A gauge theory
ofmeasurement-based
quantum computation
Robert Raussendorf, UBC
joint work with Gabriel Wong (Harvard) and Bartek Czech (Tsinghua)Outline÷÷i÷÷÷
based quantum

computation⑧Computational

phases ofCSPTO)

MBQC

Part I:
MBQC

The 2D cluster state is a computationally universal “material”
Quantum states can have computational powerMeasurement-based quantum computation

Unitary transformation
Z

Projective measurement
ZpX
YX
1-p

Y
deterministic,
reversible

probabilistic,irreversible

Measurement-based quantum computation

measurement of Z ( ), X ("), cos ↵ X + sin ↵ Y (%)• Information written onto the resource state, processed and read out by one-qubit measurements only.
• Universal computational resources exist:
cluster state, AKLT state.
R. Raussendorf, H.-J. Briegel, Physical Review Letters 86, 5188 (2001).

The MBQC “elevator thought experiment”
Consider a 2D cluster state:
preparation

measurement

quantum gates

At the end of any MBQC, all that is left is a tensor product state
and the measurement record s.
• The computational output o is extracted from the measurement record s = (s1, s2, .., sN ).
• Individually, each bit si of measurement record is completelyrandom; information is only in the correlations.

The MBQC “elevator thought experiment”
Consider a 2D cluster state:

preparation

measurement

quantum gates
Some cluster qubits simulate the preparation of a quantum register, some its measurement, and some implement quantum gates.
Do these di↵erent types of qubits contribute di↵erently to theMBQC output?

No!

The MBQC “elevator thought experiment”
Consider a 2D cluster state:

preparation

measurement

quantum gates
Some cluster qubits simulate the preparation of a quantum register, some its measurement, and some implement quantum gates.
Do these di↵erent types of qubits contribute di↵erently to theMBQC output?

No!

The classical side-processing relations
Every MBQC has a classical side-processing relation that convert
measurement record s into
• computational output o, and• the choice q of local measurement bases,o = Zsmod 2, q = T s

mod 2.

(1)

Every bit of computational output is a parity
of individual local measurement outcomes.

What determines the support of these parities?
Which measurement outcomes si contributeto what output bit oj ?

The classical side-processing relations
Every MBQC has a classical side-processing relationso = Zsmod 2, q = T s

mod 2.

Every bit of computational output is a parity
of individual local measurement outcomes.

What determines the support of these parities?
The measurement record s is a gauge field, and each bit of
the computational output o is a holonomy of the gauge field.

How the side-processing comes about... it describes how to counteract the randomness of measurement outcomes.
randomheralded+random
heraldedrandomheraldedrandomheraldedrandomheraldedUz qM ZUxqM XUz q
M ZUxqM XUz q
M Z

1z

2x3z

4x5z

X meas.

• Propagate the random heralded byproduct operators forward
in time past the end of the computation.

Part II:
A short history of“computational phases of quantum matter”

Symmetry-protected phases of quantum mattersymmetry

spin latticein 2D

zero temperature
Image credit Tzu-Chieh Wei... all across support quantum computation.

Motivation: MBQC and symmetry

Can MBQC schemes be classified by symmetry, in a similar way
as, say, elementary particles can?
If so, does this have a bearing on quantum algorithms?

I. Symmetry protects computation

A. Miyake, Phys. Rev. Lett. 105, 040501 (2010).

II. Symmetry-protected wire in MBQC

• Computational wire persists throughout symmetry-protected
phases in 1D.
• Imports group cohomology from the classification of SPT
phases.
D.V. Else, I. Schwartz, S.D. Bartlett and A.C. Doherty, PRL 108 (2012).
F. Pollmann et al., PRB B 81, 064439 (2010); N. Schuch, D. Perez-Garcia, and I. Cirac,
PRB 84, 165139 (2011); X. Chen, Z.-C. Gu, and X.-G. Wen, PRB 83, 035107 (2011).

-w←in theUnder the hood ..

-☒[i-1=symmetry-☒t- respecting
basisTheorem. Consider a symmetry-protected phase characterized by a finite abelian symmetry group and a maximally noncommutative cohomology class [!]. In the symmetry-respecting
basis B = {|ii}, the MPS tensor of a given state in the SPT
phase described by [!] has the form--A[i] = Bi ⌦ Ajunk[i].
The byproduct operators Bi are all unitary and constant throughout the phase.
D.V. Else, I. Schwartz, S.D. Bartlett and A.C. Doherty, PRL 108 (2012).

Resource state Id> :☒-☐b-☒'-☐

MBQC on 10>⑥ • ↳← local↳☒-☐b-☒'-☐mmE¥

III. First quantum computational phase• 1-qubit universal MBQC on a chain of spin-1 particles protected by an S4 symmetry.

J. Miller and A. Miyake, Phys. Rev. Lett. 114, 120506 (2015).

IV. The SPT)MBQC meat grinder

A classification of MBQC schemes by symmetry in 1D.
A. Prakash and T.-C. Wei, Phys. Rev. A (2016).
RR, A.Prakash, D.-S. Wang, T.-C. Wei, D.T. Stephen, Phys. Rev. A (2017).

Under the hood ..

cohomology class [!]#The given SPT phase determines thebyproduct operators [Else et al., RRL 2012]

Bi#The byproduct operators determine MBQC
computational power [RR et al., RRA 2017]set of gates
Byproduct operators: What entered as a nuisance that could
fortunately be dealt with, becomes the central object governing
MBQC computational power.

• The symmetries of the phase areX
X
X
X
X
X

X
X
X
X
X

virtual quantum register

V: Computationally universal SPT phase in 2D

X
X

X

XXXX

XX

X X

X
circuit model time

• The 2D cluster state is inside the phase
Result. For a spin-1/2 lattice on a torus all ground states in the2D cluster phase, except a possible set of measure 0, are universal
resources for measurement-based quantum computation.
R. Raussendorf, C. Okay, D.S. Wang, D.T. Stephen, H.P. Nautrup, A computationally universal quantum phase of matter, Phys. Rev. Lett. 122, 090501 (2019).

Part III:A gauge theory of MBQC

Motivation
The discussion of computational phases of quantum matter isongoing. It has so far taught us• The computational capability of MBQC schemes is determined and classified by symmetry.

• To harness this capability, however, a breaking of symmetry
is required, namely through the local measurements.
We want a complete picture of the role of symmetry in MBQC.
Gauge symmetry seems part of that.

Result #1
Recall the MBQC elevator thought experiment:
preparation

measurement

quantum gates

All cluster qubits contribute in the same fashion to the computational output, namely through parities.
What determines the support of these parities?
Answer: The measurement record s can be regarded as a gauge
field, and the computational output o corresponds bit-wise toholonomies of that gauge field.

Slightly modified setting—dodging boundaries

cluster state on achaia :¥#0-9-0-01=0/-0:*
::

:

÷

;

:

:

corresponds !

; ;

to :

¥#i.

>-☒-☒-☒-☒-☒-☒tthit

dustarstateouariug,corresponds
to ::What is implemented on the cluster ring?

:*|☐_÷t÷±÷÷÷÷÷÷÷÷÷±¥Byestate""

Bellstate

prep"

ftp.#u*=*-*.FFeasnromenta-implementedunitary[=✗Qz"-byproduct op.

What do the two output bits represent?

Bellstate

prep

ftp.i#uz*-**MBFeasnroment-implemented

unitaryU = ↵I + X + Z + Y.

outcome00
01
1011occurs with probability|↵|2| |2| |2| |2

How are the two output bits obtained from s?
Belli¥%÷|☐_ ¥¥¥É¥¥÷¥+÷±÷±"

projection

gate sequence

Bell

state

prep|µ¥u*¥;y#u

Teasmament"

Efren ted
unitaryo1 =X?⃝

i evensi

mod 2, o2 =X

i odd

si

mod 2.

(2)

Recap: cluster state stabilizer⑦

YE

IK;1eTPauli/ stabilizer action on s and q"÷Localobservables measured :O↵(q) = cos ↵ X + ( 1)q sin ↵ Y.

.

X † O↵(q)X = O↵(q 1),
Z † O↵(q)Z =O↵(q). a-2-flipssGauge transformations
Recall:X † O↵(q)X = O↵(q 1), flips 9
Z † O↵(q)Z =O↵(q).
flips sandKj =Zj_iXj Ziti:

hC|O↵i (qi)|Ci = hC|Kj O↵i (qi)Kj |Ci, 8i, j insertion
(i)

†(i)

←ofkj

hasnoeffed !

Thus we have the following equivalence/ gauge transformations
sj 1 !
7
sj 1 1,
7
sj+1 1,
Kj : sj+1 !
qj !
7
qj 1.

(3)

• Those equivalence transformations are local.
• For j = 2, .., 2N
1 those equivalence transformations preserve the classical processing relations.

Cohomological interpretation%>⑦③

Meas record.

basisadaph-u.to#$-s,s-•

919Th
✗

→1+90

on sites

☒
! ¥É%

6+5
•

in

H'(c) H'(E)'-4of,

priimalchain
complex

on

1%4 "

•

a2

dudechaincomplexedges ,

Gauge transformations in cohomological form
6cos-4

7€

]

8,

2

chain

primalcomplexstandard formsj 1 !
7
sj 1 1,
7
sj+1 1,
Kj : sj+1 !
qj !
7qj 1.

7%3
2gdudechain
complexcohomological forms !
7 sq !
7 qs 7! sq !
7 qd⇤,⇤,d⇤,⇤.

Gauge-invariant quantities
Find a 1-chain e such that, for allRequiress(e) 7! [sd⇤](e) = s(e).

d⇤(e) = 0.
Hence,and thus

0 = d⇤(e) = ⇤(@e), 8⇤,@e = 0.

The 1-chain e is, in fact, a 1-cycle.
• s(e) is the corresponding holonomy.
• s(e) is gauge invariant.

• Have the same invariant s(e) on the dual complex C.

+Gauge-invariant quantities•°É
priimalchaincomplex?s7

sduaiechaiu
complex• s(e) is gauge invariant.

• Have the same invariant s(e) on the dual complex C.
These holonomies have previously been identified as the computational output. Recall:

o1 =

X

i evensi

mod 2, o2 =

X

i odd

si

mod 2.

TETE

The MBQC “elevator thought experiment”
Consider a 2D cluster state:
preparation

measurement

C

quantum gates

Every bit of computational output is a parity
of individual local measurement outcomes.
In 1D at least, these parities represent holonomies of a gauge field.

Outlookbased quantumcomputation

MBQC( MBQC)⑧Computational

phases ofquantum{matterOutlook☒ Extend gauge theoreticaldescription
SPTfrom fixed-pointstate to-surrounding--

phaseProblem : Can no longer use the stabilizers

goodsign:

classical
relationsextend
processing
1into thephase unchanged

Summary• In 1D so far, MBQC can be understood in terms of gauge
theory, with the measurement record as the gauge field.

• In that picture, the MBQC computational output arises asholonomies of the gauge field.

• Extension to computational phases of quantum matter issuggestive.
Joint work with Gabriel Wong (Harvard)
and Bartek Czech (Tsinghua)