SciPost Phys. 14, 129 (2023)

Measurement-based quantum simulationof Abelian lattice gauge theories
Hiroki Sukeno1⋆ and Takuya Okuda2
1 Department of Physics and Astronomy, State University of New York at Stony Brook,
Stony Brook, NY 11794-3840, USA
2 Graduate School of Arts and Sciences, University of Tokyo, Komaba,
Meguro-ku, Tokyo 153-8902, Japan⋆ hiroki.sukeno@gmail.com

Abstract
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 ofquantum 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 singlequbit 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 tothe 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 andpost-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.
Copyright H. Sukeno and T. Okuda.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 28-10-2022
Accepted 29-03-2023
Check forupdatesPublished 25-05-2023
doi:10.21468/SciPostPhys.14.5.129

Contents
1 Introduction
1.1 Summary of results
1.2 Organization of the paper

33
6

2 Lattice models and resource states
2.1 Cell complex notation
2.2 Model M(d,n)

6
6
7
1

SciPost Phys. 14, 129 (2023)

Example 1: M(3,1) (Ising model in 2 + 1 dimensions)
Example 2: M(3,2) (Z2 gauge theory in 2 + 1 dimensions)
Generalized cluster state gCS(d,n)

8993 Measurement-based quantum simulation of gauge theory
3.1 Simulation of M(3,1)
3.1.1 Measurement pattern
3.2 Simulation of M(3,2)
3.2.1 Measurement pattern
3.2.2 Enforcing gauge invariance
3.3 MBQS of imaginary time evolution

10
10
10
13
13
14
16

4 Enforcement of Gauss law constraint against errors
4.1 Gauss law enforcement by error correction
4.1.1 Wave function without correction
4.1.2 Direct effect on time evolution
4.1.3 Extra byproduct operators
4.1.4 Syndromes and symmetry of resource state
4.1.5 Final simulated state
4.2 Gauss law enforcement by energy cost

18
19
19
19
20
21
22
24

5 Generalizations(ZN )5.1 MBQS of M(d,n)

24
25

2.3
2.4
2.5

5.1.1
5.1.2
5.1.3
5.2
5.3

(Z )

N
State gCS(d,n)

25

(ZN )
Model M(d,n)

26

(ZN )
Hamiltonian formulation of M(d,n)(ZN )MBQS protocols for M(d,n)

5.1.4
Euclidean path integral and Hamiltonian MBQS
Kitaev Majorana chain

27
27
29
30

6 SPT order of the generalized cluster state
6.1 Symmetries of the generalized cluster states
6.2 Gauging map
6.3 Mapping the generalized cluster states
6.4 Brane operators and projective representation

33
34
34
35
36

7 Symmetries in measurement-based quantum simulation and a bulk-boundary
correspondence
7.1 Ising model M(3,1)
7.2 Gauge theory M(3,2)

37
37
38

8 Conclusions & Discussion
8.1 Discussion: comparison to circuit-based quantum simulation
8.2 Conclusions and future directions

39
39
40

A Proof of equations

42
(Z )

N
B Continuous-time limit of model M(d,n)

43

C Gauge group R

44

2

SciPost Phys. 14, 129 (2023)(R)

C.1 Hamiltonian formulation of M(d,n)

44

C.2

45

C.3

(R)
MBQS of M(d,n)(R)
A formal correspondence between gCS(d,n) and the Euclidean path integral

47

D Stabilizer operators as gauge transformations

48

E Gauging map via minimally coupled gauge fields

49

F

50

Tensor network representation of gCSd,nReferences

153

Introduction

Gauge theory is a foundation of modern elementary particle physics. The numerical simulationof Euclidean lattice gauge theories [1] has been a great success, even in the non-perturbative
regime that is hard to study analytically. On the other hand, there are situations such as
real-time simulation and finite density QCD where the path integral formulation of latticegauge theory suffers from the sign problem—a difficulty in the evaluation of amplitudes dueto the oscillatory contributions in the Monte-Carlo importance sampling [2–5]. In the Hamiltonian formulation, the dimension of the Hilbert space grows exponentially with the size ofthe system. The quantum computer is expected to solve this issue, enabling 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 applicationof quantum computers/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 simulation 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 anentangled resource state and (ii) single-qubit measurements with bases adapted according
to the former measurement outcomes. In the usual MBQC, resource states (such as cluster
states [15]) are constructed to achieve universal quantum computation. In MBQS, the resource states, the generalized cluster states (gCS), are tailored to simulate the gauge theories
and reflect their spacetime structure.

1.1

Summary of results

Below, we give an overview of our main results. We sometimes refer to the equations thatappear in the main text for convenience.
Our prototype examples are the (2 + 1)-dimensional Ising model and the lattice Z2 gauge
theory [20–22], which we simulate by adaptive single-qubit measurements on appropriate
generalized cluster states. Then we extend this idea to Wegner’s lattice models M(d,n) [22]that involve (n − 1)-form Z2 gauge fields in d spacetime dimensions. For example n = 1
corresponds to the Ising model, while n = 2 is the ordinary lattice gauge theory. We also
generalize the method to the model with the group G = ZN , but in this summary we focus on
G = Z2 . The partition function of M(d,n) with the inverse temperature β, which we write as3

SciPost Phys. 14, 129 (2023)| Ψ(0)⟩
post-measurement
product state

teleportation| Ψ(t)⟩gCS(SPT)

Figure 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)〉, the wave function after the evolution with the Hamiltonianof the model M(d,n) , at the boundary of the reduced lattice.
Z(d,n) , is constructed with the classical Hamiltonian — or the Euclidean action (eq. (11)) of spin
variables. We consider the quantum Hamiltonian on a (d − 1)-dimensional lattice (eq. (14)),which is related to the partition function asZ(d,n) ≃ Tr(e−∆τH(d,n) )k(k∆τ = β) ,(1)

in the limit k → ∞. The precise definition of the lattice models will be given with the notationof the algebraic topology in the main text. Our first goal is to present a method to perform thequantum simulation of the time evolution|Ψ(t)〉 = exp(−i t H(d,n) )|Ψ(0)〉 .

(2)

We aim to realize the first-order Trotter decomposition T(d,n) (t) of the unitary exp(−i t H(d,n) ).
It is common in MBQC to identify one of the spatial dimensions as time in gate-based
quantum computation. Similarly, we regard the resource state as a spacetime in which thelattice gauge theory lives. See Fig. 1 for an illustration of the concept of MBQS. The underlying resource state, the generalized cluster state gCS(d,n) , is a quantum many-body state of
qubits defined on a d-dimensional lattice. The entanglement structure of gCS(d,n) reflects thespacetime structure of the action. We take the d-th direction of the lattice open, with the pastboundary x d = 0 and the future boundary x d = L d , while the other directions are assumed tobe periodic. We claim that the Trotterized time evolution T(d,n) (t) is obtained by measuring thebulk and the past boundary of the state gCS(d,n) with a set of single-qubit bases. The outcomeof each measurement is random, thus the post-measurement state comes with Pauli operatorsthat depend on the outcomes. This type of Pauli operators are called the byproduct operators,
as in the usual MBQC. Nonetheless, by adapting the parameters in the later measurements
according to the former measurement outcomes, the Hamiltonian quantum simulation can be
performed deterministically. Namely, we will show that〈Φ({s}, {ξ})|gCS(d,n) 〉 = N × Σ({s}) × T(d,n) (t)|Ψ(0)〉 ,(3)

where |Φ({s}, {ξ})〉 is the tensor product of measurement basis states with the parameters {ξ}chosen appropriately according to the outcomes {s}, Σ({s}) is the byproduct operators, andN is an unimportant normalization constant. (Here the future boundary of the lattice is left
unmeasured, where the final state on the right hand side is induced.) The byproduct operators
can be removed at the end of the quantum simulation with appropriate post processing.
4

SciPost Phys. 14, 129 (2023)

We generalize the idea above in several directions. First, we present a generalization to
the imaginary-time evolution with exp(−τH(d,n) ) with G = Z2 . On quantum computers, it is
not straightforward to realize the Trotter steps in the imaginary-time evolution since they are
generically not unitary, and the same difficulty carries over to the measurement-based simulation. We modify the resource state gCS(d,n) with further ancillary qubits (eq. (73)) and extend
the measurement bases to two-qubit ones. For each measurement, the basis corresponding tomeasurement outcomes is modified to a set of four two-qubits states. Two of the four induce
desired operators for the imaginary quantum simulation. The other two outcomes, which occur with an exponentially small probability, give an undesired operator and the simulationneeds to be aborted. The imaginary-time quantum simulation thus can be performed with
post-selections.
Another generalization is to the quantum simulation of the Kitaev Majorana chain [23].
The resource state is made of spins and fermions on a two-dimensional lattice. The entangler
to generate the resource state is adopted and generalized from the one that is used for the one
introduced to implement the Jordan-Wigner transformation in [24].
In the Hamiltonian formulation of gauge theory, physical states are required to obey thegauge invariance condition 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 [25, 26]. 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 aneffective method to enforce the Gauss law constraint. In the case with the (2 + 1)-dimensional
Z2 gauge theory, we assume a particular error model where the error is represented as Pauli
operators acting on the resource state (eq. (84)). We identify the errors that violate the Gausslaw constraint. We then show that such errors can be detected by analyzing measurement outcomes in MBQS, and that we can construct an operator to apply on the simulated state so thatthe final wave function obeys the Gauss law constraint. We will also discuss the suppressionof the Gauss law violation by energy penalty terms.
We also discuss a relation between the generalized cluster state and the partition function,
which is a specialized version of the relation between a graph state and the Ising model foundin [27, 28]. Concretely, the partition function Z(d,n) for M(d,n) can be written as
Z(d,n) = 〈Φ|gCS(d,n) 〉 ,(4)

where |Φ〉 is a product of single-qubit basis states that are related to our measurement pattern
in MBQS. A further generalization of the relation implies that the expectation value of the
Wilson loop can be estimated via the Hadamard test with a constant-depth circuit.
Aside from studies of quantum computational methods, we present more formal aspectsof MBQS regarding symmetries. We show that a generalized cluster state possesses a nontrivial symmetry-protected topological (SPT) order [29–36] protected by higher-form symmetries [37, 38]. The symmetry generators, S(M) and S(M′ ), are Pauli operators that aresupported on closed manifolds M and M′ of some nontrivial codimensions. We demonstratethat the state gCS(d,n) possesses the SPT order with respect to such symmetries by using themethod called gauging. We further support this claim by introducing boundary to the latticeand explicitly constructing a set of symmetry generators that exhibit a projective representation at the boundary.
Further, we propose that MBQS can be regarded as a type of bulk-boundary correspondence
between the resource state and the simulated field theory. Specifically, the gauge symmetry of
the boundary simulated theory is promoted to a higher-form symmetry of the bulk resource
state. This feature is used in Section 4 for the enforcement of the Gauss law constraint in
MBQS. On the other hand, as we discuss in Section 7, the boundary simulated theory has a
global (higher-form) symmetry, while the bulk resource state can be regarded as a model in5

SciPost Phys. 14, 129 (2023)

which the boundary global symmetry is gauged. It can be seen as a new type of holographic
correspondence.
Measurement has been identified as a new ingredient that allows us to transform shortrange entangled states into long-range entangled states such as the ground states of gauge
theories [24, 39, 40]. It was also pointed out that the MBQC itself has a structure of gauge
theory, where the gauge transformation relates different patterns of parameters and measurement outcomes [41]. We construct the gauge transformation for our MBQS in Appendix D.
Our holographic interpretation in Section 7 also supports the perspective of MBQC as a gauge
theory.

1.2

Organization of the paper

This paper is organized as follows. In Section 2, we review the models M(d,n) and introduce thegeneralized cluster states gCS(d,n) . In Section 3 we provide the measurement-based protocols
for the simulation of the Ising model M(3,1) and the Z2 gauge theory M(3,2) . We explain our
measurement pattern to execute MBQS. We also study generalization to the imaginary-time
evolution. In Section 4, 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 theGauss law constraint. In Section 5, we discuss generalization to the ZN (n − 1)-form theory ind dimensions as well as the case with the Kitaev’s Majorana fermion in (1 + 1) dimensions. We
also make a connection between the Euclidean path integral and the generalized cluster state(ZN )for the model M(d,n). In Section 6, we show that the generalized cluster state is an SPT state.
In Section 7, we discuss an interplay of the symmetries between the bulk and the boundary
in our MBQS. Section 8 is devoted to Conclusions and Discussion. In the appendix we provesome equations used in the main text and discuss supplementary aspects of our MBQS and thegeneralized cluster states.

2

Lattice models and resource states

2.1

Cell complex notation

Let us consider a d-dimensional hypercubic lattice. Let ∆0 be the set of 0-cells (vertices),∆1 the set of 1-cells (edges), and ∆2 the set of 2-cells (faces), and so on. We write Ci(i = 0, 1, 2, ..., n) for the group of i-chains ci with Z2 coefficients (later this will be generalized
to general Abelian groups), i.e., the formal linear combinationsXci =a(ci ; σi )σi ,(5)
σi ∈∆iwith a(ci ; σi ) ∈ Z2 = {0, 1 mod 2}. Sometimes we regard the chain ci as the union of thei-cells σi such that a(ci ; σi ) = 1. The boundary operator ∂ is a linear map Ci+1 → Ci such that
∂ σi+1 is the sum of the i-cells that appear on the boundary of σi . We get a chain complex∂∂∂

Cn −→ · · · −→ C1 −→ C0 ,(6)

with ∂ 2 = 0. Similarly, by considering the dual lattice,1 we get the dual chain complex
∂∗

∂∗

∂∗

Cn∗ −→ · · · −→ C1∗ −→ C0∗ ,(7)

1
The dual lattice is obtained by identifying the center of the d-cells in the original lattice as 0-cells in the dual
lattice, then connecting the dual 0-cells by dual 1-cells, which intersect with (d − 1)-cells in the primary lattice,and so on.

6SciPost Phys. 14, 129 (2023)∂( σ2 ⟷

=
σ*
0) (

⟷

dual

)

∂*( σ1 ⟷ σ*)=(

⟷

dual

)

dual

dual

1

Figure 2: An illustration of boundary operators and duality for a square lattice. (Top)The 2-cell σ2 is represented by a shaded square. Its boundary ∂ σ2 is the sum of four
1-cells. (Bottom) The 1-cell σ1 is represented by a gray line. Its dual boundary is thesum of two 2-cells. In both figures, the duals of the relevant cells are indicated.
with (∂ ∗ )2 = 0. There are natural identifications of ∆i (i-cells) with ∆∗n−i (dual (n − i)-cells),∗
and Ci (i-chains) with Cn−i(dual (n − i)-chains).2 The notions of boundary operators andduality are illustrated in Fig. 2. We will often consider placing qubits on all the i-cells σi ∈ ∆ifor some i. Then on each σi we have Pauli operators X (σi ) and Z(σi ). For each i-chain ci wedefineY
X (ci ) :=X (σi )a(ci ;σi ) ,σi ∈∆iZ(ci ) :=Y(8)Z(σi )a(ci ;σi ) .

σi ∈∆i

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 coordinate x d (“time”)
specifies an artificial time slice. The boundaries x d = 0 and x d = L d , where L d is the linear
lattice size in the d-th direction, are examples. The bulk state to be introduced later will be theresource state for MBQS. As we proceed in the protocol of MBQS, the state originally defined on
the x d = 0 time slice will be teleported to a middle time slice x d = j, where j ∈ {0, 1, . . . , L d }.
∆,Throughout the paper, unless otherwise stated, we use the notation where the bold fonts (∆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 spaceof the simulated model.
A cell σ i inside a time slice x d = j is of the form
σ i = σi × { j} ,(9)while a cell σ i extending in the time direction takes the form
σ i = σi−1 × [ j, j + 1] .

(10)

Sometimes we express a point in the time direction as pt and an interval as I.

2.2

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 I is given by
X
I[{Sσ n−1 }] = −JS(∂∂ σ n ) ,(11)∆n
σ n ∈∆

2∗∗
Between Ci and Cn−i
, the intersection pairing is given by #(ci ∩ cn−i
)=

7

∗
∗
σi ∈∆i a(ci , σi )a(cn−i ; σn−i ) mod 2.

P

SciPost Phys. 14, 129 (2023)

where J is a coupling constant. Sσ n−1 ∈ {+1, −1} is a classical spin variable living on each(n − 1)-cell σ n−1 ∈ ∆ n−1 andY
σi )S(cc i ) =(Sσ i )a(cc i ;σ
,
(12)
∆i
σ i ∈∆

Pc i ; σ i )σσ i . This class of theories is called “generalized Isingfor a given i-chain c i = σ i ∈∆∆ i a(cmodels” in the literature [22], where the action (11) is viewed as the (classical) Hamiltonianof such a classical spin model. For n = 2, (11) is the Z2 version of the action of Wilson’s latticegauge 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,for σ n−1 ∈ ∂ ∗σ n−2 .

σ n−1 ) → −S(σσ n−1 ) ,Gσ n−2 : S(σ(13)

This is a higher-form generalization of the standard discrete gauge transformation, which corresponds to the case with n = 2. For n = 1, M(d,n) is the Ising model in d dimensions.
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 dualitychanges the global structure of the model. See, e.g., [42].
For each classical spin model in d dimensions, one can construct a quantum spin model
defined on a (d − 1)-dimensional spatial lattice. See [20] and Appendix B. The qubits areplaced on (n − 1)-cells σn−1 . The Hamiltonian is given byX
X
H(d,n) = −X (σn−1 ) − λZ(∂ σn ) ,(14)
σn−1 ∈∆n−1

σn ∈∆nwhere we used the notation (8) and λ is a coupling constant. Gauge-invariant states |ψ〉 mustsatisfy the Gauss law constraint
G(σn−2 )|ψ〉 = (−1)Q(σn−2 ) |ψ〉 ,(15)

for any σn−2 ∈ ∆n−2 , where G(σn−2 ) is defined as
G(σn−2 ) = X (∂ ∗ σn−2 ) ,(16)

and Q(σn−2 ) = 1 if there is an external charge on the cell σn−2 and Q(σn−2 ) = 0 otherwise.
Conjugation by the operator G(σn−2 ) generates a gauge transformation in the Hamiltonian
picture.

2.3

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

The Ising model M(3,1) in 2 + 1 dimensions has the HamiltonianX
X
H(3,1) = −X (σ0 ) − λZ(∂ σ1 ) .
σ0 ∈∆0(17)

σ1 ∈∆1The second term is the nearest neighbor interaction between two vertices connected by edges.
We have the following Trotter decomposition of the time evolution e−iH(3,1) t :! x3
YYiδt X (σ0 )iδtλZ(∂ σ1 )T(3,1) (t) :=
e
e
,
(18)
σ0 ∈∆0σ1 ∈∆1

with t = x 3 δt.
8

SciPost Phys. 14, 129 (2023)

2.4

Example 2: M(3,2) (Z2 gauge theory in 2 + 1 dimensions)

The Hamiltonian of the model M(3,2) , the Z2 gauge theory in 2 + 1 dimensions, is [22]H(3,2) = −XX (σ1 ) − λ

σ1 ∈∆1XZ(∂ σ2 ) .

(19)

σ2 ∈∆2The second sum is over plaquettes (faces) σ2 . The plaquette operator Z(∂ σ2 ) is the productof Pauli-Z operators on the four edges surrounding σ2 . The Gauss law constraint is
X (∂ ∗ σ0 ) = (−1)Q(σ0 ) ,(20)

where the left hand side is the product of Pauli-X operators on the edges attached to thevertex 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) t is given by! x3
YYT(3,2) (t) :=e i δt X (σ1 )e i δt λZ(∂ σ2 ),(21)
σ1 ∈∆1σ2 ∈∆2

with t = x 3 δt.

2.5

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|0〉 = |0〉 ,Z|1〉 = −|1〉 ,(22)

X |+〉 = |+〉 ,X |−〉 = −|−〉 .

(23)

We place a qubit
P on every (n − 1)-cell σ n−1 ∈ ∆ n−1 and on every n-cell σ n ∈ ∆ n . For eachc n ; σ n )σσ n , we definen-chain c n = σ n ∈∆∆ n a(cX (cc n ) :=Y∆nσ n ∈∆σn)(X σ n )a(cc n ;σ.

(24)

We similarly define Pauli Z operators and Pauli operators on (n − 1)-cells. A general Pauli
operator takes the form
(25)
P = e iα X (cc n )Z(cc ′n )X (cc n−1 )Z(cc ′n−1 ) ,where α is a c-number phase.
Now we define the stabilizers
σ n ) = X (σσ n )Z(∂∂ σ n ) ,K(σ∗σ n−1 ) = X (σσ n−1 )Z(∂∂ σ n−1 ) .
K(σ

(26)
(27)

The generalized cluster state |gCS(d,n) 〉 is defined by the eigenvalue equations
σ n−1 )|gCS(d,n) 〉 = K(σσ n )|gCS(d,n) 〉 = |gCS(d,n) 〉 ,K(σfor all σ n−1 ∈ ∆ n−1 , σ n ∈ ∆ n . (28)Explicitly, the cluster state can be written as∆ n−1 ⊔∆∆n )|gCS(d,n) 〉 = UC Z |+〉⊗(∆
,

9

(29)

SciPost Phys. 14, 129 (2023)(1) σ1 × {j}(2) σ0 × {j}

ℳ(A)

{j}

ℳ(X)

(3) σ0 × [ j, j + 1]ℳ(B)[ j, j + 1]
{j + 1}Figure 3: The unit cell in the generalized cluster state gCS(3,1) used to simulate themodel M(3,1) . The blue balls represent the qubits living on 1-cells σ 1 ∈ ∆ 1 , while theblack ones are those on 0-cells σ 0 ∈ ∆ 0 . The green arrows show the measurement
pattern in MBQS.
where UC Z is the entangler that applies controlled-Z gates (C Z gates) between qubits on adjacent qubits:Y
σ n−1 )a(∂∂ σ n ;σUC Z :=(C Zσ n−1 ,σ
,
(30)
σn )
∆ n−1
σ n−1 ∈∆∆nσ n ∈∆

with ∂ σ n =∂ σ n ; σ n−1 )σσ n−1 . The C Z gate is given by∆ n−1 a(∂σ n−1 ∈∆P

C Zc,t = |0〉c 〈0| ⊗ I t + |1〉c 〈1| ⊗ Z t .

(31)

It is invariant under the exchange of the control (c) and the target (t) qubits.

3

Measurement-based quantum simulation of gauge theory

In this section, we introduce the MBQS protocols for the real-time evolution of the Ising modelM(3,1) and the gauge theory M(3,2) . See [41] for a pedagogical introduction to MBQC.

3.1

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. 3 for an illustration. We describe the measurement protocol to simulate the time evolution with HamiltonianH(3,1) given in (17).
3.1.1

Measurement pattern

Our simulation protocol will involve two types (A and B) of measurements. Each measurement
realizes a desired unitary operator, multiplied by an extra Pauli operator that depends on
the non-deterministic measurement outcome. The desired operator simulates a factor in theTrotterized time evolution operator (18). The extra operator is called a byproduct operator andis determined by the measurement outcomes. As we will explain, we can adaptively choose themeasurement 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 x 3 = j, we have qubits on the vertices σ0 ∈ ∆0 and the edges σ1 ∈ ∆1 . See Fig. 3 (1). The
10

SciPost Phys. 14, 129 (2023)

qubits on the edges are entangled by the C Z gate with the adjacent qubits on ∂ σ1 . The wave
function |Ψ〉 for the qubits on the vertices is arbitrary. Our claim is that the unitary operator(1)

U(3,1) := (Z(∂ σ1 ))s e−iξZ(∂ σ1 ) ,(32)

is realized by measuring the qubit on σ1 with the basisM(A) = e iξX |s〉 s = 0, 1 .

(33)

Indeed,3
σ1 〈s| e−iξX σ1

1
C Zσa(∂,σσ1 ;σ0 ) |+〉σ1 |Ψ〉 = p (Z(∂ σ1 ))s e−iξZ(∂ σ1 ) |Ψ〉 .
1 0
2
σ0 ∈∆0Y(34)

We prove this equation in Appendix A. The Pauli operator Z(∂ σ1 )s is the byproduct operator(1)
from this measurement. Up to the byproduct operator and a choice of angle ξ, U(3,1) is essentially the time-evolution by the term Z(∂ σ1 ) in the Ising Hamiltonian (17).
1 We refer to s as
the measurement outcome, and the measurement in the basis (33) as A-type.
Next, we explain the B-type measurement. It is defined as the measurement with the basisM(B) = e iξZ |s̃〉 s = 0, 1 ,
(35)
1
1A
(1)
where |s̃〉 is the eigenvector of the X operator with the eigenvalue (−1)s :

2A

(3)

e

(36)
1

i⇠X
In other words, |0̃〉 1=|+〉 and |1̃〉 = |−〉. This measurement
implements
a gate teleportation.
(4)
eB(1)
(1)
i⇠Z
e
To see this, we consider a general state |Ψ〉1 and an ancilla|+〉 , and we entangle them with
2B(1)
(5)
ei⇠X 2 Theei⇠X
the C Z gate. Then we Z
measure the qubit 1 with the basis M(B) .(2)circuit is given by4

Z

3B(1)
s

e i⇠X

| i
2B(2)

e i⇠Z

3B(2)iX s e i⇠X H|

Z

1B(2)

(6)

Z

(3)

Z

s

(4)

s

(8)

(2)

| i

(5)

| i(9)

(3)

X
s

(7)

| i
X s e i⇠X H| (10)
i
s
i⇠X
X
e
H| i
s X i⇠X
X e
H| i
s

|+i

Here the realized unitary operator is(3)

U(3,1)| =iX s e−iξX H .
Indeed,1 〈s̃|e

(1)

(4)
(5)
(6)
(37)

X s e i⇠X H| i

(7)

1
C Z12 |Ψ〉1 |+〉2 = p (X 2 )s e−iξX 2 H2 |Ψ〉2 .
2

(38)

−iξZ1

The proof is given in Appendix A. The Pauli operator X s is the byproduct operator from thismeasurement. The special case with the angle ξ = 0 will be called X-type, and we use thenotationsM(X ) = {|s̃〉 | s = 0, 1 } ,sU(X ) = X H .

(39)
(40)

3
The wave function |Ψ〉 in the time slice x 3 = j is entangled by C Z gates with the qubits in the bulk x 3 ≥ j as
well. One can focus on the local effect by measurement as in eq. (34) since C Z gates commute with each other.
4
This technique is standard in the context of MBQC [43].

11

1

(2)

X |s̃〉 = (−1)s |s̃〉i⇠X
.

3A

1

(1)
(2)

(1)

(1

(2)

(2

(3)

(3

(3)
(4)
(5)
(6)
(7)

(4)
(5)

(4

(5

SciPost Phys. 14, 129 (2023)

For the time evolution from x 3 = j to x 3 = j + 1, the measurement pattern on the threedimensional cluster state gCS(3,1) for our MBQS protocol is as follows.

M(A) → M(X ) → M(B)
pt
pt
I
σ1
σ0
σ1
σ1
σ0
σ0
(3)
(1)
U(X )
U(3,1)
U(3,1)(ξ1 , s1 )(0, s2 )(ξ3 , s3 )

basis
layer
3d cell
2d cellsimulation(ξ, s)(41)

See Fig. 3. We have a set of measurement outcomes and angles, which depend on locationsof cells. We denote them as {s1 (σ1 × { j}), s2 (σ0 × { j}), s3 (σ0 × [ j, j + 1])} and {ξ1 (σ1 × { j}),
ξ3 (σ0 × [ j, j + 1])}, respectively, where the subscripts indicate the order of the corresponding
measurements within the time step. To avoid clutters, we often make the cell dependence of
these parameters implicit. The total unitary operator for one time step is
Y
Y
(3)
(1)
U(3,1) ({ξi }) =U(3,1) U(X )U(3,1)(42)
σ0 ∈∆0=Y

σ1 ∈∆1s3 (σ0 ) −iξ3 X (σ0 )(X (σ0 ))e

H(σ0 )(X (σ0 ))s2 (σ0 ) H(σ0 )σ0 ∈∆0Y(Z(∂ σ1 ))s1 (σ1 ) e−iξ1 Z(∂ σ1 )σ1 ∈∆1

(43)
=

Y

Y

(X (σ0 ))s3 (σ0 ) e−iξ3 X (σ0 ) (Z(σ0 ))s2 (σ0 )σ0 ∈∆0(Z(∂ σ1 ))s1 (σ1 ) e−iξ1 Z(∂ σ1 ) .

(44)

σ1 ∈∆1

As in the usual protocols of measurement-based quantum computation, the outcomes of measurements with bases M(A) and M(X ) should be collected before performing the measurementswith M(B) , and the parameter ξ3 for each σ0 should be chosen so that the unitary gate of the
X rotation is as wanted. Concretely, we choose the parameters in the first step as follows:ξ1 (σ1 ) = −λδt ,(45)
σ1 ∈∆1 s1 (σ1 )a(∂ σ1 ;σ0 )P

ξ3 (σ0 ) = −(−1)s2 (σ0 ) (−1)

δt .

(46)

We use the relationY
σ0 ∈∆0e−iξ3 X (σ0 ) (Z(σ0 ))s2 (σ0 )Y(Z(∂ σ1 ))s1 (σ1 ) =σ1 ∈∆1Y(Z(σ0 ))s2 (σ0 )σ0 ∈∆0Y(Z(∂ σ1 ))s1 (σ1 )σ1 ∈∆1Ye−iδt X (σ0 ) , (47)σ0 ∈∆0

to propagate the byproduct operators forward. Then U(3,1) ({ξi }) in eq. (44) becomesYYΣ(1) ({s})
e iX (σ0 )δte iλZ(∂ σ1 )δt ,σ0 ∈∆0(48)

σ1 ∈∆1

where Σ(1) ({s}) is the product of all the byproduct operators from the 1st time step, i.e.,YYΣ(1) ({s}) =X (σ0 )s3 Z(σ0 )s2Z(∂ σ1 )s1 ,
(49)
σ0

σ1

and the remaining product of operators is T(3,1) (δt) defined in eq. (18).
In the following steps, we continue with the same measurement pattern, except that themeasurement angles are adjusted according to the former measurement outcomes as we propagate the byproduct operators to the frontmost position. After j Trotter steps we have!
jYΣ(k) ({s}) |ψ(t)〉 , (t = jδt) ,
(50)
k=1

12

SciPost Phys. 14, 129 (2023)

with Σ(k) ({s}) being the byproduct operators coming from the k-th time step and|ψ(t)〉 = T(3,1) (t)|ψ(0)〉. Performing another time step gives us
!
!
j
j+1
Y
Y
U(3,1) ({ξi })Σ(k) ({s}) |ψ(t)〉 =Σ(k) ({s}) |ψ(t + δt)〉 ,
(51)
k=1

k=1

by choosing {ξi } appropriately according to the preceding byproduct operators. In the end,
we have
L

3
Y
(k)
Σ ({s}) |ψ(T )〉 ,(T = L3 δt) ,
(52)
k=1

and the effect of the total byproduct operator can be removed by the post processing.

3.2

Simulation of M(3,2)

In this section we present the MBQS protocol for the Z2 lattice gauge theory M(3,2) . The
resource state is gCS(3,2) , whose qubits are placed on 2-cells (faces) and 1-cells (edges), andthe entanglers are applied appropriately. This state is also known as the Raussendorf-BravyiHarrington (RBH) cluster state [44]. See Fig. 4 for illustration.
3.2.1

Measurement patternFirst, let us focus on a 2-cell (face) σ2 and the four 1-cells (edges) ∂ σ2 surrounding it, in alayer at the level x 3 ∈ {1, ..., L3 }, As a straightforward generalization of the simulation of theinteraction term in (32), we find that the simulation of the plaquette term with a byproductoperator(1)

U(3,2) := (Z(∂ σ2 ))s e−iξZ(∂ σ2 ) ,is induced by measuring the qubit on σ2 with the basisM(A) := e iξX |s〉 s = 0, 1 .

(53)

(54)

In other words,
σ2 〈s|e−iξX σ2

1
C Zσa(∂,σσ2 ;σ1 ) |+〉σ2 |Ψ〉 = p (Z(∂ σ2 ))s e−iξZ(∂ σ2 ) |Ψ〉 .
2 1
2
σ1 ∈∆1Y(55)

Here, |Ψ〉 is a general wave function of qubits defined on 1-cells at x 3 .
We have already seen in Section 3.1 that we can simulate the X -rotation gate with a teleportation. We utilize
M(B) = e iξZ |s̃〉 s = 0, 1 ,
(56)
(4)

U(3,2) := X s e−iξX H ,

(57)

and

M(X ) = {|s̃〉 | s = 0, 1} ,s

U(X ) := X H .

(58)
(59)
(G)

Now we present our measurement pattern in Table 1. The measurement basis M(3,2) ,which we will explain below, is used to enforce gauge invariance, See Fig. 4 for an illustration.
13

SciPost Phys. 14, 129 (2023)

Table 1: Measurement pattern for (2+1)d Z2 gauge theory. In the third step, M(A)is used for Method (i) (with energy cost terms) and M(X ) is used for Method (ii)(with syndromes).

3.2.2

basis
layer
3d cell
2d cellsimulation

M(A)
pt
σ2
σ2
(1)
U(3,2)(ξ, s)(ξ1 , s1 )

M(X )
pt
σ1
σ1
U(X )

→

→



M(A)

M(X )



I
σ1
σ0
Gauss law
(ξ3 , s3 ) (0, s3 )(0, s2 )

M(B)
I
σ2
σ1
(2)
U(3,2)→(ξ4 , s4 )Enforcing gauge invariance

In simulating gauge theory dynamics on real devices, a time evolution that violates gaugeinvariance, or more precisely the Gauss law constraint (20), would be induced due to thenoise that occurs to physical qubits. We consider the following two methods to enforce gauge
invariance. (i) Add an energy cost term for violating gauge invariance to the Hamiltonian of
the simulated gauge theory:
X
X
X
H =−
X (σ1 ) − λZ(∂ σ2 ) − Λ(−1)Q(σ0 ) X (∂ ∗ σ0 ) ,(60)
σ1 ∈∆1σ2 ∈∆2σ0 ∈∆0where the coefficient Λ is taken so large that the cost term becomes much more significant thanthe other terms but is small enough so that the Trotter decomposition is justified. We expect,
but do not prove, that such a cost term would suppress the violation of gauge invariance.
See [25, 45–52] for the study of similar cost terms.
(ii) Actively correct the errors based on the measurement outcomes, much in the spiritof topological quantum memory [53]. In the following we explain how the protocols for thetwo methods work in the error-free situation. In Section 4, we will explain the protocols in a
specific error model in detail.
(i) Gauss law enforcement by energy cost. In the first method, we use the measurement(G)basis M(3,2) = M(A) so that it induces
(G,i)

∗

U(3,2) := (Z(∂ ∗ σ0 ))s e−iξZ(∂ σ0 ) .

(61)

In this case, the total unitary for a single time step in a unit 2-cell in 2d becomes
Y
Y
Y
Y
(4)
(G,i)
(1)
(i)
U(3,2)({ξi }) :=U(3,2)
U(3,2)U(X )U(3,2)σ1 ∈∆1=Y

σ0 ∈∆0σ1 ∈∆1σ2 ∈∆2X (σ1 )s4 (σ1 ) e−iξ4 X (σ1 ) H(σ1 )σ1 ∈∆1×s2 (σ1 )X (σ1 )Y

H(σ1 )σ ∈∆1=

X (σ1 )Yσ1 ∈∆1Z(∂ σ2 )s1 (σ2 ) e−iξ1 Z(∂ σ2 )σ2 ∈∆2s4 (σ1 ) −iξ4 X (σ1 )eσ1 ∈∆1×∗Z(∂ ∗ σ0 )s3 (σ0 ) e−iξ3 Z(∂ σ0 )σ0 ∈∆0

Y

1
Y

Y

Y

∗

X (∂ ∗ σ0 )s3 (σ0 ) e−iξ3 X (∂ σ0 )σ0 ∈∆0Z(σ1 )s2 (σ1 )YZ(∂ σ2 )s1 (σ2 ) e−iξ1 Z(∂ σ2 ) .

σ2 ∈∆214(62)

SciPost Phys. 14, 129 (2023)ℳ(A)(1) σ2 × {j}(3) σ0 × [ j, j + 1]ℳ(X)(2) σ1 × {j}(3) σ1 × [ j, j + 1]
(4)ℳ(A) or ℳ(X)ℳ(B)

Figure 4: The protocol for the model M(3,2) . The black balls represent the qubits
living on 2-cells σ 2 ∈ ∆ 2 and the blue ones are those on 1-cells σ 1 ∈ ∆ 1 .
We write the wave function with the Trotterized time evolution as(i)|ψ(t)〉 = T(3,2)(t)|ψ(0)〉 ,
Y
Y
 Y
j
Qσ
∗
0
(i)
T(3,2)(t) =e iX (σ1 )δte iΛ(−1) X (∂ σ0 )δte iλZ(∂ σ2 )δt ,σ1 ∈∆1σ0 ∈∆0(63)(t = jδt) , (64)σ2 ∈∆2and denote the by product operators coming from the j-th time step as Σ( j) . Then we obtainthe following for ( j + 1)-th step by appropriately choosing {ξi }i=1,2,4 :(i)U(3,2)({ξi })jY!
Σ(k) |ψ(t)〉 =

k=1

j+1
Y

!
Σ(k) |ψ(t + δt)〉 .

(65)

k=1

(ii) Gauss law enforcement by error correction. The second method is well known in the(G)context of the topological MBQC [54–56]. We use the measurement basis M(3,2) = M(X ) . In
this case we obtainY
1 X
a(∂ ∗ σ0 ;σ1 )
〈s̃|
C
Z
|+〉
|Ψ〉
=
(−1)as (Z(∂ ∗ σ0 ))a |Ψ〉 .
(66)
σ0
σ0
σ0 ,σ1
2
σ ∈∆
a=0,1
1

1

Here ∂ ∗ σ0 is a summation of 1-cells (edges) that surround the 0-cell (vertex) and |Ψ〉 is a
general wave function of qubits defined on 1-cells at the time interval [ j, j + 1] (2-cells inthree dimensions). Thus we obtain(G,ii)

P(3,2) =1 X(−1)as (Z(∂ ∗ σ0 ))a .
2 a=0,1

15(67)

SciPost Phys. 14, 129 (2023)

Then the total operator for a single time step in 2d becomes
Y
Y
Y
Y
(1)
(G,ii)(4)
U(X )U(3,2)
P(3,2)
U(3,2)Y=σ1 ∈∆1σ0 ∈∆0σ1 ∈∆1σ2 ∈∆2X (σ1 )s4 (σ1 ) e−iξ4 X (σ1 ) H(σ1 )σ1 ∈∆1×

σ0 ∈∆0Ys2 (σ1 )X (σ1 )=X (σ1 )s4 (σ1 ) e−iξ4 X (σ1 )Y
σ1 ∈∆1

1 X(−1)as3 (σ0 ) Z(∂ ∗ σ0 )a2 a=0,1!

Z(∂ σ2 )s1 (σ2 ) e−iξ1 Z(∂ σ2 )(68)

σ2 ∈∆2σ1 ∈∆1×Y

H(σ1 )σ1 ∈∆1YY

1 X(−1)as3 (σ0 ) X (∂ ∗ σ0 )a2 a=0,1Y
σ0 ∈∆0Z(σ1 )s2 (σ1 )Y!

Z(∂ σ2 )s1 (σ2 ) e−iξ1 Z(∂ σ2 ) .

(69)

1 X
(−1)as(σ0 ) X (∂ ∗ σ0 )a2 a=0,1(70)

σ2 ∈∆2The operator
P(σ0 ; s) :=is a projector that restricts the measurement outcome s3 (σ0 ) to be the eigenvalue of X (∂ ∗ σ0 )of the simulated state. At the j-th time step ( j ≥ 0), assume that the measurement outcome
was s3 (σ0 × [ j, j + 1]) = x(σ0 ) with x(σ0 ) ∈ {0, 1}. Then at the ( j + 1)-th time step, since
the Z byproduct operators from the measurements at σ1 × { j + 1} may flip the eigenvalue ofX (∂ ∗ σ0 ), the measurement outcome becomesX
s3 (σ0 × [ j + 1, j + 2]) = x(σ0 ) +a(∂ ∗ σ0 ; σ1 )s2 (σ1 × { j + 1}) .
σ1 ∈∆1

Therefore we obtain the following relation for the error-free MBQS:X
s3 (σ0 × [ j, j + 1]) + s3 (σ0 × [ j + 1, j + 2]) +a(∂ ∗ σ0 ; σ1 )s2 (σ1 × { j + 1}) = 0 ,(71)

σ1 ∈∆1

with j ≥ 0. In Section 4, we introduce an error model and consider an error correction scheme,
and we will see that the left-hand side of the relation above serves as a syndrome for the error
correction.
We remark that the byproduct operators are handled in the same manner as before. The
parameters {ξ1 , ξ4 } are chosen accordingly to the former measurement outcomes.

3.3

MBQS of imaginary time evolution

The properties of the ground state of a gauge theory can be an interesting target of study. The
imaginary-time evolution with Hamiltonian H can be used to prepare the ground state via
e−τH|Ψin 〉 ,τ→∞ Tr(e −τH )|GS〉 = lim(72)

from a generic initial state |Ψin 〉. However, it is non-trivial to implement the imaginary-time
evolution on a quantum computer because e−τH is not unitary. In this subsection, we wish
to explain how we can perform the imaginary-time evolution of the gauge theory M(3,2) in
MBQS by including ancillary qubits and allowing us to implement two-qubit measurements.

16

SciPost Phys. 14, 129 (2023)

A method to realize any linear operator for qudit systems using measurements is given in theliterature, see e.g. [57–59].5
Let us consider gCS(3,2) , the cluster state on the three-dimensional lattice with qubits on
σ 2 and attach the state
the 1- and 2-cells. For each 2-cell σ 2 , we consider a qubit on its copy σ̃
|0〉σ̃
as
a
directproducttotheclusterstate:
σ2|gCS(3,2) 〉 ⊗ |0〉⊗∆2 .
˜(73)

˜ 2 is a copy of the set of 2-cells ∆ 2 .
Here ∆For the plaquette interaction with measurement at σ 2 = σ2 × pt, we generalize the singlequbit A-type measurement basis (54) to a set of the two-qubit measurement basis that includesÆ|φ1A〉 = e−α eαX a |0, 0〉a,b + sinh(2|α|)|−, 1〉a,b ,
(74)


Æ
|φ2A〉 = e−α eαX a |1, 0〉a,b − sgn(α) sinh(2|α|)|−, 1〉a,b ,(75)
where a and b refer to two qubits. They are both normalized and mutually orthogonal,A〈φiA|φ Aj〉 = δi j . Two other states |φ3,4〉 can be constructed so that the entire basis {|φ Aj〉}4j=1 isorthonormal. When the measurement is successful, i.e. if the outcome is |φ Aj〉 with j = 1 or
j = 2, the non-unitary operator
1
V(α,M ) := p e−α Z(∂ σ2 )s eαZ(∂ σ2 ) ,2(76)with s = j − 1 is implemented. Indeed, we have the equality
Aσ2 ,σ̃2 〈φ j |1
C Zσa(∂,σσ2 ;σ1 ) |+〉σ2 |0〉σ̃2 |Ψ〉 = p e−α Z(∂ σ2 )s eαZ(∂ σ2 ) |Ψ〉 ,2 1
2
σ1 ∈∆1Y(77)

where |Ψ〉 is a general wave function defined on 1-cells within a time slice. See Fig. 5 for an
illustration. Writing the cluster state with free ancillary qubits in the middle of simulation as|ϕ〉, the probability of obtaining j = 1 or j = 2 is
 e−2α
prob( j; α) := Tr |φ Aj〉(σ2 ,σ̃2 ) (σ2 ,σ̃2 ) 〈φ Aj| · |ϕ〉〈ϕ| =Tr e2αZ(∂ σ2 ) |ϕ̄〉〈ϕ̄| ,2
where |ϕ̄〉 is defined via the relationY|ϕ〉 =C Zσa(∂,σσ2 ;σ1 ) |+〉σ2 |0〉σ̃2 |ϕ̄〉 .
σ1 ∈∆1

2

1

(78)

(79)

−4α

Thus we have e 2 ≤ prob( j; α) ≤ 12 . Thus, the probability of finding either j = 1, 2 becomes
exponentially close to 1 when α is small:prob(1; α) + prob(2; α) ≳ e−4α .

(80)This means that as we take the Trotter step small, the probability of success for each measurement is exponentially close to 1. On the other hand, the success probability of the entire
algorithm becomes small as we increase the total time to be simulated.
5

Other methods for performing the imaginary-time evolution on quantum computers are discussed in, for e.g.,[60–62].

17

b
a

| Ψ⟩

SciPost Phys. 14, 129 (2023)| + ⟩a | 0⟩b

b
a

| Ψ⟩

| Ψ⟩a

a

b

| 0⟩b

c | + ⟩cFigure 5: Measurements in the imaginary time MBQS. The gray balls represent the
ancillary qubits to implement the imaginary time evolution, and the black lines indicate the C Z gates for the resource state. (Left) Setup for the A-type measurement
to implement eαZ(∂ σn ) . We measure the two qubits encircled by the purple line in
the basis |φ Aj=1,2,3,4 〉. If |φ Aj=1,2 〉 comes out, the measurement is a success. If it fails,
one should throw away the result from the simulation. (Right) Setup for the B-type
measurement to implement eαX (σn−1 ) . We measure the two qubits encircled by the| Ψ⟩|φa Bj=1,2,3,4| 0⟩b the measurement is a success when wea 〉.b Likewise,
purple line in the basis
Bobtain |φ j=1,2 〉.

| + ⟩cFor the imaginary time evolution corresponding
to the X term in the Hamiltonian (19), we
generalize the B-type measurement basis (56) to a two-qubit measurement basis {|φ Bj 〉}4j=1
includingÆ|φ1B 〉 = e−α eαZa |+, 0〉a,b + sinh(2|α|)|1, 1〉a,b ,(81)
Æ|φ2B 〉 = e−α eαZa |−, 0〉a,b − sgn(α) sinh(2|α|)|1, 1〉a,b ,(82)

cwhich satisfy 〈φiB |φ Bj 〉 = δi j . We measure with this basis the two-qubits a, b in the state|ϕ〉 = C Za,c |Ψ〉a |0〉 b |+〉c . See Fig. 5. We obtain for j = 1, 2 (s = 0, 1)
B
a,b 〈φ j |1
C Za,c |Ψ〉a |0〉 b |+〉c = p e−α (X c )s eαX c H c |Ψ〉c ,2(83)

where a = σ1 × I is a 1-cell stretching in the time direction, b = ã is its ancillary qubit,
and c = σ1 × pt is the 1-cell in the next time slice. The probability to find either j = 1, 2 isprob(1, α) + prob(2, α) ≳ e−4α .

4Enforcement of Gauss law constraint against errors

In Section 3.2, we presented two measurement patterns for the MBQS of the model M(3,2) , theZ2 gauge theory in (2 + 1) dimensions. The two, Methods (i) and (ii), differed in how to deal
with the Gauss law constraint. The protocols for the two methods were explained assumingthat there were no errors. In this section, we analyze the effect of errors with the following
simplified error model where we assume that the measurement is always perfect while theresource state may be affected by the phase and bit flip errors. Namely, we consider a faulty
resource state|ψ E 〉 := Z(ee 1 )Z(ee 2 )X (ee ′1 )X (ee ′2 )|ψ〉 ,(84)|ψ〉 = UC Z (|ψ2d 〉 ⊗ |+〉bulk ) ,

(85)

where

18

SciPost Phys. 14, 129 (2023)

and e n (ee ′n ) are the n-chains which support Z (X ) errors. We assume that the state |ψ2d 〉 isdefined on the plane x 3 = 0, and that e 1 does not include cells in ∆1 ×{L3 }∪∆0 ×[L3 −1, L3 ].6
Here we use abbreviated notations such as {σ0 ×{ j}|σ0 ∈ ∆0 } =: ∆0 ×{ j}. The ideas presented
here can be generalized to the MBQS of imaginary-time evolution.

4.1

Gauss law enforcement by error correction

Here we explain how the gauge invariance violation of the simulated state can be suppressedby syndrome measurements and error correction. The essence of ideas is adopted from theliterature [53–55, 63]. We make a connection to the symmetry of the resource state, which
will play an important role in identifying an SPT order of the state in Section 6.
We focus on the real-time evolution, and examine how the error chains affect the timeevolution operators. We claim that only the phase error on the bulk 1-chain Z(ee 1 ) leads to theviolation of the Gauss law constraint, but one can perform error corrections to suppress contributions that violate the gauge invariance. The other types of errors Z(ee 2 ), X (ee ′1 ), and X (ee ′2 ) do notaffect the eigenvalue of X (∂ ∗ σ0 ). They affect the time evolution but the faulty time evolution
operator which we denote as U E+R (t) still commutes with X (∂ ∗ σ0 ).
4.1.1

Wave function without correction(ii)
Let T(3,2)(t) (t = x 3 δt) be the time evolution unitary we wish to realize in the model M(3,2)
with Method (ii) in the error-free situation:
 x3

Y
Y
(ii)
−i ξ̃1 Z(∂ σ2 )−i ξ̃4 X (σ1 )
e
,
(86)
e
T(3,2) (t) :=σ2

σ1with ξ̃1 = −λδt and ξ̃4 = −δt. This is simply T(3,2) (t) in (21). (The parameters ξ̃1 and ξ̃4should not be confused with the bare parameters that define the measurement angles, {ξ1 , ξ4 },whose signs are chosen depending on the preceding measurement outcomes {s1 , s2 , s4 }.)With a faulty resource state, we find that the following state is induced at the new boundary
at x 3 :
(x )

′(x )

(x )

′(x )

(x )

′(x )

Z(e1 3 )X (e1 3 )Z(b1 3 )X (b1 3 )U E (t)|ψ2d 〉∆1 ×{x 3 } .

(87)Here the operators Z(b1 3 )X (b1 3 ) are the byproduct operators resulting from the former
(x )

′(x )

measurements. The error Z(e1 3 ) and X (e1 3 ) are “extra byproduct operators” caused byerrors, given later in eq. (92) and (93). The explicit form of the faulty unitary U E (t) will begiven in eq. (91). In following paragraphs, we explain details of the extra byproduct operatorsand the faulty time evolution.
4.1.2

Direct effect on time evolution

The first effect of errors is direct influence of the error chains on the unitaries. We note that,for A-type measurements, an X error changes the measurement outcome compared to the one
without it, s → s + 1, while a Z error changes the angle, ξ → −ξ. For B-type measurements,
6

The latter assumption is necessary for the final state to be gauge invariant. This is because the error chains on
the final time slice are not detected by syndrome measurements. The total error rate for the boundary would be
proportional to its area and would be asymptotically small compared to that for the bulk, which would scale with
its volume.

19

SciPost Phys. 14, 129 (2023)

Table 2: Effect of errors as changes from (ξ, s).
A-typeB-typeσ1 × { j}σ2 × { j}cells

σ1 × [ j, j + 1]
σ0 × [ j, j + 1]

X error(ξ, s + 1)(−ξ, s)Z error(−ξ, s)(ξ, s + 1)

an X error changes the angle, ξ → −ξ, while a Z error changes the eigenvalue, s → s + 1. See
Table 2. For example, the A-type measurement at σ2 × { j} is affected by a Z error asY−iξ1 X σ2
〈s|e
Z(σ
)
C Zσa(∂,σσ2 ;σ1 ) |+〉σ2 |Ψ〉σ22
2σ1 ∈∆1= σ2 〈s|(−1)s e−i(−ξ1 )X σ2Yσ1 ∈∆1

1

C Zσa(∂,σσ2 ;σ1 ) |+〉σ2 |Ψ〉21

1= p (−1)s (Z(∂ σ2 ))s e−i(−ξ1 )Z(∂ σ2 ) |Ψ〉 .
2(88)

The sign change is inherited from ξ1 here to ξ̃1 in (86). Relative to eq. (86), the sign of thej-th (from the right when the power is written as the product of x 3 factors) ξ̃1 in U E (t) is
flipped if e 2 in (84) contains σ2 × { j}. Similarly the sign of the j-th ξ̃4 is flipped if e ′2 contains
σ1 × [ j, j + 1].
4.1.3

Extra byproduct operators

The other effect is indirect and comes from the extra byproduct operators caused by errors,
which flip the signs of the angles as we strip them off to the left of the time-evolution unitaries
(x )
′(x )
(which were expressed as Z(e1 3 )X (e1 3 ) in eq. (87)). More concretely, a similar calculation
as we did in eq. (88) reveals that the X error chains on σ2 × { j} and Z error chains on σ1 × { j}give rise to extra Z-byproduct operators, and they flip the sign ξ̃4 → −ξ̃4 as we move them tothe front. Likewise, the Z error chains on σ1 × [ j, j + 1] cause extra X -byproduct operatorsthat flip the sign ξ̃1 → −ξ̃1 .
Now we give the explicit formula of the unitary U E (t). We define the faulty parameters asξ̃1E (σ2 , j) := ξ̃1 (−1)#(ee 2 ∩σ2 ×{ j})+#(ee 2 ∩Pj
k=0#(ee ′2 ∩σ1 ×[ j, j+1])+#(ee ′2 ∩ξ̃4E (σ1 , j) := ξ̃4 (−1)∂ σ2 ×[k,k+1])

Pj
k=0

∗

,

∂ σ1 ×{k})+#(ee 1 ∩

(89)
Pj
k=0

σ1 ×{k}),(90)

where the intersection pairing (see footnote below eq. (7)) should be taken by regarding oneof the pair of chains as its dual. Then the faulty evolution unitary is given by
U E (t) =


xY
3 −1
Y
j=0

e

−i ξ̃4E (σ1 , j)X (σ1 )

σ1

Y

e

−i ξ̃1E (σ2 , j)Z(∂ σ2 )


,

(91)

σ2

where the product is ordered from right to left as j increases. We also give explicit formulas

20

SciPost Phys. 14, 129 (2023)

Figure 6: (Left) The top layer corresponds to the boundary x 3 = 0. The five balls
represent the 1-cells contained in ∂ ∗σ 0 in eq. (97). The shaded planes represent thedual boundary of the dual 3-cell truncated at the boundary. (Right) The six balls
represent the 1-cells contained in ∂ ∗σ 0 in eq. (99). The shaded planes represent thedual boundary of the dual 3-cell.
for the chains of extra byproduct operators:
( j)

e1 =
′( j)
e1 =

j
X
X
k=0 σ2 ∈∆2
j
X
X

j
X
X 



# e 1 ∩ σ1 × {k} × σ1 ,# e ′2 ∩ σ2 × {k} × ∂ σ2 +

(92)



# e 2 ∩ σ1 × [k, k + 1] × σ1 .

(93)

k=0 σ1 ∈∆1

k=0 σ1 ∈∆1

Crucially, the structure of the Pauli operators that appear in exponents of U E (t) is the sameas the error-free time evolution unitary, thus the faulty time-evolution unitary does commute
with the Gauss law generator:[G(∂ ∗ σ0 ), U E (t)] = 0 .

(94)

Also note in eq. (92) and (93) that the only contribution that does not commute with G(σ0 )is caused by the phase error Z(σ1 × { j}). Besides, the syndrome measurement results areaffected only by the error Z(σ0 × [ j, j + 1]). This confirms our claim we made at the beginningof the section that only phase errors on the 1-chain Z(ee 1 ) cause the effect that is related to theviolation of the gauge invariance.
4.1.4

Syndromes and symmetry of resource stateConsider first the error-free resource state. We decompose the bulk qubits into|+〉bulk = |+〉∆0 ×[0,1] ⊗ |+〉other .
We write|ψ〉 = UC Z (|ψ2d 〉∆1 ×{0} ⊗ |+〉∆0 ×[0,1] ⊗ |+〉other ) .

(95)

We note that for each σ0 ×{0} the product of the X -basis measurement results over ∂ ∗ σ0 ×{0}and σ0 × [0, 1] is forced to be (−1)Q(σ0 ) . This is because|ψ〉 = UC Z (−1)Q(σ0 ) X (∂ ∗ σ0 )|ψ2d 〉∆1 ×{0} ⊗ X (σ0 )|+〉∆0 ×[0,1] ⊗ |+〉other= (−1)Q(σ0 ) X (∂ ∗ σ0 × {0})X (σ0 × [0, 1])|ψ〉 ,
21

(96)

SciPost Phys. 14, 129 (2023)

where we used the relation X (∂ ∗ σ0 )|ψ2d 〉 = (−1)Q(σ0 ) |ψ2d 〉. In other words, the perfect resource state satisfies|ψ〉 = (−1)Q(σ0 ) X (∂∂ ∗σ 0 )|ψ〉 ,for all σ0 = σ0 × {0} ∈ boundary 0-cell .

(97)

See Fig. 6 (Left). In terms of the measurement outcomes {s2 , s3 }, with the perfect resource
state, it should be satisfied thatXs3 (σ0 × [0, 1]) +a(∂ ∗ σ0 ; σ1 )s2 (σ1 × {0}) + Q(σ0 ) = 0 .
(98)
σ1 ∈∆1

As for the bulk, the corresponding relations are|ψ〉 = X (∂∂ ∗σ 0 )|ψ〉for all σ0 ∈ bulk 0-cell ,(99)(see Fig. 6 (Right)) andXs3 (σ0 × [ j, j + 1]) + s3 (σ0 × [ j + 1, j + 2]) +a(∂ ∗ σ0 ; σ1 )s2 (σ1 × { j + 1}) = 0 , (100)σ1 ∈∆1

which is exactly the relation eq.(71). The relation (99) can be regarded as a consequenceof a 1-form symmetry of which X (∂∂ ∗σ 0 ) is one of generators. In Section 6, we will see thatthe state gCS(3,2) belongs to a nontrivial SPT phase protected by this symmetry (together withanother 1-form symmetry).
With the Z errors on a 1-chain e 1 , the relations eq. (98) and (100) are modified as follows:Xs3 (σ0 × [0, 1]) +a(∂ ∗ σ0 ; σ1 )s2 (σ1 × {0}) + Q(σ0 ) = # ∂ ∗σ 0 ∩ e 1 ,(101)
σ1 ∈∆1

with σ0 = σ0 × {0}, ands3 (σ0 × [ j, j + 1]) + s3 (σ0 × [ j + 1, j + 2]) +Xa(∂ ∗ σ0 ; σ1 )s2 (σ1 × { j + 1}) = # ∂ ∗σ 0 ∩ e 1 , (102)σ1 ∈∆1

with σ0 = σ0 × { j + 1} and j ≥ 0.
The combination of measurement outcomes for X (∂∂ ∗σ 0 ) on the left hand side of eq. (101)and (102) serves as the syndrome for our error correction, and we use these relations to infer
the locations of the endpoints of the error chains. When the overlap of a Z error 1-chain with
∂ ∗σ 0 is even, the error does not change the sign of the eigenvalue of X (∂∂ ∗σ 0 ). However, theeigenvalue is flipped when the overlap is odd. Therefore one can identify the endpoints (0cells) of the Z error 1-chains using the X -measurement results on 1-chains (Fig. 7 (a)). Basedon the locations of the 0-cells, one can construct the recovery 1-chains r 1 ∈ ∆ 1 by connecting
them with the shortest paths. Importantly, the recovery chain satisfies ∂ (rr 1 +ee 1 ) = 0. One canuse the minimal-weight perfect matching algorithm [64, 65], for example, to find such paths(Fig. 7 (b)).
4.1.5

Final simulated state

We divide the total time steps into M sets of steps: L3 = ℓ1 + · · · + ℓ M , with M and ℓi both
positive integers. We define a stack Si of layers (consisting of cells that are relevant for error
correction) as

Si =ℓ1 +...+ℓ[ i −1(i = 1, ..., M ) ,(103)

L j = (∆0 ∪ ∆1 ) × { j} ∪ ∆0 × [ j, j + 1] .

(104)

j=ℓ1 +...+ℓi−1

Lj ,

22

SciPost Phys. 14, 129 (2023)

(a)

(b)

(c)

Figure 7: An example of the Gauss law enforcement. The time direction is verticalfrom top to bottom. (a) The purple lines represent Z(ee 1 ), the Z errors on 1-cells.
The blue balls represent 0-cells where the number of error chains surrounding it isodd, meaning that the set of X eigenvalues does not satisfies the symmetry equations:X (∂∂ ∗ σ0 ) ̸= 1 (bulk) or X (∂∂ ∗ σ0 ) ̸= (−1)Q(σ0 ) (boundary). (b) The cyan lines are the
1-chains that connect two balls with the shortest paths, which we use as recovery
chains Z(rr 1 ). (c) The error and recovery chains are projected to the final time slice.
The net contribution from the errors and their recovery chain is a 1-cycle, Z(z1 ).
Here we define ℓ0 = 0 and use abbreviated notations such as {σ0 × { j}|σ0 ∈ ∆0 } =: ∆0 × { j}.
We choose {ℓi } so that in every Si we have an even number of endpoints of Z error chains. Once
we perform the MBQS measurements on a stack Si , we analyze the measurement outcomes at
1-cells in it, as explained above. One can construct the recovery 1-chains by connecting 0-cells
at which the error is detected via relations eq. (101) and (102), within Si . When the error ison a 1-chain σ0 × [ j, j + 1] in the last layer of Si , the sum of the error chain and the recovery
chain may not be a 1-cycle. Continuing construction of recovery chains for M steps, however,the sum of the total recovery chain and the total recovery chain becomes a 1-cycle, given theassumption that there is no Z errors at ∆1 × {L3 } ∪ ∆0 × [L3 − 1, L3 ].
We treat the recovery chains constructed in Si as the same manner as the byproduct operators in the simulation after Si . In particular, the parameter ξ4 is adjusted according to the( j)recovery chain, which would suppress the impact of errors. Let r1 be the projection of r 1 tothe boundary, ∆1 × { j} (Fig. 7 (c)):
( j)

r1 =

j
X


# r1 ∩
σ1 × {k} × σ1 .

X
σ1 ∈∆1

k=0

Importantly, the sum of projected error and recovery chains is a 1-cycle on the time slice at
x 3 = L3 ,
(L )

(L )

(L )

z1 3 = r1 3 + e1 3 .

(105)

When we reach x 3 = L3 , we post-process the byproduct operator as well as the final recovery chain. The final state is thus
(L )

′(L )

|ψfin 〉 = Z(z1 3 )X (e1 3 )U E+R (L3 δt)|ψ2d 〉∆1 ×{L3 } ,

23

(106)

SciPost Phys. 14, 129 (2023)

where U E+R (t) is given by
E+R

U

(t) =


xY
3 −1
Y

e

−i ξ̃4E+R (σ1 , j)X (σ1 )

σ1

j=0

Y

e

−i ξ̃1E+R (σ2 , j)Z(∂ σ2 ),ξ̃1E+R (σ2 , j) := ξ̃1E (σ2 , j) ,(108)

ξ̃4E+R (σ1 , j) := ξ̃4E (σ1 , j) · (−1)#(rr 1 ∩λλwith t = x 3 δt and
λ

( j)

¨
:=

(107)

σ2

Pℓ( j)
k=00,

σ1 × {k} ,

( j)

)

,

( j ≥ ℓ1 ) ,( j < ℓ1 ) ,

(109)

(110)

PA j
where ℓ( j) = m=1 ℓm with A j the largest integer such that ℓ( j) ≤ j.
The quantum state prepared with the simulation with the error-correction procedure thus
satisfies the Gauss law constraint. We remark that, if we relax the assumption that there
is no error at the final layer, endpoints in ∆0 × {L3 } of error chains in the vicinity of theboundary cannot be identified. Also there can be an odd number of endpoints in the final
stack S M . (The latter should be properly handled by selecting an even number of endpoints tobe corrected.) In general, due to the presence of the error chain one of whose endpoints is not
identified, the sum of the recovery and error chain would not be a 1-cycle, but there would be
extra contributions of open 1-chains whose endpoints are at the boundary. Such open chains
will violate the Gauss law constraint. When the error rate is small, the errors that cause theviolation will be localized near the boundary x 3 = L3 .

4.2

Gauss law enforcement by energy cost

In the same setup as above, we consider MBQS with the energy cost term. Again, we focus onZ(ee 1 ). Here we consider the time x d = j and e0 (e1 ) is the restriction of e 1 to [ j, j + 1] ({ j}).
The state after one step of the procedure with the faulty resource state readsYYa(σ0 ;e0 )ξ3 X (∂ ∗ σ0 )X (σ1 )s4 (σ1 ) e−iξ4 X (σ1 )X (∂ ∗ σ0 )s3 (σ0 ) e−i(−1)σ1 ∈∆1×Y

σ0 ∈∆0s2 (σ1 )+a(σ1 ;e1 )Z(σ1 )σ1 ∈∆1Yσ2 ∈∆2Z(∂ σ2 )s1 (σ2 ) e−iξ1 Z(∂ σ2 ) Z(b1 )X (b1′ )|ψ2d 〉∆1 ×{ j} .

(111)

The P
angle ξ3 is chosen based on the former measurement outcomes b̃1 := b1 + s̃2 withs̃2 = σ1 s2 (σ1 )σ1 :∗ξ3 = −δt Λ (−1)Q(σ0 ) (−1)a(∂ σ0 ; b̃1 ) .

(112)

This choice would give us unitaries that suppress contributions with errors Z(σ1 )a(σ1 ;e1 ) , if
e0 = 0.7

5Generalizations

In this section we generalize the results in Section 3 in several directions. We generalize thegauge group from Z2 to ZN , and at the same time allow the parameters (d, n) of the model
7
However, when there is the error e0 , which affects the measurement of the Gauss law enforcement itself, thiswould flip the sign of the angle, so that it would make the simulated unitary operator a time-evolution with the(large) negative energy and cause contributions that violate the Gauss law, which is energetically favored.

24

SciPost Phys. 14, 129 (2023)

M(d,n) to be arbitrary. We also discuss a correspondence between the Euclidean path integralof the lattice gauge theory and the measurement-based quantum simulation of the model.
Finally, we propose an MBQS protocol for the Kitaev Majorana chain.

5.1

(Z )

N
MBQS of M(d,n)

In this subsection We introduce the MBQS protocol for the Abelian lattice gauge theory with(n − 1)-form fields in d spacetime dimensions. We choose the gauge group to be ZN withinteger N ≥ 2. The case with gauge group R is discussed in Appendix C.
5.1.1

(Z )

N
State gCS(d,n)

We begin by reviewing the qudit, the N -dimensional generalization of the qubit. We define
the operators Z and X by
Z|a〉 = ωa |a〉 , X |a〉 = |a + 1〉 ,(113)
with ω := e2πi/N and a ∈ {0, 1, . . . , N − 1 mod N }. They satisfy
ZN = X N = 1,

Z † = Z −1 ,X † = X −1 ,Z X = ωX Z .

(114)

We define the X -basis by
1 X −ab|ea〉 := pω |b〉 ,
N b

(115)

which satisfy X |ea〉 = ωa |ea〉. The analog of the Hadamard operator is the Fourier transformF :=1 X −ab|ea〉〈a| = p
ω |b〉〈a| .
N a,b
a

X

(116)

It satisfies X F = F Z. The controlled-Z operator is defined asX
C Z :=
ωab |a b〉〈a b| .

(117)

a,b

We generalize the state |+〉 as
1 X|+〉 := |e0〉 = p
|a〉 .
N a

(118)

Let us consider a d-dimensional hypercubic lattice C . The n-cells in this lattice can be
expressed as¨
σ n = σ α1 ,··· ,αn (x) =

x+

n
X

«
t i eα i 0 ≤ t i ≤ 1∀i ∈ {1, ..., n}

,

(119)

i=1

where 1 ≤ α1 < · · · < αn ≤ d are the directions in which the cell is stretched, x is the positionof the site at a corner of the cell, and eαi is the unit vector in the αi -th direction. We define
the boundary operator as∂ σ α1 ,··· ,αn (x) =nX(−1)i−1 σ α1 ,··· ,αÒi ,··· ,αn (x + eαi ) − σ α1 ,··· ,αÒi ,··· ,αn (x) ,

(120)

i=1

Òi means αi is removed from the subscript. One can confirm that ∂ 2 = 0 holds and that
where αthe definition of ∂ is consistent with that in simplicial homology [66].
25

SciPost Phys. 14, 129 (2023)

Arrangement based on the symmetry of the lattice

e1
e2

CZ
CZ −1

e3

time

{j}
[ j, j + 1]
{j + 1}(Z )N
Figure 8: The cluster state for M(3,2), (2+1)d ZN lattice gauge theories. (Left) The
primary lattice. The black lines represent the C Z gate, and the pink lines represent(C Z −1 ) gate. (Right) The primary and the dual lattices.

Pσ k )σσk
Let Ck ≡ Ck (C ; ZN ) be the Abelian group consisting of formal finite sums σ k ∈∆∆ k a(σσ k ) ∈ {0, 1, . . . , N − 1 mod N } ≃ ZN . We place a qudit on every (n−1)-cellwith a(σ
σ
∈
∆
n−1
n−1
P
c i ; σ i ) (withand on every n-cell σ n ∈ ∆ n . For each n-chain c n ∈ Cn with c i = σ i ∈∆∆ i a(ca(cc i ; σ i ) ∈ {0, 1, . . . , N − 1 mod N }), defineYσn)X (cc n ) :=(X σ n )a(cc i ;σ.
(121)
σ n ∈∆∆n

We similarly define Pauli Z operators on n-cells and Pauli X /Z operators on (n − 1)-cells.
The Hamiltonian that defines the generalized cluster state is now given byXX
σ n−1 ) + K(σσ n−1 )† ) −σ n ) + K(σσ n )† ) ,HC = −
(K(σ
(K(σ
(122)
∆ n−1
σ n−1 ∈∆∆nσ n ∈∆(Z )N
whose ground state is given by the cluster state |ψC 〉 = |gCS(d,n)〉 that satisfies

σ n )|ψC 〉 = K(σσ n−1 )|ψC 〉 = |ψC 〉 ,K(σ∆ n ⊔∆∆ n−1 )⊗(∆

|ψC 〉 = UC Z |+〉for all σ n ∈ ∆ n , σ n−1 ∈ ∆ n−1 ,

.

(123)
(124)

The stabilizers and the unitary UC Z are now defined asσ n ) := X σ n Z(∂∂ σ n ) ,K(σYσ n−1 )
σ n )a(∂∂ σ n ;σσ n−1 ) := X σ n−1
Z(σ
K(σ
,

(125)
(126)

∆n
σ n ∈∆UC Z :=

σa(∂∂ σ ;σY∆nσ n ∈∆∆ n−1
σ n−1 ∈∆)nn−1
C Zσ n ,σ
.
σ n−1

(127)

See Fig. 8 for an illustration with (d, n) = (3, 2).
5.1.2

(Z )

N
Model M(d,n)(Z )N
For gauge group ZN , the model M(d,n) is generalized to M(d,n)
defined by the actionX
I[{uσ n−1 }] = −J(u(∂∂ σ n ) + u(∂∂ σ n )∗ ) .

σn

26

(128)

SciPost Phys. 14, 129 (2023)(N )Table 3: Measurement pattern for model M(d,n) .
basis
layer
d-dim cell(d − 1)-dim cell#

M(ZN ,A) → M(ZN ,X ) → M(ZN ,A) → M(ZN ,B)
pt
pt
I
I
σn
σn−1
σn−1
σn
σn
σn−1
σn−2
σn−1
(1)
(2)
(3)
(4)

(Z )

N
The field uσ n−1 takes values in u ∈ {1, ω, ..., ωN −1 }. The partition function of M(d,n)
as aclassical spin model is given byX(ZN )Z(d,n)=
e−β I[{uσ n−1 }] .
(129)

config.
(Z )N
Hamiltonian formulation of M(d,n)

5.1.3

(Z )

N
As a quantum lattice model, M(d,n)
is described by the Hamiltonian

H =−

λX
1X
(X σn−1 + h.c.) −(Z(∂ σn ) + h.c.) .
2σ
2 σ
n−1

(130)

n

Suppose that we have external charge sources with charges Q s ∈ {0, 1, . . . , N − 1 mod d} associated with (n − 2)-cells s. Physical states must obey the Gauss law constraintωQ s , for σn−2 = s ,(131)
G(σn−2 ) =1,otherwise ,whereG(σn−2 ) :=Y
σn−1 ∈∆n−1(X σn−1 )−a(∂ σn−1 ;σn−2 ) .

(132)

See Appendix B for a derivation of the quantum model as a limit of the classical model.
5.1.4

(Z )

N
MBQS protocols for M(d,n)

One Trotter step consists of measurements (1)-(4) given in Table 3. These measurements havethe following effects on the wave function.
(1) σ n = σn × { j}We measure the qudit on σ n along

M(ZN ,A) := e i(ξX +h.c.) |s〉 s = 0, 1, . . . , N − 1 mod N .

(133)

Using the identity (A.3), we get
Y 1
p Z(∂ σn )s e−i(ξZ(∂ σn )+h.c.) .
N
σn

27

(134)

SciPost Phys. 14, 129 (2023)(2) σ n−1 = σn−1 × { j}We measure the qudit on σ n−1 along

M(ZN ,X ) := {|s̃〉 s = 0, ..., N − 1 mod N } .

(135)

It affects the qubit at σ n = σn−1 × [ j, j + 1]. We apply the identity (A.1). We usea(∂∂ σ n ; σ n−1 ) = (−1)n to get
Y 1

n+1
(136)
p F (−1) Z s σ .
n−1
N
σn−1
Here the state lives on the interval [ j, j + 1].
(3) σ n−1 = σn−2 × [ j, j + 1]
We measure the qudit on σ n−1 along M(ZN ,A) to implements the energy cost for Gausslaw violation. We apply the identity (A.3). We note thata(∂∂ (σn−1 × [ j, j + 1]); σ n−1 ) = a(∂ σn−1 ; σn−2 )and find
Y
σn−2



s 

1 −i(ξ Qσ Zσa(∂ σn−1 ;σn−2 ) +h.c.) Y a(∂ σn−1 ;σn−2 )
n−1
n−1
.
Zσ
p e
n−1
N
σn−1

(137)

Here the state lives on the interval [ j, j + 1].
(4) σ n = σn−1 × [ j, j + 1]
We measure the qudit on σ n along

M(ZN ,B) := {e i(ξZ+h.c) |es〉 s = 0, . . . , N − 1 mod N } .
We apply the identity (A.1) with a(∂∂ σ n ; σ n−1 ) = (−1)n−1 to get
Y 1nnX s(−1) e−i(ξX +h.c.) F (−1) σ .
p
n−1
N
σn−1

(138)

(139)

Combining (1)-(4), we get(Z )NU(d,n)({ξi })
Yn:=
X (−1) s4 e−i(ξ4 X +h.c.) σ

σn−1



×

Y Y
σn−2

σn−1


n−1

n
X σ(−1) a(∂ σn−1 ;σn−2 )
n−1

Y

e

−i(ξ3

Q

σn−1

(−1)n a(∂ σn−1 ;σn−2 )X σn−1+h.c.)

σn−2

s3 (σn−2 ) Yσn−1

Zσs2 (σn−1 )
n−1



Y

Z(∂ σn )s1 (σn ) e−i(ξ1 Z(∂ σn )+h.c.) .

σn

(140)
We used the relations F ±1 Z F ∓1 = X ±1 and F ±1 X F ∓1 = Z ∓1 . Writing the wave function with
the Trotterized time evolution as(Z )N|ψ(t)〉 = T(d,n)(t)|ψ(0)〉 ,
(141)
Q
Y
Y
j
Y
a(∂ σn−1 ;σn−2 )Qσ(ZN )iΛω n−2 ( σX+h.c.)δtn−1 σn−1
T(d,n)(t) :=e i(X +h.c.)δt/2ee iλ(Z(∂ σn )+h.c.)δt/2 ,

σ1

σn−2

σn

(142)
28

SciPost Phys. 14, 129 (2023)(t = jδt), and denoting the byproduct operators coming from j-th time step as Σ( j) , we obtain
!
!
j
j+1
Y
Y
(ZN )
U(d,n) ({ξi })Σ(k) |ψ(t)〉 =Σ(k) |ψ(t + δt)〉 ,
(143)
k=1

k=1

with appropriate choices of {ξi }i=1,3,4 .

5.2

Euclidean path integral and Hamiltonian MBQS(Z )N
Our MBQS with the quantum Hamiltonian derived from M(d,n)
is done by measurements ongCS(d,n) , and it is suggestive that the spacetime structure of the classical action I(d,n) resembles
the structure of the entangler in gCS(d,n) . The aim of this section is to make such a quantumclassical correspondence manifest in terms of the Euclidean path integral or the partition func(ZN )tion of the model M(d,n). It is a version of the correspondence found in [27], specialized to the(Z )Nmodel M(d,n). This subsection has some overlap with [67].
For the gauge group ZN , let us define
⊗∆
∆n
(ZN )
∆ n−1
|φ(d,n)〉 := e iκ(X +h.c.) |0〉
|+〉⊗∆
.

(144)

Its overlap with the generalized cluster state (124) is the probability amplitude for obtaining
the special measurement outcomes s = 0 in the measurement of qubits on n-cells with theA-type basis (133) and those on (n − 1)-cells with X -basis. Now we replace κ to an imaginary(Z )parameter κ = iβ J, then this quantity is proportional to the partition function Zd,nN definedin (129). Indeed we have
(Z )

(Z )

1

(Z )

∆ n |+ 2 |∆∆ n−1 |NN
Zd,nN = N |∆〈φ(d,n)|gCS(d,n)〉κ→iβ J.

(145)

The case with the gauge group R is discussed in Appendix C.
For the gauge group Z2 , we developed the imaginary-time MBQS in Section 3.3. If we use
the A-type measurement basis state and define (74)

⊗∆
Æ
∆n
(Z2 )
∆ n−1
|φ̃(d,n)(α)〉 := e−α eαX |0, 0〉 + sinh(2|α|)|−, 1〉
,
(146)
|+〉⊗∆
we have

(Z )

(Z )

2
2
〈φ̃(d,n)(β J)|gCS(d,n)〉∝

Xe−β I[{Sσ n−1 }] .

(147)

{Sσ n−1 }The left-hand side of (147), which is real and positive as can be seen from the right-hand side,is the probability amplitude to obtain (146) as the outcome of the simultaneous non-adaptive
measurement of all the qubits on n-cells in the A-type basis and those on (n − 1)-cells in the
X -basis.
C ), which is a generalization of the Wilson
We can also consider an extended operator W (Cloop. The operator is supported on an (n−1)-cycle C ∈ ker(∂∂ n−1 ), where ∂ n−1 is the boundaryC ) isoperator ∂ acting on n-chains C n . The unnormalized expectation value of W (CXC )〉 :=
C )e−β I[{Sσ n−1 }] .
〈W (C
S(C
(148)
{Sσ n−1 }The relation (147) generalizes to
(Z )

(Z )

2
2
C )〉 ∝ 〈φ̃(d,n)
C )|gCS(d,n)〈W (C(β J)|Z(C
〉.

(149)

This relation implies that there is a constant-depth unitary circuit UW such thatC )〉 ∝ 〈0, 0|⊗∆n 〈0|⊗∆n−1 UW |0, 0〉⊗∆n |0〉⊗∆n−1 . One can perform the Hadamard test [58]〈W (Cto estimate the matrix element of UW to obtain the the expectation value of the generalized
Wilson loop operators within polynomial time.
29

SciPost Phys. 14, 129 (2023)

5.3

Kitaev Majorana chain

Here we propose a measurement-based simulation scheme for a fermionic system, namely
Kitaev’s Majorana chain [23] defined by the Hamiltonian
HK = Hhop + H P ,

(150)

where
Hhop = w

X(−c †j c j+1 + c j c j+1 + h.c.) ,j∈Z

X
H P = −µ(c †j c j − 1/2) .

(151)

j∈Z

More precisely, we wish to implement the Trotterized time evolution operator
ne−iHhop δt e−iH P δt ,(152)

via measurements. Unlike [68], where a different scheme for the Kitaev chain was proposed,
our scheme involves measurements of fermion parities and is motivated by a relation between
the Jordan-Wigner transformation and measurements [24].
To describe the resource state used for simulation, let us consider a two-dimensional squarelattice C . As shown in Fig. 9(a), we assign some orientations to edges e ∈ ∆ 1 and faces
f ∈ ∆ 2 . On each vertex v ∈ ∆ 0 we introduce a complex fermion cv such that {cv , cv†′ } = δv v ′ ,{cv , cv ′ } = 0. It can be decomposed into a pair of Majorana fermions γv , γ′v as cv = (γv +iγ′v )/2,cv† = (γv − iγ′v )/2. We define(153)
Pv := −iγv γ′v = 1 − 2cv† cv ,and label its eigenvalue and eigenstate as (−1) p and |p〉v = (cv† ) p |0〉v respectively (p ∈ {0, 1}).
Note that, on a single vertex v , the operators (Pv , γv , γ′v ) obey the same algebraic relations as
the Pauli operators (Z, X , Y ), so that |p〉v = (γv ) p |0〉v . We also define a hermitian operator
squaring to 1,(154)
Se = iγ′v γv + ,−where the vertices v ± = v ± (ee ) are defined by the relation v + − v − = ∂ e . On each edge e ∈ ∆ 1
we introduce a qubit. Following [24], we define the operator C Se to be Se controlled by thequbit on e and set
Y
UC S :=C Se ,
(155)
∆1
e ∈∆

with the following ordering. Within a horizontal layer the operators C Se commute with eachother, and we let them appear in the product simultaneously. We order such layers so that as
we go down in Fig. 9(a) we go to the left within the product (155).8
We now define the cluster state |ψC 〉 [24] by∆0∆1|ψC 〉 := UC S |0〉⊗∆⊗ |+〉⊗∆.
(156)
Its stabilizers are9

UC S Pv UC−1S = Pv Z(∂∂ ∗ v ) ,
8

(157)

This ordering cannot be realized by a shallow circuit. A different ordering (for example, all vertical edges after
all horizontal edges) can be, but it is not clear whether such an ordering enables measurement-based simulation.
9Note the relations C Se γv + C Se−1 = Ze γv + , C Se γv − C Se−1 = γv − , C Se γ′v + C Se−1 = γ′v + , C Se γ′v − C Se−1 = Ze γ′v − for oneedge shown in Fig. 9(d), and UC S γv UC−1S = Ze + γv , UC S γ′v UC−1S = Ze − γ′v , for a general vertex v in a two-dimensionallattice, where ∂ e + = v − v ′ , ∂ e − = v ′′ − v for some v ′ and v ′′ .

30

SciPost Phys. 14, 129 (2023)

(a)

(b)

(c)

(d)

Figure 9: (a) We place fermions on vertices, and qubits on edges. (b) We introduce aboundary to be identified with a one-dimensional lattice for the Kitaev chain. (c) Anedge and its ends within a one-dimensional chain. (d) A vertical edge and its ends.
for an arbitrary vertex v ,iγ′v−|
UC S X e UC−1S = X e Se Ze ′ =Xe|Ze ′ — γ v +

,

(158)

for a vertical edge e , andγ′v — X e — iγv +−,UC S X e UC−1S = X e Se Ze ′ = |
Ze ′

(159)

for a horizontal edge e .
Using the operators S and P on a one-dimensional lattice we can rewrite the Hamiltonian
HK = Hhop + H P as
X
µ X
Hhop = w
Se ,
HP =
Pv .
(160)
2 v∈∆
e∈∆
1

0

where the vertices v ∈ ∆0 and the edges e ∈ ∆1 are those of a one-dimensional lattice. Written
in this way, it is manifest that different terms commute with each other within each of Hhopand H P , so that the evolution operator can be written as!nYYµe−iwδtSe
e−i 2 δt Pv
.
(161)
e∈∆1

v∈∆0

Let us now consider a reduced two-dimensional lattice Cred which is periodic in the 2direction and has a boundary, which is to be identified with the one-dimensional lattice for the
Majorana chain. See Fig. 9(b) and (c). We assign a non-negative integer j ≥ 0 for each layer,
so that as cells of the two-dimensional lattice, the vertices are v = v × { j}, horizontal edges
are e = e × { j}, and vertical edges are e = v × [ j, j + 1]. In the middle of simulation the stateof the total system takes the form|ψ2d 〉 = UC S Obp |ψ1d 〉 ⊗ |0〉bulk ⊗ |+〉bulk ,(162)
where Obp is a product of γ v and Pv . Let us consider a horizontal edge e = e × { j} and its
boundary vertices v ± . See Fig. 9(c). The relatione−iξX e C Se = C Se e−iξX e Se
31

(163)

SciPost Phys. 14, 129 (2023)

(a)

(b)

(c)

Figure 10: The measurement bases Mhop , M P MX Y for the indicated qubits (arrows
on edges) and fermions (dots).
implies thate 〈s|e−iξX e1
C Se |+〉e = p (Se )s e−iξSe ,2

s = ±1 ,(164)

or equivalently
1
Pe(A) (s, ξ)C Se |+〉e |ψ〉v − v + = p e iξX e |s〉e (Se )s e−iξSe |ψ〉v − v + ,
2

(165)

where

1 + (−1)s e iξX e Ze e−iξX e
.
(166)
2
Thus we can implement the hopping (Se ) term in the Hamiltonian, up to a byproduct operator(Se )s , by measuring the edge qubit with the basis
©
¦
(167)
MA = e iξX |s〉 s = 0, 1 .
Pe(A) (s, ξ) :=See Fig. 10(a).
Next let us consider an edge that extends within the bulk in the 2-direction. See Fig. 9(d).
We findp(B)
P(P)
v (t)Pe (s, ξ)C Se (γv − ) |0 + 0〉v − e v + =
−

where
P(P)
v :=

1
(Pv )s+1 e−iξPv + e iξZe (γv − ) t (γv + ) t+p |0s0〉v − e v + ,
2 +
(168)

1 + (−1) t Pv −
2

−

and
Pe(B) (s, ξ) :=

,

(169)

1 + (−1)s e iξZe X e e−iξZe
.
2

(170)

In (168), one may move (γv + ) t to the left, changing e−iξP to e−i(−1) ξP . This implies that wePt
can teleport 10 |ψ〉 = p ψ p |p〉 at v − to e−i(−1) ξP |ψ〉 at v + by measuring the fermion in the
P-basis
M P = |0〉 , |1〉 ,(171)t

at v − and by measuring the qubit on e in the eigenbasis
MB = e iξZ |±〉 ,10

See (1.32) of [69] for an analog of (168) in the purely qubit case.

32

(172)

SciPost Phys. 14, 129 (2023)

for e iξZ X e−iξZ , up to byproduct operators. See Fig. 10(b) and (c). By applying (165) and (168)repeatedly, we see that the following measurement pattern, with measurement angles ξ chosen
adaptively, realizes the time evolution (161) for the Kitaev chain:basis →
layer
2d cell
1d cell

MA → M P
pt
pt
σ1
σ0
σ1
σ0

→ MB
I
σ1
σ0

→
(173)

Let us discuss the computation of physical observables. Natural local observables of theKitaev chain are the individual terms in the Hamiltonian. The observable Pv in H P can be computed simply by measuring it at v. The computation of the observable Se is more interesting.
At the end of simulation the state is of the formY|Ψ〉 =C Se Obp |ψ〉∆0 |+〉∆1 ,
(174)
e∈∆1

where Obp is the byproduct operator. If we measure X on edge e with outcome se ∈ {0, 1}, theresulting state is!
Y
1 + (−1)se X e
1 + (−1)se X e Se|Ψ〉 =Obp |ψ〉∆0 ⊗ |+〉∆1
C Se′
2
2
e′ ∈∆1!
Y
1 + (−1)se Se
Obp |ψ〉∆0 ⊗ |+〉∆1 .
(175)
=
C Se′
2
e′ ∈∆
1

This means that the outcome of measuring X e on |Ψ〉 is the same as the outcome of measuring
Se on Obp |ψ〉∆0 . The knowledge of the measurement outcomes throughout the simulation
allows us to know whether the operator Se commutes or anti-commute with Obp . Thus we cancompute Se for the state |ψ〉 by measuring X e on edge e.

6

SPT order of the generalized cluster state

A nontrivial phase, in the sense of the topological order, is defined as the depth of the local
quantum circuit required to prepare the state from a product state being large. A wave function
in an SPT phase can be prepared with a finite depth local quantum circuit. The SPT phase isnontrivial if the state requires a large-depth local quantum circuit when we demand the circuit
to commute with the symmetry of interest.
The gauging procedure [38, 70–73] is a powerful tool to diagnose an SPT order. Upon
gauging, trivial and nontrivial SPT Hamiltonians belong to two distinct topological phases.
This method has been applied to several models to demonstrate that they have nontrivial SPT
orders. In the original argument [70], the Levin-Gu SPT state, which is an SPT with a 0-form
symmetry, was minimally coupled to a 1-form gauge field, and the gauged model was shown
to possess excitations with double-semionic braiding statisics, which differs from the braiding
statistics found in the gauged version of trivial SPT order. The method can be employed for
detecting SPT orders protected by higher-form symmetries as well [38], and the argument
suggested that the RBH cluster state has a non-trivial SPT order protected by a Z2 × Z2 1-form
symmetry [72]. Our generalized cluster state gCS(d,n) is a natural extension of these states,
and it is plausible that it has an SPT order with higher-form symmetries. In this section, we
discuss gauging the Hamiltonian that defines gCS(d,n) .
33

SciPost Phys. 14, 129 (2023)

First we will see that the generalized cluster state gCS(d,n) possesses (d − n)- and (n − 1)form Z2 symmetries. Then we define the gauging map and discuss its properties. The definition
we employ is the one discussed in [38], for example. (We can regard the map as a result of
minimally coupling gauge degrees of freedom. See Appendix E.) Then we apply the method
to our generalized cluster state to argue that it possesses a nontrivial SPT order protected by(d − n)- and (n − 1)-form Z2 symmetries.

6.1

Symmetries of the generalized cluster states

The generalized cluster state gCS(d,n) possesses higher form symmetries. They are (d − n)and (n − 1)-form Z2 symmetries generated by the following operators, S(M ) and S(M ′ ),respectively:YYσ n ) , S(M ′ ) =σ n−1 ) ,S(M ) =
X (σ
X (σ
(176)
σ n ∈M

σ n−1 ∈M ′with

M ∈ ker(∂∂ n ) = {zzn | ∂ zn = 0} ,M ′ ∈ ker(∂∂ ∗d−n+1 ) = {zzn−1 | ∂ ∗zn−1 = 0} .

(177)
(178)

We refer to M and M ′ respectively as n- and (d − n + 1)-brane operators (generalizing “membrane” operators). The existence of such symmetries can be shown by taking a product ofcorresponding stabilizers over the closed manifold:
Y
Y
Y
σn) =
σ n )Z(∂∂ σ n ) =σn) .
K(d,n) (σ
X (σ
X (σ
(179)
σ n ∈Mσ n ∈Mσ n ∈M

A similar calculation can be done for the other type of stabilizers as well. Special cases ofsymmetry generators in (176) are X (∂∂ σ n−1 ) and X (∂∂ ∗σ n−2 ). See (99).

6.2

Gauging map

We now introduce the gauging map. Let Hn−1 (Hn ) be the Hilbert space for all the qubits on(n − 1)-cells (n-cells). Let us define the symmetric subspaces
s ym

H0

= {|ψ〉 ∈ Hn ⊗ Hn−1 | S(M )S(M ′ )|ψ〉 = |ψ〉 for ∀M , M ′ } ,

(180)

and
s ym

H1

= {|ψ〉 ∈ Hn−1 ⊗ Hn | T (N )T (N ′ )|ψ〉 = |ψ〉 for ∀N , N ′ } ,

(181)

with
T (N ) =

Y

σn) ,
Z(σ

σ n ∈N

T (N ′ ) =Yσ n−1 ) ,
Z(σ

(182)

σ n−1 ∈N ′where N = {zzn | ∂ zn = 0} (N ′ = {zzn−1 | ∂ ∗zn−1 = 0 }) is a set of cycles. One can show thats ym

dim H0s ym= dim H1.

(183)

σ n )σσ n , we write |ccn 〉 = |{a(σσ n )}〉. Let us consider the
For an arbitrary chain cn = σ n a(σlinear map Γ : Hn ⊗ Hn−1 → Hn−1 ⊗ Hn defined byPΓ (|ccn 〉 ⊗ |ccn−1 〉) = |∂∂ cn 〉 ⊗ |∂∂ ∗ cn−1 〉 .
34

(184)

SciPost Phys. 14, 129 (2023)

s ym

s ym

s ym

Its restriction to H0 induces the gauging map Γ : H0 → H1 (up to a normalization factor
discussed in Appendix E). The gauging map Γ defines the transformation AΓ of a symmetry
respecting operator A via
Γ (A|ψ〉) = AΓ (Γ |ψ〉) ,s ym∀|ψ〉 ∈ H0

.

(185)

Now let us discuss some properties of Γ . Γ brings the generators of the symmetries toΓΓ

identity: S(M ) → I, S(M ′ ) → I. Namely,Γ S(M ) S(M′ )|ccn 〉 ⊗ |ccn−1 〉 = I · Γ (|ccn 〉 ⊗ |ccn−1 〉) .
The output states, Γ (|ψ1 〉) and Γ (|ψ2 〉), are identical if and only if there exists S(M ) such that
S(M)|ψ1 〉 = |ψ2 〉. Γ is locality preserving, gap preserving, bijective and isometric.
Below we will make use of the following argument [70]. Let us consider gauging two
Hamiltonians
ΓH1 → H1Γ ,

(186)

Γ

H2 → H2Γ .

(187)

If the two topological orders that H1Γ and H2Γ possess differ, then the SPT orders that H1 and H2have cannot be the same. Indeed, if there were a path [74] that connects H1 and H2 without
breaking the symmetry or closing the energy gap, then there would also be a path that connects
H1Γ and H2Γ , which is a contradiction.

6.3

Mapping the generalized cluster states

Now we consider the trivial Hamiltonian defined on n-cells and (n − 1)-cells in d-dimensions,
X
X
σn) −
σ n−1 ) ,Htrivial := −
X (σ
X (σ
(188)
∆n
σ n ∈∆∆ n−1
σ n−1 ∈∆and the Hamiltonian for the gCS(d,n) ,Xσn) −HgCS := −K(σ∆nσ n ∈∆

X

σ n−1 ) .
K(σ

(189)

∆ n−1
σ n−1 ∈∆

σ n ) is mapped to
Let us first consider gauging the trivial Hamiltonian. The operator X (σσ n−1 ) to X (∂∂ ∗σ n−1 ). Following [70] we add Z(∂∂ σ n+1 ) (if n + 1 ≤ d) andX (∂∂ σ n ), and X (σZ(∂∂ ∗σ n−2 ) (if n − 2 ≥ 0) to the Hamiltonian so that the fluxes vanish (the gauge fields areflat) in the ground states. Therefore we obtain
X
X
X
X
gaugedHtrivial = −X (∂∂ σ n ) −Z(∂∂ ∗σ n−2 ) −X (∂∂ ∗σ n−1 ) −Z(∂∂ σ n+1 ) .
∆n
σ n ∈∆

|

σ

{zn−2

∆ n−2
∈∆

∆ n−1
σ n−1 ∈∆}|=:H n−1∆ n+1
σ n+1 ∈∆

{z

=:H n

}

(190)
Here H n and H n−1 define two decoupled theories on n-cells and (n − 1)-cells, respectively.
For the generalized toric code H n , there are Nsite · d Cn physical qubits living on n-cells andk := d Cn logical qubits, where d Cn = d!/n!(d − n)!. There are k pairs of anti-commuting
logical operators, where a logical Pauli Z operator acts on n-dimensional hyperplanes and alogical Pauli X operator acts on (d − n)-dimensional hyperplanes which are non-contractable
on a torus. See [75] for details.
On the other hand, gauging the gCS Hamiltonian gives
X
X
gaugedσn) −σ n−1 ) ,HgCS = −K Γ (σK Γ (σ(191)∆n
σ n ∈∆∆ n−1
σ n−1 ∈∆

35

SciPost Phys. 14, 129 (2023)

where
σ n ) = Z(σσ n )X (∂∂ σ n ) ,K Γ (σΓ(192)∗

σ n−1 ) = Z(σσ n−1 )X (∂∂ σ n−1 ) .
K (σ

(193)

We do not need to add extra terms to make the fluxes vanish because K Γ (∂∂ σ n+1 ) = Z(∂∂ σ n+1 ),K Γ (∂∂ ∗σ n−2 ) = Z(∂∂ ∗σ n−2 ). The Hamiltonian (191) is the Hadamard transform of the originalΓ
Hamiltonian HgCS , so the ground state described by the mapped Hamiltonian HgCS
is still shortrange entangled, meaning it does not have a topological order. Since the gauged version of thetrivial Hamiltonian possesses a nontrivial topological order, the ungauged Hamiltonian HgCS
is in a nontrivial SPT phase.

6.4

Brane operators and projective representation

Another approach to probing a nontrivial SPT order is to find a projective representation11
σ n ) and K(σσ n−1 ) associated withby introducing boundaries [33, 77]. Recall that we have K(σσ n ∈ ∆ n and σ n−1 ∈ ∆ n−1 , respectively. In the bulk, we have a symmetry generators, each ofwhich is a product of stabilizers over a closed manifold. Namely, the (n − 1)-form symmetry is
generated by (d − n + 1)-brane operators, and (d − n)-form symmetry is generated by n-brane
operators. In this section, we consider a lattice with boundary and we wish to show that theaction of the symmetry generator at the boundary forms a projective representation, whichindicates that the bulk theory is an SPT protected by the symmetry.
Let us consider a d-dimensional lattice, which is periodic in the (x 2 , x 3 , . . . , x d )-directions
with length L i (i = 2, . . . , d). Consider an n-brane B extended in the (x 1 , x 2 , . . . , x n )-directions
and a (d −n+1)-brane B̃ extended in the (x 1 , x n+1 , . . . , x d )-directions. The former corresponds
to a union of n-cells within a hyperplane extended in the (x 1 , x 2 , . . . , x n )-directions. The lattercorresponds to a union of (n − 1)-cells extended in (x 2 , · · · , x n )-directions intersecting witha hyperplane extended in the (x 1 , x n+1 , · · · , x d )-directions. For definiteness, we consider thecase where the boundaries at x 1 = 0, L1 consist of (n − 1)-cells. We denote the boundary at
x 1 = 0 as L and that at x 1 = L1 as R . The same argument below holds as well when we taken-cells as boundaries.
Now the brane operators are defined as
Y
Y
Y
B) =
σ n−1 ) =σ n−1 )
σn) ,
M (B
K(σ
X (σ
Z(σ
(194)
B
σ n−1 ∈B

M̃ (B̃B̃) =YBσ n−1 ∈B

σn) =
K(σ

σ n ∈B̃YR)
σ n ∈∂∂ B ∩(LL ∪R

σn) .
X (σ

(195)

σ n ∈B̃

We consider the restriction of these operators to the boundary:12YB) =σ n ) , (i = L , R ) ,M i (BZ(σ(196)

σ n ∈∂∂ B ∩i

M̃ i (B̃
B̃) =

Y

σn) ,
X (σ

(i = L , R ) .

(197)

σ n ∈B̃
B̃∩i′

Now, since for an arbitrary pair of chains c , c ′ we have Z(cc )X (cc ′ ) = (−1)|cc ∩cc | X (cc ′ )Z(cc ), and|(∂∂ B ∩ i) ∩ (B̃ ∩ i)| = 1 for i = L , R , we obtain
B ), M̃ i (B̃{M i (BB̃)} = 0 ,
11(i = L , R ) .

(198)

For the 1-dimensional cluster state gCS(1,1) , one can exhibit a non-trivial representation of Z2 × Z2 from thematrix product state representation [76] as we review in Appendix F, where we also derive a tensor network
representation of gCS(d,n) for general (d, n).
12
We encourage the reader to see Fig. 10 of [72].

36

SciPost Phys. 14, 129 (2023)

This implies that the brane operators restricted to a boundary furnish a projective representation of Z2 × Z2 , with each factor generated by one type of brane operator. This observation
supports our claim that the gCS possesses a nontrivial SPT order protected by Z2 (d − n)- and(n − 1)-form symmetries. This generalizes a result in [72] for gCS(3,2) .
For the Majorana state (156) with fermionic symmetry, it was argued13 in [24] that one
can find a nontrivial commutation relation between the fermionic Z2F 0-form symmetry andbosonic Z2 1-form symmetry, the latter of which has fermionic operators at its endpoints. The
SPT class for such pair of symmetries is suggested to be nontrivial, although there is only one
SPT class.

7

Symmetries in measurement-based quantum simulation and abulk-boundary correspondence

In this section we study the interplay between the symmetries of the lattice models to be simulated and those of the bulk cluster states that simulate the lattice models. We propose that
MBQS is a type of bulk-boundary, or holographic, correspondence between the simulated theory M(d,n) and the system given by the generalized cluster state |gCS(d,n) 〉, generalizing [78].
We discuss the case of (d, n) = (3, 1) and (3, 2) explicitly. The reinterpretation of the stabilizers [41] as gauge symmetries discussed in Appendix D also supports our proposal.

7.1

Ising model M(3,1)

The Hamiltonian (17) of the model M(3,1) is invariant under the simultaneous sign flip ofZ(σ0 ) for all vertices σ0 ∈ ∆0 . This is an an ordinary (0-form) global Z2 symmetry generatedQ
by σ0 X (σ0 ).
In the middle of the simulation, the qubits that remain unmeasured reside on a threedimensional lattice, which is periodic in two (1- and 2-) directions and has a boundary.14 The
simulated state |ψ2d 〉 of the Ising model is defined on the qubits at the vertices of the twodimensional square lattice that is identified with the boundary of the three-dimensional lattice.
The total state |ψ3d 〉 takes the form|ψ3d 〉 = UC Z Obp |ψ2d 〉 ⊗ |+〉bulk .
(199)
Here Obp is the byproduct operator, i.e., a product of Pauli operators acting on |ψ2d 〉, whicharises from the preceding measurements. The state |+〉bulk is the tensor product of the X eigenstates with eigenvalue +1 (|+〉) over all the bulk qubits and the vertex qubits on theboundary. The entangler UC Z is the product of all the controlled-Z gates between neighboring
pairs of a vertex and an edge.
Q
Let g be the generator of the zero-form global symmetry Z2 , and U g =σ0 X (σ0 ) its
representation on the Hilbert space of the Ising model. We wish to understand the effect of
U g on |ψ3d 〉 in the coupled boundary-bulk system. We have




−1
UC Z Obp U g |ψ2d 〉 ⊗ |+〉bulk = UC Z Obp U g Obp
Obp |ψ2d 〉 ⊗ |+〉bulk−1= UC Z Obp U g Obp
UC−1Z |ψ3d 〉 .

(200)

13
The state we consider and the one considered in [24] only differ by the order of application of the entanglers,
UC S .
14
Explicitly, in terms of Cartesian coordinates (x 1 , x 2 , x 3 ), we have periodic identifications x 1 ∼ x 1 + L1 and1x ∼ x 2 + L2 (L1,2 ∈ Z>0 ), and have a boundary at x 3 = −(t/δt) ∈ Z. The vertices σ 0 ∈ ∆ 0 (Cred ) are the points(x 1 , x 2 , x 3 ) with x 1,2,3 ∈ Z and x 3 ≤ −(t/δt).

37

SciPost Phys. 14, 129 (2023)−1
We note that Obp U g Obp
equals (−1)m U g , where (−1)m is a sign determined by the out
−1comes of the preceding measurements. Therefore, the operator UC Z Obp U g Obp
UC−1Z equals
Q(−1)m σ0 K(σ0 ), where K(σ0 ) = UC Z (X (σ0 ) ⊗ Ibulk )UC−1Z for a boundary vertex σ0 .
σ 0 ) when σ 0 ∈ ∆0 \∆0 is a bulk vertex. ThisThe state |ψ3d 〉 in (199) is invariant under K(σmotivates us to defineY −1 Yσ0 )−1σ 0 )Λ(σ0 ) = (−1)mσ 0 )Λ(σU g (Λ) := UC Z Obp U g Obp
UC Z
K(σ
K(σ
,
(201)
∆0
σ 0 ∈∆∆0 \∆0σ 0 ∈∆

for an arbitrary gauge parameter (0-cochain) Λ : C0 → Z2 whose boundary value is constrained
to be Λ(σ0 ) = 1 for σ0 ∈ ∆0 . The action of U g on |ψ3d 〉 is equivalent to the action of U g on
|ψ2d 〉:
U g (Λ)|ψ3d 〉 = UC Z Obp (U g |ψ2d 〉) ⊗ |+〉bulk .
(202)
We now argue that the relation (202) is a manifestation of a new kind of bulk-boundary, or
holographic, correspondence. In such a correspondence, a global symmetry of the boundary
theory is identified with a gauge symmetry of the bulk theory. Indeed in the current set-up,
σ 0 ) = X (σσ 0 )Z(∂∂ ∗σ 0 )the symmetry generator U g of the boundary becomes the product of K(σσ 0 ) generates the gauge transformation of the bulk theory
over σ 0 ∈ ∆ 0 , and the operator K(σdefined by the HamiltonianXσ 1 )Z(∂∂ σ 1 ) ,
H =−
X (σ
(203)
∆1
σ 1 ∈∆

σ 0 )Z(∂∂ ∗σ 0 ) = 1. It is a (3 + 1)-dimensional Ising modeland the Gauss law constraint X (σσ 1 ), and has the cluster state
coupled to the topological Z2 gauge theory with gauge field X (σ|gCS(3,1) 〉 as the unique ground state.

7.2

Gauge theory M(3,2)

In this subsection, we use the asterisk (∗) to denote quantities (such as bulk i-cells σ ∗i ∈ ∆ ∗i )associated with the dual lattice. Let us consider the Z2 gauge theory defined by the Hamiltonian (19).
The electric Z2 one-form symmetry is generated by
U g (z1∗ ) := X (z1∗ ) ,(204)

for z1∗ ∈ ker(∂1∗ ). The operator X (z1∗ ) commutes with the Hamiltonians (19). Moreover, the action of X (z1∗ ) on the physical Hilbert space is invariant under a local deformation of z1∗ becauseof the Gauss law constraint (20), except when it crosses the location of an external charge
and changes its sign. This means that X (z1∗ ) is a topological defect operator that generates
the Z2 one-form symmetry under which Wilson lines are charged. For each generator of thedual 1-homology, the operator X (z1∗ ) defines a logical Pauli X operator in the toric code limit
λ → ∞ in the deconfined phase.
Next let us consider the Wilson loop operator
Z(z1 ) ,(205)

for an arbitrary 1-cycle z1 ∈ Z1 . The Wilson loop exhibits the area law in the confined phaseand the perimeter law in the deconfined phase [20, 22]. In the Z2 topological field theory(realized in the low-energy limit with λ > λc or in the toric code limit λ → ∞ of our Z2 gauge
theory), Z(z1 ) is a generator of the magnetic Z2 one-form symmetry.15 Since Z(z1 ) does notcommute with the electric term in the Hamiltonian of the gauge theory, the one-form symmetry15

See Section 4.3 of [37].

38SciPost Phys. 14, 129 (2023)

Table 4: Symmetries studied in Sections 6 and 7 as well as in Appendix D. The (n−1)and (d − n)-form symmetries of the generalized cluster state gCS(d,n) can be regarded
both as global (Section 6) and gauge (Section 7 and Appendix D) symmetries. The
simulated model M(d,n) on the boundary has distinct but related global and gauge
symmetries (if both exist). The global symmetry exists as a consequence of the Gausslaw constraint. When n = 1, M(d,n=1) is the Ising model and has no gauge symmetry.

gCS(d,n) (bulk)M(d,n) (boundary)global symmetry / generator

gauge symmetry /generator(n − 1)-form / (d − n + 1)-brane

σ n−1 = σ ∗d−n+1 )(n − 1)-form / K(σ(d − n)-form / n-braneσn)(d − n)-form / K(σ(n − 1)-form / (d − n)-brane(n − 2)-form / X (∂ ∗ σn−2 )is absent except in these limits. For each generator of the 1-homology, the operator Z(z1 )defines a logical Pauli Z operator in the toric code limit λ → ∞ in the deconfined phase.
Recall that the RBH cluster state is defined on the qubits placed on the edges and thefaces of the three-dimensional cubic lattice C , with qubits placed on edges σ 1 ∈ ∆ 1 and faces
σ 2 ∈ ∆ 2 . The RBH cluster state |ψC 〉 is the simultaneous eigenstate, with eigenvalue +1,
σ 1 ) = X (σσ 1 )Z(∂∂ σ 1 ) and K(σσ 2 ) = X (σσ 2 )Z(∂∂ ∗σ 2 ). In the middle of theof the stabilizers K(σsimulation, the system is again defined on the reduced lattice Cred . The state of the systemdiffers from the RBH cluster state only on the boundary qubits which encode the gauge theory
state and can be written again as (199), where this time |+〉 is the tensor product of copiesof |+〉 over all the bulk qubits and the face qubits on the boundary. The same argument as in
Section (7.1) applied to U g (z1∗ ) gives the relation
(206)
U g (cc ∗2 )|ψ3d 〉 = UC Z Obp (U g (z1∗ )|ψ2d 〉) ⊗ |+〉bulk ,
where
 −1
−1
UC Z
U g (cc ∗2 ) := UC Z Obp U g (z1∗ )Obp

Y

∗

∗

σ2 )
σ ∗2 )a(cc 2 ;σ=±K(σ∆∗2 \∆∗2σ ∗2 ∈∆

Y

∗

∗

σ2 )
σ ∗2 )a(cc 2 ;σ
,
K(σ

(207)

∆∗2
σ ∗2 ∈∆and c ∗2 is an arbitrary 2-chain on the dual lattice such that its restriction to the boundaryσ ∗2 ) = X (σσ ∗2 )Z(∂∂ ∗σ ∗2 ) generates the gauge transformation
coincides with z1∗ . The operator K(σof the bulk theory defined by the HamiltonianXσ 2 )Z(∂∂ σ 2 ) ,
H =−
X (σ
(208)
∆2
σ 2 ∈∆σ ∗2 )Z(∂∂ ∗σ ∗2 ) = 1. It is a (3 + 1)-dimensional generalized Isingand the Gauss law constraint X (σmodel coupled to the topological Z2 gauge theory (with 1-form gauge symmetry) with 2-form
σ 2 ), and has the RBH cluster state as the unique ground state. Thus the bulkgauge field X (σboundary correspondence discussed in Section 7.1 for the Ising model naturally generalizes to
the gauge theory.
We summarize in Table 4 the global and gauge symmetries of the resource state in the bulk
and the simulated theory on the boundary.

8

Conclusions & Discussion

8.1

Discussion: comparison to circuit-based quantum simulation

Let us compare the resource required for MBQS of M(d,n) with that for the standard circuitbased quantum simulation. Let us assume that we prepare gCS(d,n) from |+〉 by applying the39

SciPost Phys. 14, 129 (2023)

C Z gate. We have for MBQS,
MB
Nqubit= (d Cn + d Cn−1 ) × (L1 × · · · × L d ) ,

(209)

NCMB
Z = 2n × d Cn × (L 1 × · · · × L d ) ,

(210)

MB
MB
Nmeasure= Nqubit− d Cn−1 × (L1 × · · · × L d−1 ) ,(211)

where notations are self-explanatory. The execution time T MB for MBQS is governed by the
duration Tmeasure to perform the single-qubit measurement with X - or Z-basis and the duration
Trot to perform rotations. The latter needs to be executed after the former measurements are
performed since the rotation angle needs to be adapted. We have
 T0 + L d × (3Tmeasure + 2Trot ) , for n = 1 ,
MB
T =
(212)
T0 + L d × (4Tmeasure + 3Trot ) , for n ≥ 2 (energy cost) , T + L × (4T+ 2T ) , for n ≥ 2 (error correction) ,0dmeasurerot

where T0 (= TH + TC Z ) is the constant time for preparing gCS(d,n) .
For the circuit-based quantum simulation, let us assume that we have a (d −1)-dimensional
processor whose qubits are placed on n- and (n − 1)-cells. We consider the following unitaryacting on the simulated state |ψ〉 on (n − 1)-cells and an ancilla |+〉 j on an n-cell to realize themulti-body interaction, which is naturally related to our MBQS:
Y

Y

Q
(213)
C Z j,k e iξX j
C Z j,k |ψ〉|+〉 j = e iξ k Zk |ψ〉|+〉 j .
kkHere k corresponds to a qubit for |ψ〉. Then, the resource to perform the circuit-based quantum
simulation would beCB= (d−1 Cn + d−1 Cn−1 ) × (L1 × · · · × L d−1 ) ,Nqubit(214)

NCCBZ = 2n × d−1 Cn × (L1 × · · · × L d−1 ) ,

(215)

CB
CB
,
= Nqubit
Nrot

(216)

CB
where NCCBZ (Nrot) is the number of C Z (e iξX ) gates required on the processor. For n = 1, each
Trotter step has four layers of unitaries: (1) e iξX for (n − 1)-cells (2) C Z gates (3) e iξX for
n-cells (4) C Z gates. For n ≥ 2, we have further layers of unitaries to enforce the Gauss law
constraint. We find that the execution time T CB is given as follows:
for n = 1 , L d × (2TC Z + 2Trot ) ,CB
T = L d × (4TC Z + 3Trot ) ,(217)
for n ≥ 2 (energy cost) , L × (4T + 2T + T),
for n ≥ 2 (error correction) .
d

CZ

rot

measureBoth schemes demand run time linear in the number Trotter steps: MBQS has advantage
over the circuit-based approach when the measurement can be executed faster than the C Z
gate. This advantage is achieved at the cost of using more qubits: N MB ∼ L d × N CB .
As discussed in MBQC literatures, it is also interesting to pursuit advantage by shifting thedifficulty of entangling gates to the generation of cluster states: one can use beam splittersinstead of the C Z gates to generate large-scale continuous-variable cluster states.

8.2

Conclusions and future directions

In this work we introduced a family of resource states gCS(d,n) (generalized cluster states), andshowed that the Hamiltonian quantum simulation of Wegner’s model M(d,n) , with an (n − 1)form gauge field for the Z2 gauge group in d spatial dimensions, can be implemented by40

SciPost Phys. 14, 129 (2023)

specifically adapted single qubit measurements of gCS(d,n) . We devised methods to enforce
the Gauss law constraint based on syndrome measurements and the energy penalty. We also
generalized the simulation protocols to the gauge group ZN . By attaching ancillas and allowing
two-qubit measurements, we can also perform the imaginary-time quantum simulation. We
studied the correspondence (for gauge group ZN ) between the generalized cluster states andthe statistical partition functions of M(d,n) regarded as classical spin models. We also proposeda measurement-based protocol for simulating the Kitaev Majorana chain. We demonstrated
that gCS(d,n) has a symmetry-protected topological order with respect to generalized global
symmetries that are related to the gauge symmetries of the simulated gauge theories.
For the model M(d,n) with the gauge symmetry generated by G(σn−2 ), the analogue of the
1-form symmetry of the resource state gCS(3,2) is the (n − 1)-form symmetry of gCS(d,n) . We
expect that this higher-form symmetry can be used for the syndromes and the enforcement ofgauge invariance for the model M(3,2) would generalize to M(d,n) .
It is possible, at least formally, to make the gauge group continuous. In Appendix C, we
discuss an approach to simulating the non-compact U(1) gauge theory using the continuousvariable cluster state [79,80]. The method we present for this gauge group should be regarded
formal due to the divergence coming from integrating over non-compact variables, and inexperiments the related issue is the imperfection of the continuous-variable cluster state dueto the finite squeezing. Nonetheless, we draw readers’ attention to recent development ingenerating large scale cluster states using photons [81–84]. At the moment, the best approach(ZN )to simulating compact U(1) theories is to take N large in the model M(d,n). It would also be
important to generalize the MBQS scheme to non-Abelian gauge groups.
Another future direction is the measurement-based simulation of more realistic high energy
theories such as QED, QCD, the Standard Model and the Grand Unified Theories, as well as
quantum-many body systems in condensed matter physics. In this respect, quantum simulationof Kitaev’s Majorana chain we considered in this work would give us a hint on how to combine
Dirac or Weyl fermions in a resource state. It would also be interesting to relate the SPT
order of the bulk resource states to the dynamical phases of the simulated models, where thebulk-boundary correspondence we observed in our work may be useful.
The presence of an SPT order has been suggested to be an important ingredient of resource
states for the ability to perform the (universal) MBQC [85–96]. More recently, it was shownthat certain quantum states with the long-range entanglement can be obtained with constantdepth operations via measuring states in non-trivial SPT phases [24, 39, 40, 97, 98], and there
have been emerging new aspects of complexity in quantum-many body systems through thelens of measurements. In Appendix F we construct a tensor network representation of thegeneralized cluster state and show that a projective representation appears via the action ofgeneralized global symmetries. It would be nice to find a continuous deformation of the tensor
network within the corresponding SPT phase. As a related future direction, it would also beinteresting to relate the ability to perform MBQS of a quantum-many body system with asymmetry to the SPT order of an appropriate resource state.

Acknowledgement
We thank Lento Nagano for helpful discussions. HS thanks Tzu-Chieh Wei for helpful guidanceand comments on manuscripts. We also acknowledge the usefulness of the textbook [99]. HS
was partially supported by the Materials Science and Engineering Divisions, Office of Basic
Energy Sciences of the U.S. Department of Energy under Contract No. DESC0012704. The
work of TO is supported in part by the JSPS Grant-in-Aid for Scientific Research No. 21H05190.

41

SciPost Phys. 14, 129 (2023)

A

Proof of equations

In this appendix, we prove some identities related to measurements. To make the presentationcompact, we give proofs in the qudit case, which reduces to the qubit case by taking N = 2,
ω = −1, and F = H. We denote F |s〉 by |s̃〉, which are the X -eigenstates with eigenvalues ωs .
Proof of (38), (59), and (139)
Consider a general state |Ψ〉 on qudit 0. We wish to transport it to qudit 1 by a measurement in the basis f (Z)|s̃〉 s = 0, 1, . . . , N − 1 , where f is an arbitrary function such thatf (Z)† = f (Z)−1 . The choice f (Z) = 1 corresponds to basis M(X ) in (58) and (135), andf (Z) = e i(ξZ+h.c.) corresponds to M(B) in (56) and (138). We claim that the following identity
holds:
1 −ε s†ε†(A.1)
0 〈s̃| f (Z0 ) (C Z01 ) |Ψ〉0 |+〉1 = p F1 Z1 f (Z1 ) |Ψ〉1 .
N
Here ε = ±1. The equations (38), (59), and (139) follow from (A.1).
To prove (A.1), we expand |+〉 in the Z-basis and use the relations Z|s̃〉 = |sß− 1〉 and±1f
F |s〉 = |±s〉. The LHS of (A.1) becomes
N −1
N −1
N −1
1 X
1 X
1 X
εt
†
s
†
e
〈s̃|Z f (Z) |Ψ〉|t〉1 = p〈εt|Z f (Z) |Ψ〉|t〉1 = p〈t|F −ε Z s f (Z)† |Ψ〉|t〉1 ,
p
N t=0
N t=0
N t=0
(A.2)

which equals the RHS of (A.1).
Proof of (34), (55), and (134)
Suppose that the |+〉 state (118) of qudit 0 is entangled with a general state |Ψ〉 of qudits1,.
.
 . , m by CZ gates or their inverses. We consider measuring qudit 0 in the basis
f (X )|s〉 s = 0, 1, . . . , N − 1 , where f is again an arbitrary function. The choicef (X ) = e i(ξX +h.c.) corresponds to basis M(A) . The effect of such a measurement on the state|Ψ〉 is expressed as the identity
m
m
m
Y
s
†  Y
1 Y
†
εa
−εa
(A.3)
|+〉
|Ψ〉
=
Zaεa |Ψ〉1...m .
〈s|
f
(X
)
(C
Z
)
f
(Z
)
p
0
1...m
0
0
0a
a
N
a=1
a=1
a=1
The equations (34), (55), and (134) follow from (A.3).
P
To prove (A.3), we
again
expand|+〉intheZ-basis,insert1=
u |ũ〉〈ũ|, and use theprelation 〈ũ|t〉 = ω tu / N . The LHS of (A.3) becomesY1 X
u ∗ −suu
εa t|Ψ〉1...m .
(A.4)
f
(ω
)
ω
ω
Z
a
N 3/2 t,ua
Q −εThe summation over t forces ωu to equal a Za a , proving (A.3).
A qumode identity
We have†εε −imQ̂f (Q̂)† |Ψ〉 1 .
(A.5)
0 〈m p | f (Q̂ 0 ) (C Z01 ) |Ψ〉0 |0 p 〉1 = F e
The proof is similar to the one for (A.1).
Next, consider a qumode 0 in state |0 p 〉 and qumodes 1,...,m in a general state |Ψ〉. We
prove the relation
 m
s
n
Y
Y
X
†
1
†
εa
iεa Q̂ a(C Z0a ) |0 p 〉0 |Ψ〉1...m = pefεa Q̂ a |Ψ〉1...m .
(A.6)
0 〈sq | f ( P̂0 )2π a=1aa=1
The relation (A.6) can be shown by using the definition C Z0a = e iQ̂ 0Q̂ a and by noting thate−iuQ̂ a Q̂ b F ( P̂a )e iuQ̂ a Q̂ b = F ( P̂a + uQ̂ b ) ,for a general function F .
42

(A.7)

SciPost Phys. 14, 129 (2023)

B

(Z )

N
Continuous-time limit of model M(d,n)(Z )N
In this section, we give a derivation of the continuous-time limit for the model M(d,n),
generalizing the discussion for gauge theories [20, 100, 101]. Consider the ZN variableu ∈ {1, ω, ..., ωN −1 } with ω = e2πi/N . We define for a k-chain ck ,u(ck ) =Yua(ck ;σk ) .

(B.1)

X

(B.2)

σk

Consider an anisotropic action
I = −J tX
σ ntu(∂∂ σ nt ) − Jsσ snu(∂∂ σ sn ) + h.c.

σ nt ) is n-cells stretched in spatial (temporal) directions. The gauge transformation
Here, σ sn (σthat leaves the action invariant is defined for each (n − 2)-cell and given by
σ n−2 ;∂∂ σ n−1 )σ n−2 ) : uσ n−1 → ωa(σG (σuσ n−1 ,σ n−1 ∈ ∆ n−1 .
∀σ

(B.3)

We use the gauge transformation to bring temporal fields to unity,t
σ n−1) = 1,u(σt= σ i1 ,··· ,in−2 ,d (x) .
with σ n−1(B.4)

Then, using (120), the first term in the action becomes(−1)n−1
(−1)nσ i1 ,··· ,in−1 (x + ed ))σ i1 ,··· ,in−1 (x))
u(σ+ h.c.
u(∂∂ σ nt ) + h.c. = u(σσ i1 ,··· ,in−1 (x + ed ))u(σσ i1 ,··· ,in−1 (x))∗ + h.c.
= u(σ

(B.5)

Now the two factors on the right hand side are defined on the same (n − 1)-cell in respective
time slices, thus we can simply express it as( j+1)( j)u(σn−1 )u(σn−1 )∗ + h.c.

(B.6)

We can further rewrite it using a norm and we obtain


X
X
X

( j+1)
( j)
2
( j)
I =−
Jt
2 − |u(σn−1 ) − u(σn−1 )| + Js(u(∂ σn ) + h.c.)
σn−1jredef.

→

X
j


Jt

X
σn−1

σn

( j+1)( j)|u(σn−1 ) − u(σn−1 )|2 − JsX(u(∂ σ(nj) ) + h.c.) .

(B.7)

σn

P
Now, we consider the partition function Z = {u} e−β I[{u}] , and express it as the trace ofa product of transfer matrices, Z = tr[(e−τH ) T ]. We will identify the parameters and take thecontinuum limit as follows:
β Jt → ∞ ,β Js → λe
τ=e

(B.8)

−β J t C1t−β J t C1twith C1t ≡ 2(1 − cos( 2πi
N )).
43

,

,

(B.9)
(B.10)

SciPost Phys. 14, 129 (2023)

The diagonal element of the transfer matrix between j and j + 1 is identified asPX( j)〈{u}|(1 − τH)|{u}〉 ≃ eβ Js σn (u(∂ σn )+h.c.) ≃ 1 + β Js(u(∂ σ(nj) ) + h.c.) ,(B.11)

σnwhich gives us the identification for the diagonal part of the Hamiltonian with the ZN phase
operator,
X
Hdiag = −λ(Z(∂ σn ) + h.c.) .
(B.12)
σnNext, we consider a single-shift transition, {u}∆n−1 → {u′ }∆n−1 = {ω±1 uσn−1 }∪{u}∆n−1 \σn−1 .
tsThis gives the minimum energy cost 2J t (1−cos( 2πiN )) ≡ J t C1 from the temporal term and Js C1from the spatial term, the latter of which depends on specific field configurations. We identify
in the continuous-time limit as〈{u′ }|(−τH)|{u}〉 = e−β J t C1 eβ Js C1 = τ + O(τ2 ) .
st(B.13)

Transitions that involve more than one shift of the field become higher order contributions on
the right hand side. Thus we can identify
X(X σn−1 + h.c.) .
(B.14)
Hoff-diag = −σn−1

The total Hamiltonian in the continuous-time limit is then
X
X
(Z(∂ σn ) + h.c.) .
(X σn−1 + h.c.) − λH =−(B.15)

σnσn−1

Finally, we examine the gauge symmetry of this theory. We have used the gauge transformation to bring the temporal fields to unity, and there are residual gauge transformations thatleave the temporal fields invariant. That is, we consider the gauge transformation constantover time:Y( j)
G (σn−2 ) ,(B.16)j( j)

where σn−2 = σi1 ,··· ,in−2 (x, j) with x being the spatial coordinate, j being the time coordinate,
and i1 < ... < in−2 < d. Namely, the product is taken over time for (n − 2)-cells that do notstretch in the time (d-th) direction. This leaves us the gauge transformation that does notdepend on time, and it can be written asY
G(σn−2 ) =X σa(∂ σn−1 ;σn−2 ) .
(B.17)σn−1

C

Gauge group R

C.1

Hamiltonian formulation of M(d,n)

n−1

(R)

In this section, we consider the model M(d,n) with continuous groups. The model consists ofthe gauge degrees of freedom on the (n − 1)-cells σn−1 of the (d − 1)-dimensional lattice. The
group element on each σn−1 is given by
Uσn−1 = e iθσn−1 ,
44

(C.1)

SciPost Phys. 14, 129 (2023)

with the gauge field θσn−1 .
When θσn−1 ∈ [0, 2π), we refer the group as U(1). On the other hand, θσn−1 ∈ (−∞, +∞),
we refer the group as non-compact U(1) or simply R. Our MBQS considered in the following
part corresponds to the non-compact U(1) group.
We write its conjugate momentum as
Lσn−1 = −i∂.
∂ θσn−1(C.2)

The canonical commutation relation is[θσn−1 , Lσ′n−1 ] = iδσn−1 ,σ′n−1 .

(C.3)

The Hamiltonian is given byH=

g2 X
2 σ ∈∆
n−1

with θ (∂ σn ) :=

2
Lσ

n−1

−

n−1

1 X
cos [θ (∂ σn )] ,g 2 σ ∈∆
n

(C.4)

n

σ1 ∈∂ σn a(∂ σn ; σn−1 )θσn−1 . The Gauss law readsP

G(σn−2 ) = Q(σn−2 ) ,X
G(σn−2 ) := −(C.5)
Lσn−1 +(∂ σn−1 )+ ∋σn−2

X

Lσn−1 ,

(C.6)(∂ σn−1 )− ∋σn−2

for each (n − 2)-cell σn−2 ∈ ∆n−2 . Here Q(σn−2 ) ∈ R is the external charge. We add to the
Hamiltonian an energy cost term,Λ(G(σn−2 ) − Q(σn−2 ))2 ,(C.7)(Λ > 0) to enforce the Gauss law constraint.

C.2

(R)

MBQS of M(d,n)(R)

We consider MBQS of the model M(d,n) using the continuous-variable cluster state [79, 80].

We introduce the qumode basis by the eigenstate of P̂ and Q̂ operators, i.e. [Q̂, P̂] = i, suchthatP̂|t p 〉 = t|t p 〉, Q̂|sq 〉 = s|sq 〉,(C.8)

The Pauli operators are generalized to the Weyl-Heisenberg operators as
Z(s) = e isQ̂ ,X (s) = e−is P̂ .

(C.9)

X (s)|t q 〉 = |(t + s)q 〉 .

(C.10)

Note that
Z(s)|t p 〉 = |(t + s) p 〉 ,The |+〉 state corresponds to |0 p 〉 and the C Z gate is defined by
C Za,b = e iQ̂ a Q̂ b ,(C.11)and the Fourier transformation, which corresponds to the Hadamard gate, is defined by


iπ 2
2
( P̂ + Q̂ ) .
(C.12)
F = exp
445

SciPost Phys. 14, 129 (2023)

This Fourier transformation flips the basis as F |sq 〉 = |s p 〉.16 Also, the following equalities
hold:17F 4 = I, F †Q̂F = − P̂ ,F † P̂ F = Q̂ .

(C.15)The measurement pattern is as follows:(1) σ n = σn × { j}φ

Let DP = e iφ( P̂) and consider the measurement in the basis¦©
φ
M(R,A) := DP X (m)|0q 〉 m ∈ R .

(C.16)(2) σ n−1 = σn−1 × { j} Consider the measurement in the basis

M(X ) := {Z(m)|0 p 〉 | m ∈ R} .

(C.17)(3) σ 1 = σ0 × [ j, j + 1] We impose the Gauss law constraint by introducing the energy costterm. We consider the measurement basis©
¦
φ(C.18)
M(R,A) := DP X (m)|0q 〉 m ∈ R .
φφ(4) σ 1 = σ1 × [ j, j + 1] Let DQ be DQ = e iφ(Q̂) . Consider the measurement basis¦©
φ
M(R,B) := DQ Z(m)|0 p 〉 m ∈ R .

(C.19)For the A-type measurement we use the identity (A.6) and For the B-type measurement we
use the identity (A.5). The unitaries from the sequence (i)-(iv) result in the following unitary:Yn−1[F (−1) e−im4Q̂ e−iφ4 (Q̂) ]σn−1
σn−1





 p1
e ia(∂ σn−1 ;σn−2 )Q̂ σn−1
2π
σn−2
σn−1
Y

n
×
F (−1) e−im2Q̂
×

Y

Y


×
16
17





X

exp −iφ3

σn−1

a(∂ σn−1 ; σn−2 )Q̂ σ1




σn−1

σn−1

 p1
2π
σn

Y

s3 (σn−2 )Ye ia(∂ σn ;σn−1 )Q̂ σn−1s1 (σn )
exp −iφ1

σn−1



X
σn−1


a(∂ σn ; σn−1 )Q̂ σn−1F |sq 〉 = |s p 〉, F |s p 〉 = |(−s)q 〉, F |(−s)q 〉 = |(−s)
 p 〉. and F |(−s)
 p 〉 = |sq 〉.
For a general Gaussian rotation R(θ ) = exp i θ2 (Q̂2 + P̂ 2 ) for the (Q̂, P̂) quadrature, we haveR−1 (θ )Q̂R(θ ) = cos(θ ) Q̂ − sin(θ ) P̂ ,(C.13)R (θ ) P̂R(θ ) = sin(θ ) Q̂ + cos(θ ) P̂ .
The Fourier transform is a special case with F = R π2 .

(C.14)−146

SciPost Phys. 14, 129 (2023)=

Y[e−im4 (−1)

n−1P̂ −iφ4 ((−1)n−1 P̂)

e

σn−1

Y 1
×
p
2π
σn−2




Y

e

i(−1)n−1 a(∂ σn−1 ;σn−2 ) P̂σn−1



X
σn−1

×

e

s3 (σn−2 )σn−1

× exp −iφ3
Y

]σn−1

(−1)n−1 a(∂ σn−1 ; σn−2 ) P̂σ1 

−im2 Q̂ σn−1

σn−1


×

 p1
2π
σn

Y



Y

e ia(∂ σn ;σn−1 )Q̂ σn−1s1 (σn )exp −iφ1

σn−1

X
σn−1


a(∂ σn ; σn−1 )Q̂ σn−1 . (C.20)

When the measurement outcome is s1 = m2 = s3 = m4 = 0 everywhere, we choose thefunction φi (i = 1, 3, 4) as follows:
φ1 (x) = −δt1
cos(x) ,g2

φ3 (x) = Λδt (x − q)2 ,φ4 (x) = δt

g2 2
x ,
2

(C.21)

with q chosen as Q(σn−2 ) for the measurement at σn−2 ∈ ∆n−2 . By identifying the gauge field
θσn−1 with Q̂ σn−1 and the conjugate momentum Lσn−1 with (−1)n−1 P̂σn−1 , we obtain the desired
time evolution unitary for the non-compact U(1) lattice gauge theory:

Y
Y

g2 2
−iHδt
e
≃
exp −iδt Lσ
exp −iΛ δt (G(σn−2 ) − Q(σn−2 ))2
n−1
2
σn−1
σn−2


Y
1
×
exp iδt 2 cos(θσn ) .
(C.22)gσn

To deal with the byproduct operators with mi ̸= 0, we just need to shift the argument of
the function φi (s). This follows from the fact that we only have X (s) and Z(s) for byproductoperators, so arranging the unitary in the appropriate form only requires the commutation
relations φ(Q̂)X (s) = X (s)φ(Q̂ + s) and φ( P̂)Z(s) = Z(s)φ( P̂ + s) for a general function φ(x).
Experimentally, if one wishes to realize the MBQS with the measurement bases (C.21)
using photons, non-Gaussian function φ1 (x) = δt g12 cos(x) would be hard to realize. One
may instead consider a regime where the approximation cos(x) ≃ 1− 12 x 2 is valid: namely, thecontinuum limit, where the lattice spacing a is small. One can also improve the approximation
by expressing higher-order terms in cosine using an expansion with Gaussian gates. See e.g.
[102, 103].

C.3

(R)

A formal correspondence between gCS(d,n) and the Euclidean path integral

For gauge group R, we define

⊗∆
∆n
(R)
∆ n−1
|φ(d,n) (J)〉 := e−iφ( P̂) |0 p 〉|0 p 〉⊗∆.

47(C.23)

SciPost Phys. 14, 129 (2023)

We note that |0 p 〉 = (2π)−1/2(R)Rdθ |θq 〉. We find(R)〈φ(d,n) (J)|gCS(d,n) 〉
Y 1
X
∆ n−1
∆ n−1
= 〈0 p |⊗∆exp(iφ(a(∂∂ σ n ; σ n−1 )Q̂σ n−1 ))|0 p 〉⊗∆p2πσnσ n−1 ∈∂∂ σ n
Z Y
Y
X
1
′
′
dθ
dθ
〈θ
|θ
〉
exp(iφ(a(∂∂ σ n ; σ n−1 )θσ n−1 ))
=
σ
q
σ
σ
q
n−1
n−1
∆ n−1 |+|∆
∆ n |/2n−1(2π)|∆
σ n−1
σn
σ n−1 ∈∂∂ σ n
Z Y
X
X
1
dθσ n−1 exp(iφ(a(∂∂ σ n ; σ n−1 )θσ n−1 ))=∆∆ n |/2
|∆
|+|∆
n−1
(2π)
σ n−1
σn
σ n−1 ∈∂∂ σ n
Z Y
1dθσ n−1 exp(i I[{θσ n−1 }]) ,(C.24)=∆ n−1 |+|∆
∆ n |/2(2π)|∆
σn−1

with
φ(x) = J(e i x + e−i x ) .

(C.25)

We note that the right hand side of the equation (C.24) is divergent since the integration isover (−∞, +∞) and the integrand is periodic. Therefore the equation is formal and requiresa regularization (including division by the divergent volume of the gauge group) or gauge
fixing. Note that the rotation J → iβ J brings the expression to the Euclidean path integral.

D

Stabilizer operators as gauge transformations

In [41, 104], a reinterpretation of the MBQC as gauge theory is given. In particular, stabilizer
operators of the 1-dimensional cluster state (gCS(1,1) in our notation) are reinterpreted asgauge transformations. In this appendix we show that this extends very naturally to gCS(d,n)
for general d and n.
Let us consider the generalized stabilizer state |gCS(d,n) 〉 defined by the stabilizers
K(σn ) = X (σn )Z(∂ σn−1 ) ,K(σn−1 ) = X (σn−1 )Z(∂ ∗ σn−1 ) .

(D.1)(In this Appendix, we use normal fonts throughout.) Let us consider measuring the operators(corresponding to the B-type measurement defined in Section 3)
B[qn−1 (σn−1 )] = cos(2ξ)X (σn−1 ) + (−1)qn−1 (σn−1 ) sin(2ξ)Y (σn−1 ) ,

(D.2)

∗
and their measurement results sd−n+1(σ∗d−n+1 ) at σn−1 = σ∗d−n+1 ,18 as well as∗∗∗
B[qd−n(σ∗d−n )] = cos(2ξ)X (σ∗d−n ) + (−1)qd−n (σd−n ) sin(2ξ)Y (σ∗d−n ) ,(D.3)

and their measurement results sn (σn ) at σ∗d−n = σn . Recall that a k-cochain is a linear map
C k → Z2 ,(D.4)

where Ck is the group consisting k-chains. We denote the group of k-cochains by C k . The
corresponding objects on the dual lattices will be denoted by Ck∗ and (C ∗ )k . We regard various
quantities as cochains on the primal and dual lattices as cochains:
qn−1 ∈ C n−1 ,∗sd−n+1∈ (C ∗ )d−n+1 ,∗qd−n∈ (C ∗ )d−n ,18sn ∈ C n .

To simplify notations we write σ j as σ j .

48

(D.5)
(D.6)

SciPost Phys. 14, 129 (2023)

Consider gauge transformations with gauge parameters Λn−1 ∈ C n−1 and Λ∗d−n ∈ (C ∗ )d−n ,generated by the operators
Y
K(Λn−1 ) :=K(σn−1 )Λn−1 (σn−1 ) ,(D.7)
σn−1 ∈∆n−1

Y

K(Λ∗d−n ) :=∗∗K(σ∗d−n )Λd−n (σd−n ) .

(D.8)

σ∗d−n ∈∆∗d−n

Conjugation by the operator K(Λn−1 ) induces the transformations
B[qn−1 (σn−1 )] 7→ B[(qn−1 + Λn−1 )(σn−1 )] ,∗∗B[qd−n(σ∗d−n )] 7→ (−1)(dΛn−1 )(σn ) B[qd−n(σ∗d−n )] .

(D.9)
(D.10)

Similarly, conjugation by K(Λ∗d−n ) gives∗∗∗

B[qn−1 (σn−1 )] 7→ (−1)(d Λd−n )(σd−n+1 ) B[qn−1 (σn−1 )] ,(D.11)∗∗
B[qd−n(σ∗d−n )] 7→ B[(qd−n+ Λ∗d−n )(σ∗d−n )] .

(D.12)

Taken together, the gauge transformations of the parameters are
qn−1 7→ qn−1 + Λn−1 ,sn 7→ sn + dΛn−1 ,
∗
∗
∗ ∗
∗
∗
sd−n+1 7→ sd−n+1 + d Λd−n , qd−n7→ qd−n+ Λ∗d−n .

(D.13)

Next, let us consider the operators (corresponding to the A-type measurement)∗∗∗A[pd−n+1(σ∗d−n+1 )] = cos(2ξ)Z(σ∗d−n+1 ) + (−1) pd−n+1 (σd−n+1 ) sin(2ξ)Y (σ∗d−n+1 ) ,(D.14)and their measurement results t n−1 (σn−1 ) at σn−1 = σ∗d−n+1 , as well asA[pn (σn )] = cos(2ξ)Z(σn ) + (−1) pn (σn ) sin(2ξ)Y (σn ) ,(D.15)∗and their measurement results t d−n(σ∗d−n ) at σ∗d−n = σn . Considerations similar to the above
give the gauge transformations

t n−1 7→ t n−1 + Λn−1 ,pn 7→ pn + dΛn−1 ,
∗
∗
∗ ∗
∗
∗
pd−n+1 7→ pd−n+1 + d Λd−n , t d−n7→ t d−n+ Λ∗d−n .

(D.16)

We confirmed, in the (d, n) = (3, 2) case, that the simulated unitary evolution including the
byproduct operators in (62) is invariant under the gauge transformations (D.13) and (D.16)
σ 1 ) (σσ 1 = σ1 × {0}) is not ain the bulk. Note that at the boundary x 3 = 0, the stabilizer K(σsymmetry of the resource state with an arbitrary initial wave function |ψ2d 〉∆1 ×{0} , so there is
not a gauge transformation corresponding to this stabilizer.

E

Gauging map via minimally coupled gauge fields

The aim of this appendix is to derive the gauging map (184) by minimally coupling gauge
s ym
s ym
fields. For simplicity we use normal fonts and focus on the gauging map Γ : Hn → Hn−1 .
The other half of (184) can be treated in the same way.
Minimal coupling means introducing new qubits on (n−1)-cells and replacing the operatorZ(∂ ∗ σn−1 ) by the product Z(∂ ∗ σn−1 )Z ′ (σn−1 ), where the gauge field Z ′ (σn−1 ) is the Pauli Z

49

SciPost Phys. 14, 129 (2023)

operator for the new qubit on σn−1 . The product is invariant under the gauge transformation
generated by
G(σn ) = X (σn )X ′ (∂ σn ) .
(E.1)
s ym

s ym

The spaces Hn and Hn−1 consist of the states invariant under the symmetry operators
X (zn ) and Z ′ (zn−1 ), for which zn and zn−1 satisfy ∂ zn = 0 and ∂ ∗ zn−1 = 0, respectively.
Let Hinvn ⊂ Hn be the space of the states invariant under G(σn ). We define the linear map
s ym
s ym
s ym
is mapped toΞ : Hn → Hinvn ⊗ Hn−1 such that |ψ〉 ∈ Hn
Y 1 + G(σ )nΞ(|ψ〉) := N0|ψ〉 ⊗ |0〉′ ,
(E.2)
2
σ ∈∆
n

n

where N0 is a normalization constant. We can expandX|ψ〉 =α(cn )|cn 〉 ,(E.3)

cn ∈CnQ
where α(cn ) are complex coefficients. We note that inserting σn G(σn )a(cn ;σn ) in front of |cn 〉does not change (E.2). Thus we can writeX
Y 1 + G(σ )n|0〉 ⊗ |∂ cn 〉′ .
(E.4)
Ξ(|ψ〉) = N0α(cn )
2
σ ∈∆
c ∈C
n

n

n

n

Stripping off (1 + G(σn ))/2|0〉 leads to the gauging map
XΓ (|ψ〉) := Nα(cn )|∂ cn 〉′ .

(E.5)

cn ∈Cn

The new normalization constant N can be determined by computing the norm of Γ (|ψ〉):N = |Zn |−1/2 , where |Zn | is the number of n-cycles.
The gauging map of operators, A 7→ AΓ , is given by eliminating unprimed operators bysolving the Gauss law constraint G(σn ) = 1 for primed operators.

F

Tensor network representation of gCSd,nIn this appendix we study a tensor network representation of the generalized cluster state
gCSd,n . Since we only consider the d-dimensional bulk cell complex, we use non-bold symbols
to simplify notations.
Our starting point is the expression (29) in terms of the entangler. By introducing complete
sets of states, we writeXYn ;σn−1 ) |+〉⊗∆n |+〉⊗∆n−1|gCSd,n 〉 =|cn 〉〈cn |C Zσa(∂ σ,σ
σn−1 ∈∆n−1
σn ∈∆ncn ∈Cn

=

X

n−1

n

|cn 〉〈cn |+〉⊗∆n |∂ cn 〉(X )cn ∈Cn=

1

X

2|∆n |/2 c ∈Cn|cn 〉|∂ cn 〉(X ) ,

(F.1)

n

where the superscript (X ) indicates that the state is in the X -eigenbasis. To introduce tensors, we reexpand thePexpression in the X -eigenbasis for both n- and (n − 1)-cells by applyingP′ (X )(X ) ′(X )(X )
|c
〉
〈c
|
and
〈cn−1 | from the left. We obtain′ccn−1 |cn−1 〉nnn|gCSd,n 〉 =1

X2|∆n |/2 c ,c ′ ,c
n

n

(X )

〈cn′ |cn 〉(X ) 〈cn−1 |∂ cn 〉(X ) · |cn′ 〉(X ) |cn−1 〉(X ) .

n−1

50

(F.2)

SciPost Phys. 14, 129 (2023)′
σn γσn σn , cn =σn βσn σn , and cn−1 =mod 2
denote the Kronecker delta δ•,• . Then we have

We write cn =

(X )

P

P

〈cn′ |cn 〉 = 2−|∆n |/2

σn−1 ασn−1 σn−1 . We use δ(•, •) to

P

Y
Y
X
(−1)βσn γσn (X ) 〈cn−1 |∂ cn 〉(X ) =δ(ασn−1 ,a(∂ σn ; σn−1 )γσn ) ,
σn

σn−1

σn

(F.3)
so that|gCSd,n 〉 =1

X2|∆n | α,β,ρTσn [βσn ]{ρσ ,σnn−1|σn−1 ⊃∂ σn } Tσn−1 [ασn−1 ]{ρσn ,σn−1 |∂ σn ⊃σn−1 }× |{βσn }〉(X ) |{ασn−1 }〉(X ) .

(F.4)

Here the sum is over α ∈ {0, 1}|∆n−1 | , β ∈ {0, 1}|∆n | , and ρ ∈ {0, 1}|{(σn ,σn−1 )|∂ σn ⊃σn−1 }| . The
tensors are given explicitly asXYTσn [βσn ]{ρσ ,σ |σn−1 ⊂∂ σn } =(−1)βσn γσn
δ(ρσn ,σn−1 , γσn ) ,
(F.5)
n

n−1

γσn =0,1

σn−1 ⊂∂ σnand!
Tσn−1 [ασn−1 ]{ρσ ,σnn−1|∂ σn ⊃σn−1 } = δX

ασn−1 ,

ρ(σn , σn−1 ).

σn :∂ σn ⊃σn−1

For the the tensor Tσn [βσn ] at the n-cell σn , we have index ρσn ,σn−1 for each (n − 1)-cell
contained in the boundary of σn . For the tensor Tσn−1 [ασn−1 ] at the (n − 1)-cell σn−1 , we havesuch an index for each n-cell that contains σn−1 in its boundary.
As an example, let us consider gCS1,1 , i.e., the standard one-dimensional cluster state. The
above reduces to the matrix product state representation given by the matrices
Te [0] = 1 ,Te [1] = σz ,Tv [0] = 1 ,Tv [1] = σ x ,(F.6)

for e ∈ ∆1 and v ∈ ∆0 . For neighboring v and e, we have
Tv [0]Te [0] = 1 , Tv [0]Te [1] = σz ,(F.7)

Tv [1]Te [0] = σ x , Tv [1]Te [1] = σ x σz ,where σ x and σz are Pauli matrices. We have the relation(−1)aαv (−1) bβe Tv [α v ]Te [βe ] = U(a, b) Tv [α v ]Te [βe ] U(a, b)−1 ,(F.8)

where U(a, b) = σza σ xb is a non-trivial projective representation of Z2 × Z2 . As noted in [85],such a relation can be used to show that the cluster state is in a non-trivial SPT order.
Coming back to the general (d, n), for given σn−1 , let σ′n be any of the n-cells such that′
∂ σn ⊃ σn−1 . The tensors Tσn and Tσn−1 satisfy the relations(−1)βσn Tσn [βσn ]{ρσ ,σ|σn−1 ⊂∂ σn } = Tσn [βσn ]{ρσn ,σn−1 +1|σn−1 ⊂∂ σn } ,(F.9)|∂ σn ⊃σn−1 } = Tσn−1 [ασn−1 ]{ρσn ,σn−1 +δσ ,σ′ |∂ σn ⊃σn−1 } ,(F.10)nn−1

andTσn−1 [ασn−1 + 1]{ρσ ,σ
n

n−1

n

51

n

SciPost Phys. 14, 129 (2023)

Figure 11: An example of the set-up considered in the analysis of the projective representation. We depict a three dimensional unit lattice with x 2 - and x 3 -directions
taken periodic. The red vertical plane represents the 2-brane, and the blue horizontal one represents the dual 2-brane. The 2-cell (red ball) and the 1-cell (blue ball)
are transformed with the global symmetry. The two branes intersect along the x 1 direction and the product of two tensors have two indices in this direction (left andright) on which Z2 × Z2 acts in a projective representation.
which guarantee the invariance of |gCSd,n 〉 under K(σn ). Similarly, for given σn , let σ′n−1 beany of the (n − 1)-cells such that σ′n−1 ⊂ ∂ σn . The same tensors also satisfy the relations
Tσn [βσn + 1]{ρσ ,σnρ
′= (−1) σn ,σn−1 Tσn [βσn ]{ρσ ,σ |σn−1 ⊂∂ σn } ,
|σ
⊂∂
σ
}
n−1
n
n n−1
n−1

(F.11)

and(−1)ασn−1 Tσn−1 [ασn−1 ]{ρσ ,σnn−1|∂ σn ⊃σn−1 } =Y∂ σn ⊃σn−1(−1)ρσn ,σn−1 Tσn−1 [ασn−1 ]{ρσ ,σnn−1|∂ σn ⊃σn−1 } ,(F.12)
which guarantee the invariance of |gCSd,n 〉 under K(σn−1 ).
Let us consider a d-dimensional lattice, which is periodic in the (x 2 , x 3 , . . . , x d )-directions
with length L i = 1 (i = 2, . . . , d). Let us consider a unit hypercube {(x 1 , . . . , x d ) | 0 ≤ x i < 1 for
i = 1, . . . , d} in the lattice. (Note that we allow x i to be 0 but do not allow it to be 1.) We wish
to consider an n-brane operator extended in the (x 1 , x 2 , . . . , x n )-directions and a (d − n + 1)brane operator extended in the (x 1 , x n+1 , . . . , x d )-directions. The former corresponds to then-cell σn (unique in the periodically identified hypercube) extended in the (x 1 , x 2 , . . . , x n )directions, while the latter corresponds to the unique (n − 1)-cells σn−1 (at x 1 = 0) extended
in the (x 2 , . . . , x n )-directions. Let us consider the corresponding product of tensors Tσn [βσn ]and Tσn−1 [ασn−1 ] with the shared indices such as ρσn ,σn−1 summed over the values 0 and 1.
This product satisfies a relation that naturally generalizes (F.8), as a consequence of the relations (F.9)-(F.12). The projective representation U(a, b) acts on one index of Tσn−1 [ασn−1 ]corresponding to the negative x 1 -direction and another index of Tσn [βσn ] corresponding to
the positive x 1 -direction. We note that in (F.9) and (F.12), the transformation of the tensors
is expressed in terms of indices, which contain those in directions other than the x 1 -direction.
One can confirm in (F.4) that such effect can be absorbed by redefinition of summed indices
52

SciPost Phys. 14, 129 (2023)

of other tensors associated with qubits living on the n- and (d − n + 1)-branes. We expect that
more general tensor networks obeying the same relation provide deformations of states awayfrom |gCS(d,n) 〉 within the SPT phase, and speculate that they possess the ability to simulate
the lattice model M(d,n) via measurements.

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