Exploring Quantum Phases of Matteron Quantum Processors
Frank Pollmann
Technische Universität München
With: Yujie Liu,TUM
Bernhard Jobst, TUM
Adam Smith,TUM → Nottingham
Michael Knap, TUM
Andrew Green, London
Christina Knapp, Microsoft
Kevin Satzinger, Google
Pedram Roushan, Google

TRR 80Novel Quantum States inCondensed Matter 2022
Kyoto Nov. 21, 2022

Complexity of Quantum Many-Body Systems
Computational Complexity grows
exponentially with system size!
~40 qubits

H| i = E| i~50 qubits

Noise Intermediate ScaleQuantum (NISQ)
Identify problems that are hard on classical computers
but doable on near term NISQ devices!

Classical versus Quantum Computer
Classical Computer:

Quantum Computer:| ψi⟩| ψf ⟩Exploring Quantum Phases of Matteron Quantum Processors
Crossing a Symmetry Protected Topological (SPT) phasetransition on quantum processor[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)][Liu, Smith, Knap, FP (in preparation)]

Realizing and characterizing Topologically Ordered States
on a quantum processor[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 6572 (2021)]

Exact quantum circuit crossing a phase transition

Exact quantum circuit
connecting the SPT
and the trivial phase| ψ0⟩ =

[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)]

Circuit Construction
Exact quantum circuit| ψ0⟩ =…

…

…

Thermodynamic limit: Dominant eigenvector as circuit:=[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)]

Detecting a topological phase transition
Cluster state model:∑xz
σjzσj+1
σj+2−glim ⟨σkx⟩ = 0

H=j|i−j|→∞

∏

i<k<j

Symmetric∑jσjx
lim ⟨|i−j|→∞

∏

i<k<j

Symmetricσkx⟩ ≠ 0g/J

Non-local “string”order parameter [den Nijs and Rommelse ’89, FP and Turner ‘12]
[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)]

Quantum circuit crossing an SPT phase transition
Results on a 20 qubit IBM-Q device:[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)]

Quantum circuit crossing an SPT phase transition
Results on a 20 qubit IBM-Q device:‣ Accurate representation on real machine!
‣ Generalization to the entire one-dimensional BDI class[Jones et al., PRR 3, 033265 (2021)][Smith, Jobst, Green, FP, PRR 4, L022020 (2022)]

Quantum Convolutional Networks (QCNN)QCNN instead of string order parameter [Cong et al. 2018][Smith, Jobst, Green, FP, PRR 4, L022020 (2022)]

Model independent learning
Mask short range correlations
with symmetric noise[Liu, Smith, Knap, FP (in preparation)]

Model independent learning
Example: Custer state with two layers of noise
Unitcell (l = 2)

[Liu, Smith, Knap, FP (in preparation)]

Model independent learning
Benchmark on physical models
H0

H0 + gxx∑i

Xi Xi+1[Liu, Smith, Knap, FP (in preparation)]

Exploring Quantum Phases of Matteron Quantum Processors
Crossing a Symmetry Protected Topological (SPT) phasetransition on quantum processor[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)][Liu, Smith, Knap, FP (in preparation)]

Realizing and characterizing Topologically Ordered States
on a quantum processor[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 6572 (2021)]

Intrinsic topological order and anyons
Toric code model

HT C =J[Kitaev ’03]

X

Avv

Av =YJX

Bp , J > 0pzi , Bp =i2v

Yxii2p[Av , Bp ] = 0 ➡ Exactly solvablez| 0i = |= ±1

‣ Z2 topological orderi+|i + ...

Realizing the toric code on a quantum processor
Toric code ground state(1 + Bp) | 0⟩960.

|0⟩⊗310000000000000H

H

H

H

H

H

H

H

H

H

H

0000

H
0000000000000.96∏
pB

4| G⟩ ∝00z4Z parity, ⟨A⟩

0.98

60.
99-101X parity, ⟨B⟩|G⟩‣

Linear depth in system width[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 1237 (2021)][See also Semeghini et al., Science 374, 1242 (2021)]

Realizing the toric code on a quantum processor| G⟩ ∝∏
pA

0.96

0.
98

0.98

0.98
0.93

0.96
0.96
0.87

0.96
0.78
0.93
0.78

0.87
0.90

0.98

0.96
0.94

0.98
0.98

0.94
0.95
0.78
0.86

0.78
0.81
0.90
0.81

98
0.

0.98
0.93

0.94
0.94

0.95
0.94
0.86
0.93

0.81
0.96
0.81
0.79

98
0.

0.93

0.94
0.96

HH

HH0 0 0 0 00
H000 0 0 0 00
00
000 0 0 0 00
H000 0 0 0 0000000 0 0 0 00
0000000.96
0000HH

HH

HHHHH

HH

HH

HHH

H

H

00.93
0.96

Z parity, ⟨A⟩
Z parity, ⟨A⟩

0.98

-1
0.96

0.95

000B
⊗31
00
00
|0⟩

00.94
0.98

0.96
0.95
0.79

|0⟩⊗31096
0.

0.91
0.96

B

0.94

0.
99

0.91

96
0.

0.
98

A

(1 + Bp) | 0⟩0.
99

Toric code ground state

01X parity, ⟨B⟩-101X parity, ⟨B⟩

31 qubits, average stabilizer delity 0.92 ±0.06fi[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 1237 (2021)]

Probing topological entanglement
Topological entanglement entropy S = ↵L
[Kitaev and Preskill ’06, Levin and Wen ‘06]

BA

Subtraction scheme cancels
area law boundary terms!

CStopo = − γ = SA + SB + SC − SAB − SAC − SBC + SABC
[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 1237 (2021)]

Probing topological entanglement
Toric code: Stopo = − ln 2
Full state tomography
for 4 and 6 qubits
Randomized
measurements
for 9 qubits
Average over locationand orientation[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 1237 (2021)]

Anyonic braiding statistics

Exchange: U-matrix
Can take rational
phases other than ±1

Mutual: S-matrix
No analogue forfundamental fermions/bosons in 3D

Simulating anyonic statistics[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 6572 (2021)]

Simulating anyonic statistics[Jiang et al. ’08]

[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 1237 (2021)]ZL mZSurface code logical qubits

X parity, ⟨B⟩

XL measurement

XLTowards logical qubits
CDRawZL

Boundary conditions lead to A
ground state degeneracy

ZLPerform state injection /readout over Bloch sphere

ZL measurement

Z parity, ⟨A⟩

BXL

Corrected

10

100

10

10

1001111011XL 0011110

1

10

100

10

10

100

10

1001

YL

Linear quantum circuits for string-net models
CDand quantum gates for braiding abelian andnon-abelian anyons [Liu, Smith, Shtengel, FP, arXiv:2110.02020]
ZL

Corrected

XL measurement

X parity, ⟨B⟩

Raw

5|+L⟩ raw

3|1L⟩ raw

XLYL

⟩ corrected, dec[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 1237|+ (2021)]
LExploring Quantum Phases of Matteron Quantum Processors
Crossing a Symmetry Protected Topological (SPT) phasetransition on quantum processor[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)][Liu, Smith, Knap, FP (in preparation)]

Realizing and characterizing Topologically Ordered Stateson a quantum processor[K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 6572 (2021)]

Thank you!

Thank you!

Yujie Liu

Adam Smith

Andrew Green

Bernhard Jobst

Michael Knap

Christina KnappKevin Satzinger

Pedram Roushan[Smith, Jobst, Green, FP, PRR 4, L022020 (2022)][K. J. Satzinger, Y. Liu, A. Smith, C. Knapp et al., Science 374, 6572 (2021)]