have imperfections and to model these imperfections
we describe quantum gates with completely positive trace preserving (CPTP) maps, which arean equivalent description to using Kraus operators for open system dynamics. While the unitarydescription of the ideal quantum gates is a bounded operator acting on the Hilbert space of states of
the quantum computer, the CPTP maps are instead maps on bounded operators themselves. This
space is known as Hilbert-Schmidt space, and vectors in this space are denoted with double angle
brackets | ·⟩⟩.
In addition to imperfect gates, the state preparation and measurement will also be imperfect.
To model imperfect state preparation and measurement (SPAM), we write ⟨⟨Ei | and | ρj ⟩⟩ [246].
The advantages of moving to this superoperator space is that quantum gates act as linear operatorson the states, and the probabilities associated with the different measurement outcomes are given

145
by inner products with the SPAM effects ⟨⟨Ei | . We call the collection of gates and SPAM effects
available to a quantum computer a gate set.
We will use ⟨⟨0| and | 0⟩⟩ to mean the measurement of the zero state and the preparationof the zero state respectively and {Gk } to denote a gate set. A circuit C, consisting of gates aswell as a measurement effect Ei and state preparation ρj , therefore has associated with it a successprobability pij given by
pij = ⟨⟨Ei | C| ρj ⟩⟩ = ⟨⟨Ei | Gkn ...Gk1 | ρj ⟩⟩,(7.1)

where the indices k1 through kn index into the gate set.
When the gates are noisy, as considered in this paper, the probability will instead be givenas˜ ρj ⟩⟩ = ⟨⟨Ei | G̃k ...G̃k | ρj ⟩⟩,p̃ij = ⟨⟨Ei | C|
n
1

(7.2)

where G̃ is meant to represent an imperfect implementation of G.
In particular, in this paper, we will consider the specific noise model of Markovian errors, andso we are able to express G̃ as
G̃ = GeL ,(7.3)

for an error generator L. Characterizing gates in this way, as a perfect gate composed with anerror process is a convenient way of understanding errors in quantum gates. Therefore, in thispaper, we are interested in the related question of how to understand quantum circuits as perfectcircuits, followed by error processes. Unfortunately, the general difficulty of characterizing circuits
is further confounded by a gauge degree of freedom. To introduce what we mean by a gauge degreeof freedom, we first need to give some representation of the gates in our gate set. Because our errors
are Markovian, we can choose our particular representation as the Pauli transfer matrices [245].

146
As discussed in [245], such a process eL can be expressed as


T
1
0








L

,
e =

 m

R





(7.4)

where the top row of all trace-preserving (TP) error maps is fixed to {1, 0, 0, 0, · · · } and the remainderof the first column, m, describes any deviations from unitality.
Error generators, L, as introduced in Eq. (7.3) are discussed in detail in [244] and are thelogarithms of error processes. Thus unitary error processes can be generated by Hamiltonian error
generators, while stochastic error processes can be generated by stochastic error generators. The
classification is in general more complicated, and for this paper it will suffice to give a short introduction of the relevant sectors of error generators. Returning to the taxonomy in [244], the H
sector is most familiar from the Von Neumann equation and is given by Hermitian operators thatact via commutation
HP (ρ) = −i[P, ρ],(7.5)

while the stochastic generators are given by
SP (ρ) = P (ρ)P − ρ,

(7.6)

where P is a Pauli matrix in both equations. The stochastic errors appear as diagonal elements inthe transfer matrix representation.
The final class of errors which we will talk about are active errors. These generate parts of
non-unital errors, such as amplitude damping, and have the specific form
1
aP Q (ρ) = i(P ρQ − QρP + {[P, Q], ρ}),
2

(7.7)

for Pauli matrices P , Q. To gain intuition for these errors, note that we can express a small
amplitude damping channel, Γ1→0 (ρ) as
Γ1→0 (ρ) = esx +sy +ax,y | ρ⟩⟩.

(7.8)

147

Figure 7.1: On the left, the circuit has three layers, with errors (denoted as E1 and E2 ) after each of
the first two layers. It is possible to factor the errors to the right hand side of the circuit (the third′layer), but the errors will in general be transformed. This is denoted on the right with the error E12
after the third layer of the circuit.

It then becomes clear, upon writing out each circuit, that there is a gauge degree of freedom
given by any invertible matrix M , which we call a gauge transformation. The probabilities in
Eq. (7.1) transform as
pij = ⟨⟨Ei | M M −1 Gkn ...Gk2 M M −1 Gk1 M M −1 | ρj ⟩⟩,(7.9)

so that the gates in the gate set transform as G → M −1 GM and the SPAM effects transform as⟨⟨Ei | → ⟨⟨Ei | M and | ρj ⟩⟩ → M −1 | ρj ⟩⟩. Because these gauge transformations do not affect any
observable quantities, like the survival probabilities discussed in this paper, they are themselves unobservable and unphysical. Because of this, directions that are variant under these transformations
are also not physically well-defined directions. We call directions which are first-order gauge invariant directions - directions that remain the same under gauge transformations - FOGI directions[247]. For a local quantity on the tangent space to be physical, it must be supported on the FOGI
subspace.
In particular, consider a gauge transformation M as before. A function is first-order gauge
invariant at 0 ifD f (G̃all ) |g=0 g = D f (G̃all (M)) |g=0 g,(7.10)

where D is the derivative, and g is a vector that parameterizes the errors on the gate set.

7.4

Error Factorization in Quantum Circuits
As we have discussed, a common method for understanding the impact of errors on a gate

is to factor the noisy gate into an ideal gate followed by an error channel. In Sec. 7.3 we have148
given an explicit example of this in Eq. 7.3 for the Markovian errors we consider in this paper. In
this section we will introduce and discuss the generalization of this idea to factoring errors out of
quantum circuits. An example of this is shown in Fig. 7.1, with the errors denoted as Ek , with kdenoting the layer of the error and Ek′ with the prime denoting the error having been factored tothe end of the circuit. For the errors discussed in this paper it is always possible to factor the errors
to the end of the circuit, but they will in general be modified. Formally, for any noisy circuit C˜ as
before, we haveC˜ = G̃kn ...G̃k1 = Gkn En ...Gk1 E1 = CE ′(7.11)

where we have defined E ′ := C † Gkn En ...Gk1 E1 . A strong motivation for such a representation is given
by quantum error correction [248]. While this is not the topic of this paper, we elaborate on theidea to highlight the similarities to our problem. The noisy gate model of quantum error correction
considers the circuit and the error correction circuit as consisting of faulty gates. Thresholds forthis model, while potentially complicated to compute, are the most accurate. The channel coding
model is used largely in quantum information, and considers the computation being error-corrected
to be a channel, and the error correction circuit’s syndrome extraction to introduce no errors. Thismodel, while unphysical, is equivalent to the first model in expressive strength if we can factor theerrors out of the circuit, and has the advantage of being simpler to work with.
While this is a single important example, such an error factorization technique would findutility in nearly every area of quantum information science where noisy quantum circuits are studied,such as in metrology or characterization. This is due in part to the interpretability of the model.
By factoring the error to the end of the circuit, it is physically intuitive to tune parameters of thenoise model and understand their effect as corrupting an otherwise perfect circuit. In the remainderof this section, we discuss how to perform this factorization on a specific family of random circuits.

7.4.1

Factoring Errors out of Random Circuits

Efficiently factoring errors out of circuits can in general, however, be quite challenging. A
cursory analysis would suggest that at least the circuit structure or noise structure must be simple.

149
We have chosen a relatively simple noise model in this paper, thus we are now interested in choosing
a simple circuit model. One way to simplify the circuit structure is to consider ensembles of circuits.
To this end, in the remainder of this paper we will consider collections of random circuits.
There are many choices of random ensembles of circuits in quantum information science, suchas matchgate circuits, XEB circuits, RB circuits and more recently DRB circuits [246]. DRB gives
a particularly powerful family of circuits for random circuits as compared to traditional Clifford
randomized benchmarking, since it can in fact consider (almost) any set of gates that generate
the Clifford group, which in turn is precisely the gates that are used by stabilizer error correction
circuits.
The reason we might expect random circuits to make the factorization of errors easier is inpart because they rapidly scramble errors - it is this rapid error scrambling that washes out coherent
errors and generally affects a twirl superchannel (discussed in [246] and the previous chapter, see
Eq. (3)). Intuitively, while a state twirl scrambles the information in a state, a twirling superchannelrandomizes a channel. By considering a Markovian error model, we will see how the DRB error
rates comes from this error scrambling of the Markovian errors.

7.4.2

DRB as Random Circuits

While the theory developed in [246] was motivated by DRB, it encompasses a larger class ofrandom circuits. The restriction of DRB circuits to Clifford gates is so that a randomly sampled
sequence can be efficiently inverted. However, the analysis itself does not depend on this property,
and can be applied to the fidelity of any random circuit. In particular, we will consider circuits whoselayers consist of gates sampled from a gate set {Gk } given by a sequence asymptotic 2-design [246].
To estimate the fidelity, however, it is necessary to introduce state preparation and measurement
circuits - these circuits in turn will allow us to represent the effective noise in the circuit in a very
simple way. A simple subset of these circuits are those that constitute DRB experiments, howeveralmost any randomly sampled circuit that composes to the identity is in this class. This is because,in particular, almost any gate set is universal [249]. It therefore generates a 2-design and thus is150
also a sequence asymptotic 2-design.
To use the work in [246] we will, however, consider DRB circuits and assume that thesecircuits have perfect stabilizer preparation, however that work additionally shows that the circuits
are robust to imperfections in this SPAM, and in general, without the SPAM preparation, the formof the noise is altered to non-uniform stochastic noise. To see this, consider that in [246] the authors
show that the survival probabilities are given as
Sd = ⟨⟨0| LC,C̃,Π (LdG,G̃,Ω (I))| 0⟩⟩ + δd ,(7.12)

where we have fixed the SPAM effects to preparing and measuring in the zero state, which ispossible for DRB, as we will discuss in the remainder of this paper. In this equation, we have usedthe L−matrix
LG,G̃,Ω =X
G∈GΩ(G)G(G) ⊗ G̃(G),(7.13)˜Ω(C)C(C) ⊗ C(C),(7.14)

for a gate set G and
LC,C̃,Ω =X
C∈C

for the Clifford group C and stabilizer circuits C. In Eq. (7.12) we have interpreted these matrices
as operators on the space of process matrices, otherwise known as “super-duper” operators.
The final superchannel twirl comes from the stabilizer preparation in DRB. If we remove this,as would be the case in truly random circuits, then as mentioned previously, we find that the channel
is in general stochastic. We leave the study of this more general setting to future work. For thiswork, we consider the circuits in [246], which give the explicit form of the post-factored noise as adepolarizing channel Dγ . That is, the circuits can be rewritten approximately as
˜ 0⟩⟩ ≈ ⟨⟨0| Dγ SGin ...Gi | 0⟩⟩ = ⟨⟨0| Dγ C| 0⟩⟩,⟨⟨0| C|1(7.15)

where S is the stabilizer preparation. Thus what remains interesting for this work is determining γ.

151

7.5

The Impact of Markovian Errors on the Depolarization Rate
The random circuits we consider in this section are characterized by the sampling distributionΩ, the gate set {Gk } and the error model, which is described by the 3|G| coefficients of the Hamiltonian contributions to the error rate. The idea that random quantum circuits, especially over gatesthat have some property close to being a two-design, average or twirl errors into a depolarizing
channel is not a new idea. However, the normal intuition that this twirling depolarizes or dephases
P
coherent errors in a simple way can easily be shown to be wrong. For a single error
αi Li , we
expect that we can twirl the Hamiltonian errors separately so that, for instance, the Hamiltonian
errors hx are sent to stochastic sx errors of order h2x , and so that the Hamiltonian error rates add
in quadrature.
Contrast this with the the gate set we will consider in Sec. 7.5.4, consisting of an RX(π/2) andRY (π/2) gate on a single qubit. In these random circuits there is a single free sampling parameter,p, corresponding to the probability of sampling the RX(π/2) gate (with 1 − p being the rate ofsampling RY (π/2)). Intuitively, assuming that coherent errors on the gates add in quadrature, wewould expect a formula for the decay rate rγ , to be
rγ =

X
j

2

2

phjx + (1 − p)hjy ,(7.16)

where hjx is the rate [244] of Hamiltonian j errors on the RX(π/2) gate and hjy is the rate of
Hamiltonian j errors on the RY (π/2) gate.
This is a simple linear function in p, and is demonstrated in Fig. 7.2 to be incorrect. Fig. 7.2
demonstrates the survival probabilities of a DRB experiment on these two gates with a hamiltonian
error model given as hyx − hzx on the RX(π/2) gate and hxy + hzy on the RY (π/2) gate. The figure
shows the survival probabilities as a function of p using two methods. First, the points are fromnumerically simulating DRB experiments, and fitting the decay constant. The solid line is given byusing the L−matrix theory from [246]. This plot thus shows good agreement between the theoryin [246] and fits from numerical experiments even near the extremal points where the theory is notproven to be valid from the results in the previous chapter and [246]. The error we have chosen

152that contrasts with our intuition is a relational error [247] and motivates a further understandingof how precisely Hamiltonian errors affect the DRB decay rate.
Using the theory in [246] we now prove a few technical results about the eigenvalue of theL−matrix in the presence of perturbative Markovian errors. In particular, in what follows we willconsider an expansion of the form
γ≈

X

cj 2j

(7.17)jwhere j ranges over the two gates, and c is the coefficients for the rates j associated with the
Hamiltonian error generators [244]. Because linear terms in these rates are amplitudes, we haveassumed (and it can be shown) that these rates do not enter linearly in the error rate as amplitudes
are not observable. After proving these results, we will look at an example using the ideas we havedeveloped, to further understand the physics of Markovian errors on random quantum circuits.

7.5.1

Active Errors Don’t Contribute to the Error Rate

In this section, we refer to the representation in Sec. 7.3 of Markovian processes with process
matrices. The top row of all trace-preserving (TP) error maps is fixed to {1, 0, 0, 0, · · · } and theremainder of the first column, m, describes any deviations from unitality, which arise from and only
from active errors. Because of the zeroes in the top row, any composition of these maps will only
ever have contributions from the first column of the matrix in the first columns.
There is a gauge in which the errors are completely depolarizing, and thus the survival probabilities are given by a trace, since for depolarizing noise the survival probabilities are given by the
fidelity. Because the trace is basis independent, and active errors don’t contribute to the diagonal
elements in the Pauli transfer basis, we find that in the depolarizing gauge the contribution from theactive errors is identically zero - the expression for the survival probabilities, a gauge invariant quantity, is functionally independent of the active errors. Because gauge transformations act as affine
transformations on the vector of active generators, it remains that in any gauge the contribution
from the active generators will be identically zero.
In other words, the contribution to the survival probabilities from the active generators is given

153

Figure 7.2: A simple gate set with a single relational error [247]. Intrinsic errors on either gateincrease the decay rate in proportion with the fraction of the gate being introduced. Relational
errors, on the other hand, can form an “inverse arch” - they contribute the most when ratio of gatesis uniform. This is evidence that the errors are truly properties of the gate set, rather than of any
individual gate. This figure also demonstrates that the L−matrix theory is descriptive in this case
despite not being “benchmarking-compatible” [246] near p = 0 and p = 1, giving evidence that thetheory in that work is more reliable than proven in the previous chapter, and is applicable to this
work.

154
by a power series. Since the survival probabilities are functionally independent of these generators,
the coefficients in this power series must be zero. A gauge transformation acts only by an affine
shift on the generator values, and thus the power series still evaluates to zero in any gauge.

7.5.2

The Ground Space

We note that the dimension of the ground space of the Hessian is independent of the numberof gates in the gate set. This is because the ground space corresponds to the set of directions in thevector space of error generators that the DRB error rate is insensitive to, which includes the gauge
directions. This is because we expect that non-trivial randomized circuits will have some sensitivity
to all non-gauge directions. By counting, we see that the physical error directions are an intrinsic
error on each gate, and the relative error between each pair of gates. After the first two gates,
this is set by specifying a relative error to two different gates in the gate set, and a single intrinsic
error, and therefore contributes three physical directions per gate. Thus, after the first two gates,
additional gates add on no additional gauge directions. In the case of two gates, there are threegauge directions since there are only three physical errors - two intrinsic, and one relational.
To understand this in more detail, we recall that gauge transformations act as affine transformations on the vector of error generators. This is essentially because the gauge transformations
have essentially linear structure, but do not necessarily map 0 to 0. So if you consider (e.g.) the
6-dimensional h vector, then a gauge transformation t does not act as a linear transformation thatmaps h → G(t)h where G(t) is a matrix. Instead, it maps h → h + g(t), where g(t) is a vector.
The same is true for the s, a and c vectors. The only way we can be sure that such affine shifts byg(t) will leave the DRB decay rate r invariant, given the formula r = hT Mh, is if every possible
g(t) lies in the kernel of M. Or, to put the other way around, if M = 0 on the subspace spannedby all “gauge directions” g(t).

155
7.5.3

Gauge Action Determines Lower Eigenvalues

In the previous argument we assumed that r = hT Gh. In the case of the stochastic sector
we have instead r = sT g + hT Gh. Indeed, if the action of the gauge transformations is to produce
affine shifts of the error generators, and the RB error rate is gauge invariant, then we have thatsT h + hT Gh = sT (h + g) + (h + g)T G(h + g),(7.18)

where s is the first-order coefficients for the error generators, h is the v sector of the generators,
and g is the gauge action.
For sectors that aren’t stochastic s = 0 and so for all pairs g, h, generated by gauge group,
0 = g T Gh + hT Gg + hT Gh(7.19)−sT g = 2gGh + λhT h(7.20)

Now we findfor all pairs g, h, where we have written h in terms of its eigenvalue and used the symmetry of G.
h is a normalized eigenvector, so
λ=

−sT g
2g T h + 1(7.21)

for an eigenvector h that transforms under a gauge transformation by an affine shift g. Thus we seethat two eigenvalues in the gauge directions are determined by the gauge action on the eigenvectors.
In particular, if s = 0 then the gauge directions correspond to zero eigenvalues.

7.5.4
7.5.4.1

Example: RX(π/2), RY(π/2)
The model

In this section we consider a simple gate set on a single qubit, consisting of RX(π/2) andRY (π/2). We will consider the Hamiltonian sector consisting of just gate has 3 error generators,
on each gate, hi , i ∈ {x, y, z}. We sample with the distribution Ω = (p, 1 − p).

156
7.5.4.2

Discussion of Eigenvectors and Eigenvalues

We start with a simple gate set, consisting of a RX(π/2) gate and a RY(π/2) on a single
qubit, with all Markovian error generators, corresponding to the C, A, S and H sectors, for equal
weighting. While we discuss only the Hamiltonian sector in this section for the reasons previously
mentioned in this paper, we numerically verify our full formula from perturbation theory in Fig. 7.3.
We analyze the spectrum of this matrix to see which direction it picks out as special in FOGI
space. We focus on the Hamiltonian sector because(1) Active errors do not contribute.
(2) Stochastic error contribute trivially at first order.
(3) Pauli-correlated errors, which introduce correlations between stochastic errors, are qualitatively similar to stochastic errors.
If we consider the Hamiltonian sector of the Hessian, we find that the three nonzero eigenvectors are supported completely by the Hamiltonian FOGI vectors. The intrinsic FOGI directions aregauge invariant to all orders, however these three directions are the directions in which the DRB
error rate is diagonal - by describing the system with these error rates, it is possible to additively
and independently compute the error rates. In other words, by increasing one of the error rates, it
is possible to increase the DRB error rate independent of the others. Labeling these directions as
F1 , F2 , F3 , at p = 1/2, these combinations are:(1) The sum of the FOGI directions,

P

Fi .

(2) The difference of the intrinsic FOGI directions. F1 − F2(3) 2F3 − (F1 + F2 )We can interpret the eigenvectors at equal weighting via the random walk discussion given
previously. We do this below:

157(1) The easiest to understand is the bottom eigenvalue. The bottom eigenvalue corresponds
to the direction of least variance in the random walk space. This occurs when there iscancellation in the trajectories. So rather than coherent cancellation on a single gate, we
are interesting in seeing the expected coherent cancellation on any trajectory. This must bebalanced against the actual error introduced from the axis error.
(2) The top eigenvalue is the maximal variance is the maximal ravine is formed by always overrotating, and narrowing the ravine so that the errors can flow more rapidly. Exactly in thesame way that the error can cancel if the relative error is made larger, they can amplify
when the error is made larger.
(3) This leaves one more direction, which is the difference of intrinsic errors. This is of course
the unique remaining orthogonal FOGI direction, and hence we are done. However, it alsosuffices to show that this is a local minimum via the following argument. The coefficienton each must be equal by symmetry. The error rate is proportional to the variance of therandom walk carried out by the gates. By increasing the angle, the errors line up andtherefore the variance is decreased. With the angles at ninety degrees, there is opportunity
for coherent amplification and therefore ballistic trajectory. By decreasing the angle, therandom walk becomes closer to a classical random walk and therefore the variance is also
decreased. Because the error rate is proportional to the variance, this shows that this is alocal optima.
While this decomposition is dependent on the weighting, we might expect that these arenatural from symmetry properties at Ω = ( 21 , 12 ), and this give operational meaning to the FOGI
directions. However, any choice of Ω = (p, 1 − p) gives, a priori, a physically meaningful operational
definition. We can parameterize this by the sampling distribution, and get the following Hessian
for the Hamiltonian sector of errors.

158

Figure 7.3: We sample 500 random Markovian error models, with leading order errors of size ϵ, for
the RX(π/2), RY (π/2) gate set, at Ω = (1/2, 1/2), and demonstrate that our formula is correct.
The residuals are quadratic, demonstrating that we have the correct formula to first order. The fit
line is proportional to x3/2 , demonstrating that the error in our estimate is O(ϵ3/2 ).

159



2
(p − 1)p3

− 31 2p2
 1
 (−2)(p − 1)p
 3
H=
 1
 3 (−2)(p − 1)22(p − 1)p3
2(p − 1)p3

1(−2)(p − 1)p3

1(−2)(p − 1)232(p − 1)p32(p3 +p)3(p−1)

2p2
32(p − 1)p3− 13 2p2

2p2
32(p − 1)p32(p − 1)2
3

1(−2)(p − 1)p32(p − 1)p32(p − 1)232(p−1)((p−2)p+2)
3p1(−2)(p − 1)2
3− 13 2p21(−2)(p − 1)p3

1(−2)(p − 1)232(p − 1)p3− 13 2p21(−2)(p − 1)p3

1(−2)(p − 1)232(p − 1)p3− 13 2p22(p − 1)p
3









1

(−2)(p−1)p3
 (7.22)2 1(−2)(p
−
1)
3


2

(p − 1)p

3

2
(p
−
1)p
3
− 13 2p2



We plot the spectral properties of this matrix in Fig. 7.4. One immediate strength of thismodel which is clear to see from these figures is that we can tune the probabilities to near 0 or 1. In
this limit, the model becomes closer to a more tightly constrained family of circuits, whose errors
consist primarily of over-rotations.
As mentioned, non-zero eigenvectors are supported directly on the (physical) FOGI directions.
In the first three plots, we show the decomposition of the three eigenvectors with non-zero eignvalues
into the their FOGI components. This analysis demonstrates exactly what each quadrature in the
DRB experiment measures, as a function of the sampling distribution.
In addition to this, we plot the eigenvalue of each eigenvector, normalized across all valuesof the sampling parameter p in each plot, and give the relative contributions of each eigenvalue inthe final plot. The final plot shows that at the extremal points the lower gap closes at 0. This isof course obvious - the sole contributing component becomes the divergently large coherent overrotations on the single gate being sampled.
The middle eigenvector is also particularly interesting, as it measures purely the relational
error at the extremes of the sampling distribution, which naively there should be no sensitivity to.
Moreover, we see that the contribution is opposite in sign at the two extremes, which we can takeadvantage of to construct a measure of relational errors, which we have shown previously corrupt
our intuitive notion of average gate fidelity.

160

Figure 7.4: Interestingly, in the limit of p = 0 or p = 1, each eigenvector has support on a single
FOGI direction. The largest eigenvalue diverges at the extremal sampling distributions, since inthese limits coherent over-rotations on a single gate can coherently amplify to very large errors.
The smallest eigenvalue, conversely, is seen to have support only on the other gate in the extremal
limits, since these errors appear vanishingly often. Finally, the intermediate eigenvalue is completely
supported by the relational error at the extremal points.

161

7.6

Conclusion
In this paper, we have studied random quantum circuits that asymptotically form two-designs.

By assuming a perturbative Markovian error model, we were able express the impact of the noiseon the circuit as a single post-circuit channel. This allowed us to produce intuitive error models
for the impact of physical errors on the performance of the circuit. The characterization of circuits
is an important next step for characterizing quantum computers. Developing techniques, such as
those discussed in this paper, that take advantage of existing gate-level characterizations as well as
the structure of the circuits being studied, are particularly useful. For instance, while conventional
wisdom suggests that Hamiltonian errors should coherently accumulate linearly in the depth of acircuit, we’ve shown that can be false for simple relational errors [247].
Furthermore, even in the simple example considered in this paper, we have been able to argue
that the entire class of active errors do not change the error rate, and have been able to further
understand how the gauge freedom on a gate set can affect the contributions of physical errors to the
decay rate. Thus the results and example we have worked through in this paper, for this particularfamily of circuits and error models, make substantial progress towards the useful characterization ofquantum circuits. In particular, our results also make progress towards the characterization of errors
on error correction circuits, since these consist of the gates used in conventional DRB experiments.
However, this analysis leaves open several interesting further steps. By including a more
complex error model, we may be able to understand the connection between experimental systemparameters, such as the anharmonicity in superconducting transmons, or laser noise in atomic
systems, and circuit survival probabilities. Removing the randomness or idealized SPAM effects
would give a more faithful model of error correction circuits, which would then let us understand thefunctional dependence of the performance of error correction on simple, experimentally controllable
parameters. For general algorithmic circuits, we would also like to develop a technique for factoring
the errors out of specific circuit structures or families. This is particularly useful for both designing
diagnostic circuits for characterizing quantum computing systems and for predicting how gate set

162
errors will impact specific use cases of quantum computers.

Chapter 8

Summary and conclusions

In this thesis we have explored a wide range of topics, primarily in the subfields of quantum
information that are quantum computation, quantum metrology and quantum characterization. In
the first two sections, our work on quantum computation in both Penning traps and with neutral
atoms demonstrates just some of the challenges associated with trying to perform quantum computations on some technologies that are purportedly scalable. Furthermore, given the fidelity andperformance requirements in these works, we see that near term devices are still in their infancy.
Despite this, in chapter three we propose a quantum error correction experiment that, withtoday’s devices, should be able to demonstrate bona fide fault tolerance - a milestone for quantum
information technologies. Chapter four and five give a brief venture into quantum metrology andshow that there is still much work to be done, and opportunities to be found, in using quantum
systems for metrological purposes. By using qubits as sensors for tasks like axion detection andnoise spectroscopy, we find that the rapid detection of signals may be possible. Furthermore, wehave demonstrated the dependence of the maximal performance of a reservoir computer on thenoise the reservoir experiences - suggesting a place where quantum reservoir computing might yieldadvantages. These works show that there is still fertile ground to explore in using quantum systems
and devices for more than just quantum computation. We concluded with two projects exploring direct randomized benchmarking. By rigorously defining the properties of direct randomized
benchmarking in chapter six, we were able to leverage those mathematical details in chapter sevento study the propagation of a simple class of errors, Markovian errors, through a simple family of164
circuits, random circuits.
In all seven of these chapters, we find the same important conclusion - by using ideas inquantum information to understand physical systems we are able to extract more insight than weotherwise might. In the case of the last chapters, this is obvious - direct randomized benchmarking
itself is a construction of quantum information. However, for the metrological examples with axion
detection and reservoir computation, as well as for manipulating atomic and ionic systems withmany degrees of freedom, we see that by thinking of the systems as qubits, controlled by gates, weare able to observe interesting properties of the systems that we may otherwise be blind to.

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