UT-Komaba22-3 Measurement-based quantum simulation of Abelian lattice gauge theories Hiroki Sukeno

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UT-Komaba/22-3

Measurement-based quantum simulation of Abelian lattice gauge theories

Hiroki Sukeno

Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA

Takuya Okuda

Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan

(Dated: October 21, 2022)

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and

their quantum simulation is expected to become a major application of quantum computers in the

future. In this work, for an Abelian lattice gauge theory in dspacetime dimensions, we deﬁne an

entangled resource state (generalized cluster state) that reﬂects the spacetime structure of the gauge

theory. We show that sequential single-qubit measurements with the bases adapted according to

the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the

gauge theory on the boundary. Our construction includes the (2 + 1)-dimensional Abelian lattice

gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the

simulation of Wegner’s lattice models M(d,n)that involve higher-form Abelian gauge ﬁelds. We

demonstrate that the generalized cluster state has a symmetry-protected topological order with

respect to generalized global symmetries that are related to the symmetries of the simulated gauge

theories on the boundary. Our procedure can be generalized to the simulation of Kitaev’s Majorana

chain on a fermionic resource state. We also study the imaginary-time quantum simulation with

two-qubit measurements and post-selections, and a classical-quantum correspondence, where the

statistical partition function of the model M(d,n)is written as the overlap between the product of

two-qubit measurement bases and the wave function of the generalized cluster state.

CONTENTS

I. Introduction 2

II. Lattice models and resource states 3

A. Cell complex notation 3

B. Model M(d,n)3

C. Example 1: M(3,1) (Ising model in 2 + 1

dimensions) 4

D. Example 2: M(3,2) (Z2gauge theory in 2 + 1

dimensions) 4

E. Generalized cluster state gCS(d,n)4

III. Measurement-based quantum simulation of

gauge theory 5

A. Simulation of M(3,1) 5

1. Measurement pattern 5

B. Simulation of M(3,2) 7

1. Measurement pattern 7

2. Enforcing gauge invariance 7

C. MBQS of imaginary time evolution 10

IV. Enforcement of Gauss law constraint against

errors 11

A. Gauss law enforcement by error correction 11

1. Wave function without correction 11

2. Direct eﬀect on time evolution 11

3. Extra byproduct operators 12

4. Syndromes and symmetry of resource

state 13

5. Final simulated state 14

B. Gauss law enforcement by energy cost 15

V. Generalizations 16

A. MBQS of M(ZN)

(d,n)16

1. State gCS(ZN)

(d,n)16

2. Model M(ZN)

(d,n)17

3. Hamiltonian formulation of M(ZN)

(d,n)17

4. MBQS protocols for M(ZN)

(d,n)17

B. Euclidean path integral and Hamiltonian

MBQS 19

C. Kitaev Majorana chain 19

VI. SPT order of the generalized cluster state 21

A. Symmetries of the generalized cluster

states 22

B. Gauging map 22

C. Mapping the generalized cluster states 23

D. Brane operators and projective

representation 23

VII. Symmetries in measurement-based quantum

simulation and a bulk-boundary

correspondence 24

A. Ising model M(3,1) 24

B. Gauge theory M(3,2) 25

VIII. Conclusions & Discussion 25

Acknowledgement 26

References 26

A. Proof of equations 30

B. Continuous-time limit of model M(ZN)

(d,n)31

arXiv:2210.10908v1 [quant-ph] 19 Oct 2022

C. Gauge group R32

1. Hamiltonian formulation of M(R)

(d,n)32

2. MBQS of M(R)

(d,n)32

3. A formal correspondence between gCS(R)

(d,n)

and the Euclidean path integral 34

D. Stabilizer operators as gauge transformations 34

E. Gauging map via minimally coupled gauge

ﬁelds 35

F. Tensor network representation of gCSd,n 36

I. INTRODUCTION

Gauge theory is a foundation of modern elementary

particle physics. The numerical simulation of Euclidean

lattice gauge theories [1] has been a great success, even in

the non-perturbative regime that is hard to study ana-

lytically. On the other hand, there are situations such

as real-time simulation and ﬁnite density QCD where

the path integral formulation of lattice gauge theory suf-

fers from the sign problem—a diﬃculty in the evaluation

of amplitudes due to the oscillatory contributions in the

Monte-Carlo importance sampling [2–5]. In the Hamil-

tonian formulation, the dimension of the Hilbert space

grows exponentially with the size of the system. The

quantum computer is expected to solve this issue, en-

abling us to simulate the quantum many-body dynamics

in principle with resources linear in the system size [6, 7].

The quantum simulation of gauge theory is thus one of

the primary targets for the application of quantum com-

puters/simulators, whose studies are fueled by the recent

advances in NISQ quantum technologies [8–14].

The goal of this paper is to present a new quantum sim-

ulation scheme for lattice gauge theories. Our scheme,

which we call measurement-based quantum simulation

(MBQS), is motivated by the idea of measurement-based

quantum computation (MBQC) [15–19]. Just as in the

common MBQC paradigm, our procedure consists of two

steps: (i) preparation of an entangled resource state and

(ii) single-qubit measurements with bases adapted ac-

cording to the former measurement outcomes. In the

usual MBQC, resource states (such as cluster states [15])

are constructed to achieve universal quantum computa-

tion. In MBQS, the resource states, the generalized clus-

ter states (gCS), are tailored to simulate the gauge theo-

ries and reﬂect their spacetime structure.

Our prototype examples are the (2 + 1)-dimensional

Ising model and the lattice Z2gauge theory [20–22] sim-

ulated on appropriate generalized cluster states. Then

we extend this idea to Wegner’s lattice models M(d,n)[22]

that involve higher-form Z2gauge ﬁelds. It is common in

MBQC to identify one of the spatial dimensions as time in

gate-based quantum computation. Similarly, we regard

the generalized cluster state as a space-time in which the

post-measurement

product state teleportation

|Ψ(t)⟩

|Ψ(0)⟩

gCS

(SPT)

FIG. 1. The concept of MBQS. We start from gCS with

the initial wave function at the boundary. After applying

single-qubit measurements based on a measurement pattern,

we obtain |Ψ(t)i, the wave function after the evolution with

the Hamiltonian of the model M(d,n), at the boundary of the

reduced lattice.

lattice gauge theory lives. See Fig. 1 for an illustration

of the concept of MBQS. We also discuss a relation be-

tween the generalized cluster state and the partition func-

tion, which is a specialized version of a relation between

a graph state and the Ising model found in [23, 24]. Our

relation implies that the expectation value of the Wil-

son loop can be estimated via the Hadamard test with a

controlled constant-depth circuit (See e.g. [25]).

In the Hamiltonian formulation of gauge theory, physi-

cal states are required to obey the gauge invariance condi-

tion called the Gauss law constraint. In noisy simulations

it is expected to be especially important to minimize the

eﬀects of errors that violate gauge invariance [26, 27].

In this work we combine the well-known error correcting

techniques in MBQC with the analysis of symmetries of

the gauge theory and the resource state to formulate an

eﬀective method to enforce the Gauss law constraint.

We also present generalizations to gauge group ZN.

Our generalized cluster state is expressed using the cell

complexes, and the construction naturally leads us to the

simulation of the model M(ZN)

(d,n), the ZNgeneralization of

Wegner’s Ising model. As another non-trivial generaliza-

tion of our MBQS, we present an approach to simulat-

ing Kitaev’s Majorana chain [28] on a fermionic resource

state.

Aside from studies of quantum computational meth-

ods, we present more formal aspects of MBQS regarding

symmetries. We show that a generalized cluster state

possesses a non-trivial symmetry-protected topological

(SPT) order [29–36] protected by higher-form symme-

tries [37, 38]. Further, we propose that MBQS can be

regarded as a type of bulk-boundary correspondence be-

tween the resource state and the simulated ﬁeld theory.

Speciﬁcally, the gauge symmetry of the boundary sim-

ulated theory is promoted to a higher-form symmetry

of the bulk resource state. This feature is used in Sec-

tion IV for the enforcement of the Gauss law constraint in

MBQS. On the other hand, as we discuss in Section VII,

the boundary simulated theory has a global (higher-form)

symmetry, while the bulk resource state can be regarded

as a model in which the boundary global symmetry is

gauged. It can be seen as a new type of holographic

correspondence.

This paper is organized as follows. In Section II, we

review the models M(d,n)and introduce the generalized

cluster states gCS(d,n). In Section III we provide the

measurement-based protocols for the simulation of the

Ising model M(3,1) and the Z2gauge theory M(3,2). We

explain our measurement pattern to execute MBQS. We

also study generalization to the imaginary-time evolu-

tion. In Section IV, we discuss a procedure to detect

certain types of errors and correct them based on the

higher-form symmetry of the resource state, enabling us

to enforce the Gauss law constraint. In Section V, we

discuss generalization to the ZN(n−1)-form theory in

ddimensions as well as the case with the Kitaev’s Ma-

jorana fermion in (1 + 1) dimensions. We also make a

connection between the Euclidean path integral and the

generalized cluster state for the model M(ZN)

(d,n). In Sec-

tion VI, we show that the generalized cluster state is an

SPT state. In Section VII, we discuss an interplay of the

symmetries between the bulk and the boundary in our

MBQS. Section VIII is devoted to Conclusions and Dis-

cussion. In the appendix we prove some equations used

in the main text and discuss supplementary aspects of

our MBQS and the generalized cluster states.

II. LATTICE MODELS AND RESOURCE

STATES

A. Cell complex notation

Let us consider a d-dimensional hypercubic lattice. Let

∆0be the set of 0-cells (vertices), ∆1the set of 1-cells

(edges), and ∆2the set of 2-cells (faces), and so on. We

write Ci(i= 0,1,2, ..., n) for the group of i-chains ciwith

Z2coeﬃcients (later this will be generalized to general

Abelian groups), i.e., the formal linear combinations

ci=X

σi∈∆i

a(ci;σi)σi(1)

with a(ci;σi)∈Z2={0,1 mod 2}. Sometimes we re-

gard the chain cias the union of the i-cells σisuch that

a(ci;σi) = 1. The boundary operator ∂is a linear map

Ci+1 →Cisuch that ∂σi+1 is the sum of the i-cells that

appear on the boundary of σi. We get a chain complex

∂

−→ ··· ∂

−→ C1

∂

−→ C0(2)

with ∂2= 0. Similarly, by considering the dual lat-

tice [39], we get the dual chain complex

C∗

∂∗

−→ ··· ∂∗

−→ C∗

∂∗

−→ C∗

0(3)

with (∂∗)2= 0. There are natural identiﬁcations of ∆i

(i-cells) with ∆∗

n−i(dual (n−i)-cells), and Ci(i-chains)

with C∗

n−i(dual (n−i)-chains) [40]. We will often con-

sider placing qubits on all the i-cells σi∈∆ifor some

i. Then on each σiwe have Pauli operators X(σi) and

Z(σi). For each i-chain ciwe deﬁne

X(ci) := Y

σi∈∆i

X(σi)a(ci;σi),

Z(ci) := Y

σi∈∆i

Z(σi)a(ci;σi).(4)

For MBQS we consider a hypercubic lattice in d-

dimensions, with the (1,2, ..., d −1)-directions periodic

and the d-th direction open. The value of the d-th coor-

dinate xd(“time”) speciﬁes an artiﬁcial time slice. The

boundaries xd= 0 and xd=Ld, where Ldis the linear

lattice size in the d-th direction, are examples. The bulk

state to be introduced later will be the resource state

for MBQS. As we proceed in the protocol of MBQS,

the state originally deﬁned on the xd= 0 time slice

will be teleported to a middle time slice xd=j, where

j∈ {0,1, . . . , Ld}. Throughout the paper, unless oth-

erwise stated, we use the notation where the bold fonts

(∆

∆

∆, σ

σ,∂

∂

∂, etc.) represent “bulk” quantities related to the

d-dimensional lattice, whereas the normal fonts (∆, σ,∂,

etc.) are used for the (d−1)-dimensional lattice identiﬁed

with the space of the simulated model.

A cell σ

σiinside a time slice xd=jis of the form

σi=σi× {j},(5)

while a cell σ

σiextending in the time direction takes the

form

σi=σi−1×[j, j + 1] .(6)

Sometimes we express a point in the time direction as pt

and an interval as I.

B. Model M(d,n)

We consider a class of theories described by classical

spin degrees of freedom living on (n−1) cells in the d-

dimensional hypercubic lattice whose action Iis given

I[{Sσ

σn−1}] = −JX

σn∈∆

∆

∆n

S(∂

∂

∂σ

σn),(7)

where Jis a coupling constant. Sσ

σn−1∈ {+1,−1}is a

classical spin variable living on each (n−1)-cell σ

σn−1∈

∆

∆n−1and

S(c

ci) = Y

σi∈∆

∆

∆i

(Sσ

σi)a(c

ci;σ

σi)(8)

for a given i-chain c

ci=Pσ

σi∈∆

∆

∆ia(c

ci;σ

σi)σ

σi. This class of

theories is called “generalized Ising models” in the litera-

ture [22], where the action (7) is viewed as the (classical)

Hamiltonian of such a classical spin model. For n= 2, (7)

is the Z2version of the action of Wilson’s lattice gauge

theory [1], whose degrees of freedom are 1-form gauge

ﬁelds. When n≥2, the theory is described by (n−1)-

form gauge ﬁelds, and the action is invariant under a

local transformation at each (n−2)-cell,

Gσ

σn−2:S(σ

σn−1)→ −S(σ

σn−1) for σ

σn−1∈∂

∂

∂∗σ

σn−2.

(9)

This is a higher-form generalization of the standard dis-

crete gauge transformation, which corresponds to the

case with n= 2. For n= 1, M(d,n)is the Ising model in

ddimensions.

On inﬁnite lattices, the models M(d,n)and M(d,d−n)

are dual to each other [22], generalizing the Kramers-

Wannier duality of the two-dimensional Ising model. On

ﬁnite lattices, the duality changes the global structure of

the model. See, e.g., [41].

For each classical spin model in ddimensions, one can

construct a quantum spin model deﬁned on a (d−1)-

dimensional spatial lattice. See [20] and Appendix B. The

qubits are placed on (n−1)-cells σn−1. The Hamiltonian

is given by

H(d,n)=−X

σn−1∈∆n−1

X(σn−1)−λX

σn∈∆n

Z(∂σn),

(10)

where we used the notation (4) and λis a coupling con-

stant. Gauge-invariant states |ψimust satisfy the Gauss

law constraint

G(σn−2)|ψi= (−1)Q(σn−2)|ψi(11)

for any σn−2∈∆n−2, where G(σn−2) is deﬁned as

G(σn−2) = X(∂∗σn−2),(12)

and Q(σn−2) = 1 if there is an external charge on the

cell σn−2and Q(σn−2) = 0 otherwise. Conjugation by

the operator G(σn−2) generates a gauge transformation

in the Hamiltonian picture.

C. Example 1: M(3,1) (Ising model in 2 + 1

dimensions)

The Ising model M(3,1) in 2 + 1 dimensions has the

Hamiltonian

H(3,1) =−X

σ0∈∆0

X(σ0)−λX

σ1∈∆1

Z(∂σ1).(13)

The second term is the nearest neighbor interaction be-

tween two vertices connected by edges. We have the

following Trotter decomposition of the time evolution

e−iH(3,1)t:

T(3,1)(t) := Y

σ0∈∆0

eiδtX(σ0)Y

σ1∈∆1

eiδtλZ(∂σ1)!x3

(14)

with t=x3δt.

D. Example 2: M(3,2) (Z2gauge theory in 2 + 1

dimensions)

The Hamiltonian of the model M(3,2), the Z2gauge

theory in 2 + 1 dimensions, is [22]

H(3,2) =−X

σ1∈∆1

X(σ1)−λX

σ2∈∆2

Z(∂σ2).(15)

The second sum is over plaquettes (faces) σ2. The pla-

quette operator Z(∂σ2) is the product of Pauli-Zoper-

ators on the four edges surrounding σ2. The Gauss law

constraint is

X(∂∗σ0)=(−1)Q(σ0),(16)

where the left hand side is the product of Pauli-Xoper-

ators on the edges attached to the vertex σ0.Q(σ0)∈

{0,1}is the external charge placed at σ0∈∆0. The

ﬁrst-order Trotter approximation of the time evoltion

e−iH(3,2)tis given by

T(3,2)(t) := Y

σ1∈∆1

ei δt X(σ1)Y

σ2∈∆2

ei δt λZ(∂σ2)!x3

(17)

with t=x3δt.

E. Generalized cluster state gCS(d,n)

Here we describe the resource state which we call the

generalized cluster state, gCS(d,n).

We deﬁne the eigenvectors of the Pauli operators by

Z|0i=|0i, Z|1i=−|1i,(18)

X|+i=|+i, X|−i =−|−i .(19)

We place a qubit on every (n−1)-cell σ

σn−1∈∆

∆

∆n−1

and on every n-cell σ

σn∈∆

∆

∆n. For each n-chain c

cn=

Pσ

σn∈∆

∆

∆na(c

cn;σ

σn)σ

σn, we deﬁne

X(c

cn) := Y

σn∈∆

∆

∆n

(Xσ

σn)a(c

cn;σ

σn).(20)

We similarly deﬁne Pauli Zoperators and Pauli operators

on (n−1)-cells. A general Pauli operator takes the form

P=eiαX(c

cn)Z(c

n)X(c

cn−1)Z(c

n−1),(21)

where αis a c-number phase.

Now we deﬁne the stabilizers

K(σ

σn) = X(σ

σn)Z(∂

∂

∂σ

σn),(22)

K(σ

σn−1) = X(σ

σn−1)Z(∂

∂

∂∗σ

σn−1).(23)

The generalized cluster state |gCS(d,n)iis deﬁned by the

eigenvalue equations

K(σ

σn−1)|gCS(d,n)i=K(σ

σn)|gCS(d,n)i=|gCS(d,n)i

for all σ

σn−1∈∆

∆

∆n−1, σ

σn∈∆

∆

∆n.(24)

Explicitly, the cluster state can be written as

|gCS(d,n)i=UCZ |+i⊗(∆

∆

∆n−1t∆

∆

∆n).(25)

where UCZ is the entangler that applies controlled-Z

gates (CZ gates) between qubits on adjacent qubits:

UCZ := Y

σn−1∈∆

∆

∆n−1

σn∈∆

∆

∆n

(CZσ

σn−1,σ

σn)a(∂

∂

∂σ

σn;σ

σn−1)(26)

with ∂

∂

∂σ

σn=Pσ

σn−1∈∆

∆

∆n−1a(∂

∂

∂σ

σn;σ

σn−1)σ

σn−1. The CZ

gate is given by

CZc,t =|0ich0| ⊗ It+|1ich1| ⊗ Zt.(27)

It is invariant under the exchange of the control (c) and

the target (t) qubits.

III. MEASUREMENT-BASED QUANTUM

SIMULATION OF GAUGE THEORY

In this section, we introduce the MBQS protocols for

the real-time evolution of the Ising model M(3,1) and the

gauge theory M(3,2). See [42] for a pedagogical introduc-

tion to MBQC.

A. Simulation of M(3,1)

For the simulation of the model M(3,1), we use gCS(3,1)

as the resource state. This is a cluster state whose qubits

are placed on 1-cells (edges) and 0-cells (vertices). See

Fig. 2 for an illustration. We describe the measurement

protocol to simulate the time evolution with Hamiltonian

H(3,1) given in (13).

1. Measurement pattern

Our simulation protocol will involve two types (A and

B) of measurements. Each measurement realizes a de-

sired unitary operator, multiplied by an extra Pauli oper-

ator that depends on the non-deterministic measurement

outcome. The desired operator simulates a factor in the

Trotterized time evolution operator (14). The extra op-

erator is called a byproduct operator and is determined

by the measurement outcomes. As we will explain, we

can adaptively choose the measurement bases according

to the previous outcomes so that the simulated unitary

operator is deterministic.

Let us explain the A-type measurement as part of the

MBQS. In a two-dimensional layer at x3=j, we have

qubits on the vertices σ0∈∆0and the edges σ1∈∆1.

See Fig. 2 (1). The qubits on the edges are entangled by

the CZ gate with the adjacent qubits on ∂σ1. The wave

function |Ψifor the qubits on the vertices is arbitrary.

Our claim is that the unitary operator

U(1)

(3,1) := (Z(∂σ1))se−iξZ(∂σ1)(28)

is realized by measuring the qubit on σ1with the basis

M(A)=eiξX |sis= 0,1.(29)

Indeed [43],

σ1hs|e−iξXσ1Y

σ0∈∆0

CZa(∂σ1;σ0)

σ1,σ0|+iσ1|Ψi

√2(Z(∂σ1))se−iξZ(∂σ1)|Ψi.(30)

We prove this equation in Appendix A. The Pauli oper-

ator Z(∂σ1)sis the byproduct operator from this mea-

surement. Up to the byproduct operator and a choice

of angle ξ,U(1)

(3,1) is essentially the time-evolution by the

term Z(∂σ1) in the Ising Hamiltonian (13). We refer to

sas the measurement outcome, and the measurement in

the basis (29) as A-type.

Next, we explain the B-type measurement. It is deﬁned

as the measurement with the basis

M(B)=eiξZ |˜sis= 0,1,(31)

where |˜siis the eigenvector of the Xoperator with the

eigenvalue (−1)s:

X|˜si= (−1)s|˜si.(32)

In other words, |˜

0i=|+iand |˜

1i=|−i. This measure-

ment implements a gate teleportation. To see this, we

consider a general state |Ψi1and an ancilla |+i2, and we

entangle them with the CZ gate. Then we measure the

qubit 1 with the basis M(B). The circuit is given by [44]

1A(1)

2A(2)

3A(3)

1B(1) (4)

2B(1) (5)

3B(1) (6)

1B(2) (7)

2B(2) (8)

3B(2) (9)

|+i(10)

ei⇠X(1)

Z(2)

s(3)

| i(4)

Xsei⇠XH| i(5)

ei⇠X(1)

Z(2)

s(3)

| i(4)

Xsei⇠XH| i(5)

ei⇠X(1)

Z(2)

s(3)

| i(4)

Xsei⇠XH| i(5)

ei⇠X(1)

ei⇠Z(2)

Z(3)

X(4)

s(5)

| i(6)

Xsei⇠XH| i(7)

ei⇠X(1)

ei⇠Z(2)

Z(3)

X(4)

s(5)

| i(6)

Xsei⇠XH| i(7)

Here the realized unitary operator is

U(3)

(3,1) =Xse−iξX H . (33)

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UT-Komaba/22-3Measurement-basedquantumsimulationofAbelianlatticegaugetheoriesHirokiSukenoDepartmentofPhysicsandAstronomy,StateUniversityofNewYorkatStonyBrook,StonyBrook,NY11794-3840,USATakuyaOkudaGraduateSchoolofArtsandSciences,UniversityofTokyo,Komaba,Meguro-ku,Tokyo153-8902,Japan(Dated:October21,2...

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