Quantum Gauge Networks A New Kind of Tensor Network

2025-04-29 0 0 1.57MB 21 页 10玖币

Quantum Gauge Networks: A New Kind of Tensor Network

Kevin Slagle1,2,3

1Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005 USA

2Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

3Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology,

Pasadena, California 91125, USA

Although tensor networks are powerful

tools for simulating low-dimensional quantum

physics, tensor network algorithms are very

computationally costly in higher spatial

dimensions. We introduce quantum gauge

networks: a diﬀerent kind of tensor network

ansatz for which the computation cost of

simulations does not explicitly increase for

larger spatial dimensions. We take inspiration

from the gauge picture of quantum dynamics

[1], which consists of a local wavefunction

for each patch of space, with neighboring

patches related by unitary connections. A

quantum gauge network (QGN) has a similar

structure, except the Hilbert space dimensions

of the local wavefunctions and connections

are truncated. We describe how a QGN

can be obtained from a generic wavefunction

or matrix product state (MPS). All 2k-point

correlation functions of any wavefunction for

Mmany operators can be encoded exactly

by a QGN with bond dimension O(Mk). In

comparison, for just k= 1, an exponentially

larger bond dimension of 2M/6is generically

required for an MPS of qubits. We provide

a simple QGN algorithm for approximate

simulations of quantum dynamics in any

spatial dimension. The approximate dynamics

can achieve exact energy conservation for

time-independent Hamiltonians, and spatial

symmetries can also be maintained exactly.

We benchmark the algorithm by simulating

the quantum quench of fermionic Hamiltonians

in up to three spatial dimensions.

Contents

1 Introduction 1

2 Quantum Gauge Networks 3

2.1 QGN from Truncation Maps . . . . . . 4

2.2 Truncation Map Construction . . . . . 5

2.2.1 Warmup Example . . . . . . . 5

2.2.2 Generic Case . . . . . . . . . . 6

2.2.3 Long-Range Correlation Func-

tions............... 6

3 Time Evolution Algorithm 7

3.1 Fermion Quench . . . . . . . . . . . . 8

4 Outlook 11

Acknowledgments 12

References 12

A Higgsed Lattice Gauge Theory 15

B Matrix Product State Mapping 15

B.1 MPS Review . . . . . . . . . . . . . . 15

B.2 Quantum Gauge Network from MPS . 16

C QGN Examples 16

C.1 Mixed State Example . . . . . . . . . 16

C.1.1 Kronecker Product Operators . 17

C.2 Cat State Example . . . . . . . . . . . 17

C.3 Bosonic Coherent States . . . . . . . . 18

C.4 Fermion Slater Determinants . . . . . 18

C.5 Rainbow State . . . . . . . . . . . . . 19

D Energy Conservation 19

E Ising Model Quench 19

F Modiﬁed Runge-Kutta 20

1 Introduction

Tensor network algorithms [2,3,4,5,6,7] are very

useful for simulating strongly-correlated quantum

physics. In one spatial dimension, matrix product

state (MPS) algorithms [3,4,5] are often the best

available tool for this task. Although still useful,

tensor network algorithms in higher dimensions [3,

5,8,9,10,11,12,13,14,15,16] typically suﬀer

from computational costs that scale as a high power

of the bond dimension. Therefore, we are motivated

to study a diﬀerent kind of tensor network ansatz

that is more computationally eﬃcient in many spatial

dimensions.

We take inspiration from the gauge picture of

quantum dynamics [1], which adds “gauge ﬁelds” to

Schr¨odinger’s picture in order to make spatial locality

explicit in the equations of motion. In the gauge

Accepted in Quantum 2023-09-04, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.12151v5 [quant-ph] 11 Sep 2023

picture, one ﬁrst chooses a collection of possibly-

overlapping patches of space that cover space. For

example, one could choose the patches to be pairs of

nearest-neighbor sites on a lattice. We use capital

letters, I,J, or K, to denote a spatial patch.

Each patch is assigned a local wavefunction |ΨI⟩

(with the same Hilbert space dimension as the usual

wavefunction), and the Hilbert spaces of neighboring

patches are related by unitary connections ˆ

UIJ (which

act on the entire Hilbert space), as depicted in Fig. 1.

In the simplest setting, the Hamiltonian is written as

a sum over terms ˆ

HIthat act within a single patch I:

H=X

HI(1)

The local wavefunctions and connections time-evolve

according to

∂t|ΨI⟩=−iˆ

H⟨I⟩|ΨI⟩

∂tˆ

UIJ =−iˆ

H⟨I⟩ˆ

UIJ +iˆ

UIJ ˆ

H⟨J⟩

(2)

where

H⟨I⟩=J∩I̸=∅

UIJ ˆ

HJˆ

UJI (3)

is the sum of local Hamiltonian terms supported on

patches that overlap with patch I. Typically, we

initialize ˆ

UIJ (0) = ˆ

1and |ΨI(0)⟩=|Ψ(0)⟩at time

t= 0, where ˆ

1is the identity operator and |Ψ(t)⟩

is the usual wavefunction in the Schr¨odinger picture.

The expectation value ⟨Ψ|ˆ

AI|Ψ⟩of a local operator

AIthat only acts within the patch Ican be evaluated

in the gauge picture as ⟨ΨI|ˆ

AI|ΨI⟩. To calculate

an expectation value ⟨Ψ|ˆ

AIˆ

BJ|Ψ⟩for a product of

operators acting on diﬀerent patches, a connection

UIJ must be inserted in the gauge picture, as in

⟨ΨI|ˆ

AIˆ

UIJ ˆ

BJ|ΨJ⟩. Such correlation functions can

be used to calculate the density matrix from the

gauge picture local wavefunctions and connections.

For example, for a system of nqubits, the density

matrix is

ˆρ= 2−nX

µ1···µn

ˆσµ1

1· · · ˆσµn

n(4)

⟨Ψ1|ˆσµ1

1ˆ

U1,2ˆσµ2

2· · · ˆ

Un−1,n ˆσµn

n|Ψn⟩

where ˆσµ

iare Pauli operators, and here we take each

patch to consist of just a single qubit (for simplicity)

so that I= 1, . . . , n indexes the qubits/patches along

some path. The local wavefunction |ΨI⟩is local in

the sense that its dynamics are local and connections

are required to extract information about operators

outside the patch I. Time evolution preserves the

following network of relations:

UIJ |ΨJ⟩=|ΨI⟩

UIJ ˆ

UJK =ˆ

UIK

(5)

along with ˆ

U†

IJ =ˆ

UJI and ˆ

UII =ˆ

ψIψJ

Figure 1: An example of a chain of qubits (black dots)

and spatial patches (colored ovals) consisting of pairs

of neighboring qubits. In the gauge picture, a local

wavefunction |ΨI⟩is associated with each patch I, and the

Hilbert spaces of neighboring patches are related by unitary

connections ˆ

UIJ . A quantum gauge network analogously

consists of local wavefunctions |ψI⟩in truncated Hilbert

spaces that are related by non-unitary connections VIJ .

In this work, we truncate the Hilbert spaces of the

local wavefunctions and connections in the gauage

picture so that approximate quantum dynamics

simulations can be performed on a computer. The

utility of the gauge picture for approximating

quantum mechanics is that locality is explicit in both

the time dynamics and the structure of the local

wavefunctions and connections. Furthermore, the

connections allow us to utilize diﬀerent truncated

Hilbert spaces for diﬀerent patches of space. We call

the resulting network of truncated local wavefunctions

and connections a quantum gauge network (QGN).

An advantage of quantum gauge networks is that

unlike traditional tensor networks (e.g. PEPS [3]),

a QGN only involves matrices and vectors (rather

than tensors with many indices) regardless of the

spatial dimension. As such, it is natural for a

QGN algorithm to only require a computation time

(e.g. CPU time) that scales as O(χ3), regardless

of the number of spatial dimensions, where χis the

dimension of the truncated Hilbert spaces. χcan be

viewed as the bond dimension of the QGN. Thus, the

natural O(χ3)computation time for a QGN algorithm

is the same as for MPS algorithms, which are very

eﬃcient in one spatial dimension (1D). In three spatial

dimensions (3D), the most computationally eﬃcient

tensor network in previous literature may be the

isometric tensor network [8,9,10], for which the

computation time of the TEBD3algorithm is O(χ12)

[11].1Thus, QGN algorithms can require signiﬁcantly

less computation time for ﬁxed bond dimension. This

is useful since larger bond dimensions allow more

correlations to be transported through the tensor

network.

The computation time per variational parameter

also scales favorable for QGN algorithms. For

a QGN, the number of parameters scales as χ2

(due to the matrix-valued connections). Therefore,

the computation time per parameter scales as

1More generally, the TEBD3computation time is

O(D10χ2) + O(Dχ6) when the bond dimension χof the central

bonds is diﬀerent from the bond dimension Dof the other

bonds. [11] In 2D, the TEBD2computation time scales as

χ7. [8]

Accepted in Quantum 2023-09-04, click title to verify. Published under CC-BY 4.0. 2

χ3/χ2=χ1.5for a QGN. This is the same ratio

as MPS algorithms, which are very eﬃcient in 1D.

For the isometric tensor network TEBD3algorithm

in 3D, this ratio scales as χ12/χ6=χ2[11], which is

remarkably eﬃcient but not as good as the ratio for

a QGN or MPS.

Although most tensor networks typically directly

encode a wavefunction or density matrix, quantum

gauge networks depart from this habit. However,

a density matrix can be computed from a QGN

similar to Eq. (4)for the gauge picture. Unlike a

generic PEPS [3] but similar to a matrix product state

(MPS) or isometric tensor network [8,9,10], local

expectation values can be eﬃciently computed from

a QGN. A disadvantage of quantum gauge networks

is that unphysical states can also be encoded. As

such, using a QGN to variationally optimize a ground

state is not as straight-forward as for tensor networks

that directly encode a wavefunction. We leave QGN

ground state optimization algorithms to future work.

In this work, we focus on QGN fundamentals and time

dynamics alorithms.

In Sec. 2, we discuss basic properties of quantum

gauge networks and how a QGN can be constructed.

We also show that for an arbitrary wavefunction

(including fermionic wavefunctions), all 2k-point

correlation functions of Mmany operators can be

encoded exactly by a QGN with bond dimension

O(Mk)[Eq. (32)], while an MPS of qubits can require

an exponentially larger bond dimension 2M/6for

k= 1. In Sec. 3, we present a QGN algorithm

for approximately simulating quantum dynamics in

the gauge picture. We benchmark the algorithm

using simulations of fermionic Hamiltonians in spatial

dimensions up to three.

2 Quantum Gauge Networks

To deﬁne a quantum gauge network (QGN), we ﬁrst

choose a collection of possibly-overlapping patches

of space that cover space. For example, one could

choose the patches to consist of just a single lattice

site. Another natural choice is to take the patches

to have the same support as the Hamiltonian terms.

That is, we might choose patches that are pairs of

nearest-neighbor sites if the Hamiltonian terms act

on nearest-neighbor sites. A QGN then consists of

(1) a local wavefunction |ψI⟩for each spatial patch;

(2) non-unitary connections VIJ =V†

JI to relate the

Hilbert spaces of nearby patches, as depicted in Fig. 1;

and (3) a collection of truncated operators to act on

the truncated Hilbert space at each patch.

The local wavefunctions and connections are similar

to the those within the gauge picture [1], except the

Hilbert space is truncated. If the full Hilbert space

has dimension N, then |ΨI⟩and ˆ

UIJ in the gauge

picture have dimensions Nand N×N, respectively.

Since Nis exponentially large in system size, it is

useful to truncate the full Hilbert space dimension

for approximate simulations. Therefore, we consider

truncated local wavefunctions |ψI⟩and connections

VIJ , which have truncated dimensions χIand χI×χJ,

respectively, where typically χI≪N. We use capital

and lower-case Greek letters (e.g. |ΨI⟩vs |ψI⟩)

for wavefunctions in the full and truncated Hilbert

spaces, respectively. Similarly, we place hats on

operators that act within the full Hilbert space (e.g.

UIJ ), while operators within the truncated Hilbert

space (e.g. VIJ ) do not have hats.

In order to calculate expectation values of local

operators ˆ

AIin the original Hilbert space, we

must also deﬁne truncated operators, i.e. χI×χI

matrices AI, that act on the truncated Hilbert space.

Throughout this work, ˆ

AIalways denotes an operator

that acts within a patch I, and similar for ˆ

BJ,

etc. The truncated operators AIare notationally

distinguished from the original operators ˆ

AIby the

lack of a hat. In Sec. 2.1, we present a concrete

mapping to obtain a QGN and truncated operators.

However, in many cases (e.g. Appendix C.1.1 and E)

the truncated operators can be taken to be a simple

Kronecker product, such as σµ

I=1⊗σµ, where 1is

an identity matrix and σµis a 2×2Pauli matrix.

Ideally, we want the quantum gauge network to

accurately encode approximate expectation values.

For example, if the QGN is an approximation for a

wavefunction |Ψ⟩, then we would like the QGN to

accurately encode local expectation values; i.e. we

want ⟨ψI|AI|ψI⟩ ≈ ⟨Ψ|ˆ

AI|Ψ⟩. Similarly, we typically

also want expectation values of string operators to

also approximately match, e.g.

⟨ψI|AIVIJ BJVJ K CK|ψK⟩≈⟨Ψ|ˆ

AIˆ

BJˆ

CK|Ψ⟩(6)

Note that in order to express a string operator that

acts on multiple spatial patches using a QGN, it is

essential to insert connections VIJ between operators

and wavefunctions associated with diﬀerent spatial

patches.

Similar to the gauge picture of quantum dynamics,

a density matrix can be extracted from a QGN. Thus,

a QGN most generally encodes a mixed state rather

than a pure state. For example, analogous to Eq. (4)

for a system of nqubits, a density matrix can be

approximately extracted from a QGN via

ˆρ≈2−nX

µ1···µn

ˆσµ1

1· · · ˆσµn

n(7)

⟨ψ1|σµ1

1V1,2σµ2

2· · · Vn−1,n σµn

n|ψn⟩

where ˆσµ

iare Pauli operators. Here, we take each

patch to consist of just a single qubit (for simplicity),

and we use I= 1, . . . , n to index the qubits/patches

along a string of nearest-neighbors, e.g. as in Fig. 2.

However, due to the approximations induced by the

QGN, diﬀerent paths (e.g. those in Fig. 2) can yield

diﬀerent density matrices.

Accepted in Quantum 2023-09-04, click title to verify. Published under CC-BY 4.0. 3

(a) (b) (c)

Figure 2: Three diﬀerent paths that cover all lattice sites.

Also similar to the gauge picture, quantum gauge

networks exhibit a local gauge symmetry:

|ψI⟩ → ΛI|ΨI⟩

VIJ →ΛIVIJ Λ†

AI→ΛIAIΛ†

(8)

where ΛIis a unitary matrix. Expectation values

must be invariant under this symmetry.

Since local expectation values ⟨Ψ|ˆ

AI|Ψ⟩ ≈

⟨ψI|AI|ψI⟩are encoded in the local wavefunctions

|ψI⟩, we can think of the local wavefunction as a

puriﬁed reduced density matrix for a spatial patch.

The connections VIJ encode long-range correlations

between the spatial patches.

It is desirable for a QGN to at least approximately

obey the following consistency conditions

VIJ |ψJ⟩ ≈ |ψI⟩

VIJ VJ K ≈VIK

(9)

Although it is easy to make the ﬁrst relation exact,

the second will typically only hold approximately.

Typically, a QGN will only possess a VIJ for nearby

patches Iand J(and not for far away patches).

Thus, the second relation only applies if all three

connections (VIJ ,VJ K , and VIK ) are contained in

the QGN. The connections VIJ should have singular

values less than or equal to 1 (to ensure that

expectation values are never larger than the largest

eigenvalue of the measured operator).

When VIJ |ψJ⟩=|ψI⟩holds exactly, QGN

connected correlation functions obey the usual

identity (with a VIJ inserted):

ψIAI− ⟨AI⟩VIJ BJ− ⟨BJ⟩ψJ

=⟨ψI|AIVIJ BJ|ψJ⟩−⟨AI⟩ ⟨BJ⟩(10)

where we abbreviate ⟨AI⟩=⟨ψI|AI|ψI⟩and

⟨BJ⟩=⟨ψJ|BJ|ψJ⟩. Therefore, if the QGN

connected correlation function [Eq. (10)] is small, we

are guaranteed that ⟨ψI|AIVIJ BJ|ψJ⟩ ≈ ⟨AI⟩ ⟨BJ⟩,

as one should expect. The equations in this paragraph

also hold for longer chains of connections; e.g. they

also hold if we replace VIJ with VI K VKLVLI .

original

Hilbert space

truncated Hilbert spaces

Figure 3: Each truncation map QImaps a subspace of

the original Hilbert space on to the truncated Hilbert space

associated with patch I.

2.1 QGN from Truncation Maps

In this subsection, we study a concrete construction

to obtain a quantum gauge network. The input for

this QGN construction is a wavefunction |Ψ⟩, along

with a truncation map QIfor each patch of space. If

we want to construct a QGN from a density matrix

instead, then |Ψ⟩should be chosen to be a puriﬁcation

of the density matrix. The truncation maps are χI×N

matrices (where Nis the dimension of the full Hilbert

space) that satisfy:

QIQ†

I=ˆ

Q†

IQI|Ψ⟩=|Ψ⟩(11)

Therefore Q†

Iis an isometry matrix whose image

includes the wavefunction. (A matrix Mis isometric

if M†M=1.) Intuitively, each QImaps a select

subspace of states into a truncated Hilbert space, as

depicted in Fig. 3. In the next subsection, we will

explain how one can obtain useful truncation maps.

With this data, we can construct the following

QGN:

|ψI⟩=QI|Ψ⟩

VIJ =QIQ†

(12)

Note that VII =1due to Eq. (11). Local operators ˆ

with support on a spatial patch Iare also truncated:

AI=QIˆ

AIQ†

I(13)

Note that this equation can be used for both bosonic

and fermionic operators. In the limit that the bond

dimension χI→Napproaches the full Hilbert space

dimension N, this truncation mapping will result in a

QGN that encodes correlation functions [e.g. Eq. (6)]

exactly.

The operator norm of VIJ is bounded by

||VIJ ||op ≤ ||QI||op||Q†

J||op = 1. Thus, the resulting

connections VIJ have singular values that are less than

or equal to 1. One can verify that

VIJ |ψJ⟩=|ψI⟩(14)

from Eq. (9)holds exactly.

Accepted in Quantum 2023-09-04, click title to verify. Published under CC-BY 4.0. 4

In practice, the truncation mapping can only be

directly applied for wavefunctions that are simple

enough such that the truncation can be eﬃciently

computed. However, the construction is also useful for

theoretically understanding how a QGN can encode a

wavefunction.

A trivial example of a QGN can be obtained from

a product state wavefunction |Ψ⟩=|ψ1⟩ ⊗ |ψ2⟩ ⊗

· · · |ψn⟩. The truncation maps can be chosen to be

QI=|ψI⟩ ⟨Ψ|, with I= 1, . . . , n. This results in a

QGN with local wavefunctions |ψI⟩and connections

VIJ =|ψI⟩⊗⟨ψJ|. The truncated operators simply

act within the on-site Hilbert space. Additional

examples of quantum gauge networks can be found

in Appendix C.

Note that not all quantum gauge networks can be

obtained from the truncation mapping. For example,

generating a QGN by sampling random numbers for

|ψI⟩and VIJ will result in a very unphysical QGN

that is not consistent with any wavefunction. This

is in contrast to MPS or PEPS tensor networks, for

which a random number initialization still returns a

physical (although unnormalized) wavefunction.

In Appendix B, we show how to obtain truncation

maps from the canonical form of a matrix product

state (MPS) [3,4,5]. If the MPS has bond dimension

χ, then the QGN will have bond dimensions equal

to dχ2, where dis the Hilbert space dimension at

each site. (d= 2 for qubits.) The idea of the

mapping is that the canonical form of an MPS can

consist of a center tensor at Ithat is surrounded by

isometric tensors. The isometric tensors are then used

to construct a truncation map QI, while the center

tensor is a local wavefunction |ψI⟩.

2.2 Truncation Map Construction

The accuracy of the truncation critically depends on a

good choice of truncation maps QI. Suppose we want

to choose truncation maps such that the quantum

gauge network exactly encodes the expectation values

of a chosen collection of operator strings. For

example, the expectation value of ˆ

AIˆ

BJˆ

CKis encoded

exactly if

⟨ψI|AIVIJ BJVJ K CK|ψK⟩=⟨Ψ|ˆ

AIˆ

BJˆ

CK|Ψ⟩(15)

Below, we show that this exact encoding can be

achieved by choosing truncation maps such that the

image of each Q†

Iis the span of certain strings of

operators involved in the expectation values.

In general, we will ﬁnd that the required bond

dimension χIfor each patch Iis bounded by

χI≤1 + 2pI[Eq. (23)], where pIis the number of

chosen operator strings (which we want to encode)

that act on the patch I. We will also ﬁnd that

bond dimension O(Mk)[Eq. (32)] is suﬃcient to

exactly encode all 2k-point correlation functions of

Mdiﬀerent operators.

2.2.1 Warmup Example

Before presenting a generic algorithm for obtaining

the truncation maps, let us ﬁrst discuss an instructive

example. Suppose we want to ensure that the QGN

encodes the expectation value in Eq. (15)exactly for

a particular choice of local operators ( ˆ

AI,ˆ

BJ, and

CK) and spatial patches I̸=J̸=K. Below, we show

that any choice of truncation maps with the following

images is suﬃcient:

im(Q†

I) = span|Ψ⟩,ˆ

A†

I|Ψ⟩

im(Q†

J) = span|Ψ⟩,ˆ

A†

I|Ψ⟩,ˆ

CK|Ψ⟩

im(Q†

K) = span|Ψ⟩,ˆ

CK|Ψ⟩(16)

span|Ψ1⟩,...,|Ψm⟩denotes the vector space

spanned by the vectors |Ψ1⟩,...,|Ψm⟩. We set

the bond dimensions to be equal to the vector

space dimension of the images: χI=dim(im(Q†

I)).

Specifying these images determines the truncation

maps QIup to a unitary gauge transformation

QI→ΛIQI, where ΛIis a unitary matrix. The choice

of gauge does not aﬀect QGN expectation values. For

examples of a quantum gauge networks obtained in

this way, see Appendix C.

On a computer, a Q†

Iwith the desired image

can be calculated from the compact singular value

decomposition MI=Q†

ISIRI. Here, MIis a matrix

of column vector, which each encode one of the

wavefunctions contained in the span of im(Q†

I).SIis

aχI×χIdiagonal matrix of nonzero singular values,

and R†

Iis an isometry matrix.

Note that if |Φ⟩ ∈ im(Q†

I), then Q†

IQI|Φ⟩=|Φ⟩,

which is a useful property. This follows because

|Φ⟩ ∈ im(Q†

I)implies that there exists |ϕ⟩such

that Q†

I|ϕ⟩=|Φ⟩, which then implies that

Q†

IQI|Φ⟩=Q†

IQIQ†

I|ϕ⟩=Q†

I|ϕ⟩=|Φ⟩where the

middle equality follows from QIQ†

I=ˆ

1[Eq. (11)].

For each patch, the image must contain |Ψ⟩so that

Eq. (11)is satisﬁed. Next, we show that Eq. (16)is

suﬃcient to exactly encode the expectation value in

Eq. (15):

⟨ψI|AIVIJ BJVJ K CK|ψK⟩

=⟨Ψ|Q†

I(QIˆ

AIQ†

I)VIJ BJVJ K (QKˆ

CKQ†

K)QK|Ψ⟩

=⟨Ψ|ˆ

AIQ†

IVIJ BJVJ K QKˆ

CK|Ψ⟩

=⟨Ψ|ˆ

AIQ†

I(QIQ†

J)BJ(QJQ†

K)QKˆ

CK|Ψ⟩

=⟨Ψ|ˆ

AIQ†

JBJQJˆ

CK|Ψ⟩

=⟨Ψ|ˆ

AIQ†

J(QJˆ

BJQ†

J)QJˆ

CK|Ψ⟩(17)

=⟨Ψ|ˆ

AIˆ

BJˆ

CK|Ψ⟩

We used the identities |ψI⟩=QI|Ψ⟩and

VIJ =QIQ†

J[Eq. (12)], AI=QIˆ

AIQ†

I[Eq. (13)], and

Q†

IQI|Φ⟩=|Φ⟩whenever |Φ⟩ ∈ im(Q†

I)in Eq. (16).

Note that even if all the local operators commute,

the path of the string matters. For example, Eq. (16)

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QuantumGaugeNetworks:ANewKindofTensorNetworkKevinSlagle1,2,31DepartmentofElectricalandComputerEngineering,RiceUniversity,Houston,Texas77005USA2DepartmentofPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA3InstituteforQuantumInformationandMatterandWalterBurkeInstituteforTheoretical...

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Quantum Gauge Networks A New Kind of Tensor Network

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