Tensor Network Computations That Capture Strict Variationality, Volume Law Behavior, and the Efficient Representation of Neural Network States

Tensor Network Computations That Capture Strict Variationality, Volume Law

Behavior, and the Efficient Representation of Neural Network States

Wen-Yuan Liu ,

Si-Jing Du ,

Ruojing Peng,

Johnnie Gray ,

and Garnet Kin-Lic Chan

†

Division of Chemistry and Chemical Engineering,

California Institute of Technology

, Pasadena, California 91125, USA

Division of Engineering and Applied Science,

California Institute of Technology

, Pasadena, California 91125, USA

(Received 21 May 2024; accepted 27 November 2024; published 30 December 2024)

We introduce a change of perspective on tensor network states that is defined by the computational graph

of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network

functions, inherit the conceptual advantages of tensor network states while removing computational

restrictions arising from the need to converge approximate contractions. We use tensor network functions to

compute strict variational estimates of the energy on loopy graphs, analyze their expressive power for

ground states, show that we can capture aspects of volume law time evolution, and provide a mapping of

general feed-forward neural nets onto efficient tensor network functions. Our work expands the realm of

computable tensor networks to ones where accurate contraction methods are not available, and opens up

new avenues to use tensor networks.

DOI:

10.1103/PhysRevLett.133.260404

Introduction

—

Simulating and representing quantum

many-body systems is a central task of physics. For this,

tensor network states have emerged as a fundamental lan-

guage and numerical tool

–

, with applications across

diverse areas such as condensed matter physics

–

12]

statistical physics

[13

–

16]

, quantum field theory

[17

–

19]

and quantum information theory

[20

–

22]

, as well as adjacent

disciplines such as quantum chemistry and machine learning

[23

–

26]

Tensor networks represent the amplitude of a quantum

state on an

site lattice, denoted

n

Ψ

≡

Ψ

n

≡

Ψ

;

...

, as the contraction of a set of tensors. As

a simple example, we consider

tensors connected in a graph

[Fig.

1(a)

],eachassociatedwithalatticesiteHilbertspace;this

is known as a projected entangled pair state (PEPS)

[27]

.Each

tensor contains

elements (

is the number of bonds

which connect tensors to adjacent tensors):

is the local

Hilbert space dimension and

controls the expressivity. For

tree graphs, the resulting contraction over the shared tensor

bonds can be computed exactly in

poly

time

[28]

Forgeneral graphs,however, both

n

Ψ

and

Ψ

ˆ

Ψ

cannot

be exactly contracted efficiently: the cost is exponential in the

tree width of the graph

[21]

. Approximate contraction

algorithms have thus been introduced whereby during the

contraction process, the large intermediate tensors that are

formed are compressed, via singular value decomposition

(SVD), to a controlled size,

poly

,where

is termed the

auxiliarybonddimension

[2,29]

.Onlyinthelimitof

→

∞

the contraction exact. The resulting approximate contraction

has a cost

poly

, with an error with respect to

the exact contraction controlled by

From the perspective of approximate contraction, tensor

network computations should seek to be converged with

respect to

, otherwise the results are not meaningful. For

example, for a Hamiltonian

ˆ

,if

Ψ

ˆ

Ψ

Ψ

Ψ

is com-

puted by approximate contraction, with insufficient

the

energy may violate the variational theorem. The requirement

that computations should use a sufficiently large

(such that

the evaluation of

Ψ

ˆ

Ψ

n

Ψ

is no longer changing

significantly with

) places restrictions on the types of graphs

that can be used to define the tensor network. For example, in

many-body physics simulations, it is believed that only highly

(b)(c)

(a)

(d)

(e)

FIG. 1. (a) PEPS on a general graph with

site tensors.

(b) PEPS on a

square lattice. (c) Insertion of a pair of

isometries for approximate tensor contractions; the isometries can

be obtained with standard tensor network techniques by SVD

[2,3]

. (d) Approximate PEPS contractions by inserting isometries

for a given configuration

n

(gray circles). (e) For a given input

vector

n

i¼ð

;

...

, a tensor network function outputs a

unique value

Ψ

n

These authors contributed equally to this work.

†

Contact author: gkc1000@gmail.com

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structured graphs with limited entanglement, such as (near-)

area law states, support an efficient contraction scheme.

Because of the latter requirement, it has been argued that,

in practical use, tensor networks cannot replicate the expres-

sivity of other flexible types of many-body ansatz, such as

neural network quantum states

[30

–

36]

In the current work, we observe that the above limi-

tations on tensor network computations arise due to an

assumption, namely that the tensor network ansatz is only

meaningfully defined in the exact contraction limit. We can,

however, adopt an alternative perspective, namely, to

consider the approximate contraction scheme for any

as itself part of the tensor network ansatz for the amplitude

Ψ

n

. This view is natural and in some ways obvious when

using tensor networks in variational Monte Carlo (VMC)

calculations

[37

–

41]

, but we will argue that this is not a

manifestation of tensor networks in a particular numerical

algorithm, but rather a change of perspective that alters and

broadens the definition and utility of tensor networks. We

refer to the larger set of parameterized many-body states

that arise from

“

consistent

”

contraction schemes (to be

defined later) as

tensor network functions

. This wider class

of states removes many restrictions that are assumed to

apply to tensor networks. We demonstrate that tensor

network functions provide strictly variational estimates

of the energy, allow for practical tensor network compu-

tations on graphs that are highly challenging to contract

accurately, including those that describe volume law

physics, and expand the set of states that can be efficiently

represented by tensor networks to cover other classes of

ansatz, such as neural network quantum states.

Tensor network functions

—

A tensor network function

(TNF) is a consistent contraction of the tensor network

representation of the amplitude

Ψ

n

. To illustrate the basic

idea, we first consider a standard PEPS for a two-dimensional

lattice,illustratedinFig.

1(b)

.Becausethegraphhasloops,the

PEPS amplitude cannot be computed efficiently according to

Fig.

1(b)

. Instead, in practice, the amplitude is defined by a

different tensor network illustrated in Fig.

1(d)

.Inthisfigure,

isometries (triangles) with bond dimension

have been

inserted, where the values of the isometries are determined

by an SVD of the contraction of neighbouring tensors [see

Fig.

1(c)

]

[2,3]

. The final amplitude is obtained by contracting

the tensors and isometries together. The pattern of isometries

used here corresponds to a specific approximate contraction

method (boundary contraction

[2,27]

) and other choices can

be made

[29,42]

, but the essential purpose is to make the

contraction of the modified graph containing the original

tensors and the isometries efficient.

Note, however, that although the structure of the tensor

networkwith isometries in Fig.

1(d)

nowappearspseudo-one-

dimensional, this does not mean that its entanglement

structure (for

) is reduced to that of a matrix product

state (MPS). This is because the isometries are themselves

functions of the configuration

n

, i.e., different values of the

isometriesareobtained,viaSVD,fordifferentconfigurations.

However, we choose to keep the same

positions

of the

isometries in the graph for any configuration (even though

the entries are different); we refer to this as using

fixed

(position) isometries, which defines a consistent contraction.

Because of the configuration dependence of the isometry

entries, the graph with isometries no longer represents a

multilinear ansatz like the original tensor network state; it is

only a multilinear ansatz in the limit that

→

∞

(and the

isometries become identities). Instead, it defines a

tensor

network function

: for any given

,any

n

is mapped to a

single value, the amplitude, through an efficient contraction.

The efficient TNF representation of

Ψ

n

does not guarantee

deterministic efficiency of computing

Ψ

ˆ

Ψ

;butsuch

expectations may be evaluated stochastically by sampling

with efficient sample complexity.

We emphasize that herewe aredefining a new classofstate,

i.e., the TNF, whose amplitudes are exactly given by a certain

contraction graph of a tensor network. The essential

differences between TNF and conventional tensor network

computationsarefurtherillustratedinthefollowingexamples,

and discussed also in Sec. 1 in Supplemental Material

[43]

Tensor network functions and approximate contraction

—

VMC algorithms also compute

Ψ

n

via approximate

contraction (for a brief review of tensor network VMC

(TN-VMC) algorithms, see Supplemental Material

[43]

Refs.

[37

–

41]

) and it is instructive to consider the differences

between the standard TN-VMC approximation of

Ψ

n

and

the definition of a TNF above. There are two seemingly small,

but significant, differences (i) in TN-VMC,

is viewed as a

parameter (typically

const ×

, with const

)where

is increased to convergence for a fixed

,and(ii) the positions

of the isometries are not fixed but dependent on the amplitude

thatisbeingcomputed(werefertothisas

dynamic

isometries),

inordertoallowforthereuseofcomputationswhencomputing

amplitudes of configurations which differ by only a few

substitutions

[41]

. See Supplemental Material for details

[43]

The most important distinction from TNF is that the use

of dynamic isometries in TN-VMC

does not lead to a

consistent function

. In other words, given the same con-

figuration

n

Ψ

n

needs not be the same, as it depends on

the prior configuration and which components of the tensor

network contraction are being reused. Only in the limit of

→

∞

are the amplitudes consistent and the function

defined, which underlies the usual need to converge

TN-VMC calculations. In contrast, it is clear that with

using fixed isometries, the corresponding TNF is consistent

for any

and any

. See Supplemental Material for

details

[43]

Tensor network functions and variational energies

—

an immediate consequence of the use of fixed isometries to

define a TNF, we can always obtain a variational energy in

Monte Carlo energy evaluation (up to statistical error).

In Fig.

2(a)

we show the energy of an optimized PEPS

(

, various

) TNF for the ground state of the open-

boundary condition (OBC) spin-

square-lattice

Heisenberg model defined using the boundary contraction

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computational graph, see Supplemental Material

[43]

compared to that of a PEPS in standard TN-VMC using

dynamic isometries, where

controls the error of approxi-

mate contraction

[41]

. We see immediately that the PEPS

TNF energy is variational for all

. In contrast, viewing the

PEPS amplitude as an (inconsistent) approximate contrac-

tion as in standard VMC leads to a nonvariational energy,

where the nonvariationality is typically regarded as the

“

contraction error.

”

This behavior is even more clear for a

lattice that is usually considered more difficult to contract.

Using the square-lattice

–

model with periodic boun-

dary conditions (PBC) as an example, in Fig.

2(b)

we show

the same comparison between TNF and standard TN-

VMC. In addition to the variational versus nonvariational

nature of the energies, we also see a large variance of the

TN-VMC energy arising from the inconsistent definition of

amplitudes. This leads to substantial difficulties in the

stochastic optimization of the ansatz. Overall, we see that

lack of variationality is not intrinsic to approximate con-

traction: it results from the perspective on the definition of

the ansatz.

Expressivity of tensor network functions with small

—

general, a TNF is parametrized by two bond dimension

parameters,

and

. We already know that as

becomes

large, the TNF reduces to the standard tensor network state

from which it is derived, but TNFs are well defined also in

the limit of small

(e.g.,

) and it is important to ask

whether they are expressive in this limit.

In Figs.

3(a)

and

3(b)

, we show results from approxi-

mating the ground-state of a spin-

square-lattice

–

Heisenberg model for

, and an

Hubbard model with

U=t

and

hole doping, using

bosonic and fermionic PEPS TNFs for OBC systems

[27,55

–

57]

, respectively. We minimize the energy for many

different

combinations using gradient descent, and

we plot the computational cost of the amplitude against the

obtained accuracy. For a specified relative accuracy, we can

then search for the

pair with the lowest computa-

tional cost. We see that contrary to the standard situation

with tensor network states where one uses

, the most

powerful TNFs for a given computational cost have

For example, for

∼

relative accuracy in the Hubbard

model, using

Þ¼ð

;

yields the same expressivity

Þ¼ð

;

, but with lower cost. In no case is it

computationally more efficient to use

. The insets of

Fig.

presenting the minimized energy for every ansatz

Ψ

Þi

, show that the most rapid improvement in energy

is obtained at small

, rather than saturating

for a given

confirming the above conclusion.

Tensor network functions and volume laws

—

As an

example of a new kind of application that we can consider

with TNFs, we consider the case where the tensor network

graph corresponds to chaotic time evolution leading to a

volumelaw state.Thenthestandardview isthatapproximate

contraction should be performed with

∼

exp

const ×

(where

is evolution time) to capture the volume law.

However, if we think of the TNF formed by a random

circuit, we might expect that the set of amplitudes, defined

using fixed position but configuration dependent and essen-

tially random isometries, are also scrambled as if by an

approximate unitary design, regardless of

, capturing

(some) elements of the volume law behavior.

To illustrate this numerically, we consider the Floquet

dynamics of an

-site 1D kicked Ising chain

[58

–

60]

(a)

(b)

FIG. 3. Time cost (in seconds) versus accuracy of the tensor

network function

Ψ

Þi

, for the

−

model on the

lattice at

(a) and Fermi-Hubbard model on the

lattice at

U=t

with hole doping

(b). Refer-

ence energies

ref

taken from PEPS

[

−

490 478

] and DMRG [

10 000

with SU(2) symmetry,

Hubbard model,

−

694 391

]. The time cost is for comput-

ing

n

Ψ

Þi

for a given

n

on a single core. Highlighted

symbols (connected by a black line) indicate the optimal

pairs that have smallest time cost for a given accuracy. Here

–

and

–

is used for the

–

model and

–

and

–

for the Hubbard model. The insets present the energy

variation of the TNF

Ψ

Þi

with

and

-0.68

-0.67

-0.66

-0.65

-0.64

2345678

-0.510

-0.505

-0.500

-0.495

-0.490

32 OBC

Energy

Fixed-iso

Dynamic-iso

Reference

(a)

(b)

Energy

Fixed-iso

Dynamic-iso

Reference

8 PBC

FIG. 2. For a given PEPS, energies (per site) obtained from

fixed (tensor network function) and dynamic (standard VMC)

isometry contractions. (a) OBC square-lattice

spin-

Heisenberg model with PEPS

, and 15000 Monte Carlo

sweeps are used for each data point. The dashed line denotes the

numerically exact energy

[41]

. (b) PBC square-lattice

spin-

frustrated

–

model at

with PEPS

and 5000 Monte Carlo sweeps are used for each data point. The

dashed line denotes the energy from fixed-isometry contraction

with

using

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Expressing the Floquet evolution operator

as a matrix

productoperator(MPO)(seeSupplementalMaterial

[43]

),the

time evolution corresponds to a

Dtensornetwork,

shown in Fig.

4(a)

. The tunable parameters in the model

include the Ising interaction strength

, transverse field

and

longitudinal field

. We focus on the maximally chaotic

regime

[61]

J; g; h

Þ¼ð

;

, which exhibits vol-

umelawphysics ina shortevolutiontime.Foreachtime step

we compute the bipartite von Neumann entropy

Þ¼

−

log

of the state

Ψ

Þi ¼

⊗

,where

thereduceddensitymatrixofthehalf-chain.(Computing

not an efficient operation, but we do so to illustrate the physics

being captured). To define a useful TNF, we construct a

contraction path that leads to configuration dependent iso-

metries: we consider two natural ones, namely, (i) contraction

in the inverse time direction, and (ii) (transverse) contraction

along the spatial direction. We then compare against the

standard MPS-MPO compression to a bond dimension

Results from (i), as well as from amplitude-independent

MPO-MPO compression in the inverse time direction, are

shown in the Supplemental Material

[43]

The evolution of

is depicted in Fig.

4(b)

, with exact

resultsincluded.Forthemaximallychaoticdynamics,which

might be thought of as closest to the random circuit intuition

above, the entanglement entropy grows linearly from the

beginning until half-width saturation. The required conven-

tional MPS bond dimension to describe the full dynamics

growsexponentiallywithtime(hereforafinite

chain

the maximum

needed would be

128

). However,

volume law physics is already recovered using a TNF with

.The

TNFaccuratelycapturesthe features ofthe

fullevolution,includingthemaximumpointintheentropyat

. Similar results are seen for the TNF defined in the

inverse time-direction (see Supplemental Material

[43]

although it is quantitatively less accurate. Notably, the

entanglement spectrum (inset of Fig.

) for the TNF, even

for small

, has a long tail and captures the broad features of

the spectrum, rather than exhibiting the sharp cutoff of a

conventional truncated MPS. Similar features for the less

chaotic regime

J; g; h

Þ¼ð

;

are also observed;

see Supplemental Material

[43]

Tensor network functions can efficiently represent

neural network quantum states

—

As discussed, TNFs only

require a consistent and efficient contraction of

Ψ

n

, and

there exist TNFs for which this is the case without any

(amplitude dependent) isometries, but for which the effi-

cient contraction of

Ψ

ˆ

Ψ

is still not possible. [We note

that tensor networks where computing

Ψ

ˆ

Ψ

is efficient

but

Ψ

n

is not are used in the form of the multi-scale

entanglement renormalization ansatz

[62]

]. Here we

describe one such application of TNFs without amplitude

dependent isometries.

It has previously been observed that common tensor

networks that can be efficiently exactly contracted for

Ψ

n

and

Ψ

ˆ

Ψ

, i.e., MPS and tree tensor network

states, can be mapped onto efficient neural networks by

constructing polynomial size neural nets that emulate tensor

network contraction, establishing that exactly contractible

tensor networks are a formal subclass of efficient neural

quantum states

[36]

. It has also been suggested that neural

networks may not have an efficient tensor network repre-

sentation, because the well known contractible tensor net-

works satisfy area laws, while neural networks need not do

[36]

However, if one allows for general tensor network

geometries, it is straightforward to construct a TNF (with-

out using isometries) that represents a neural network

quantum state without the restriction of an area law. We

consider the explicit example of a feed forward neural

network (FNN), a popular universal architecture with

layers of neurons, where the outputs of each layer

−

form the inputs

to the next layer, and the

th neuron

(activation) function in layer

ðf

;

where

are the neuron parameters. Assuming that all

are

implemented without recursion, then the computational

graph of a FNN is a directed acyclic graph. Then we

observe that (i) if the FNN is efficient, the classical binary

circuit implementation of the neural network function is of

polynomial size in logic and copy gates, where a binary

logic gate is a tensor with two binary input legs and one

binary output leg, while copy takes one binary input and

returns two binary outputs (nonlinearity of the activation

functions enters through the copy operator) (ii) given the

input product state

n

(i.e., a bitstring), the application of

each gate

produces another product state

n

(i.e.,

another bitstring). Thus contracting the resulting tensor

network in the appropriate order from inputs to the outputs

(for example, using sparse tensor contraction, or by

introducing SVDs to discover the product state structure

n

, see Supplemental Material for more details

[43]

)

can always be done efficiently.

(a)

(b)

FIG. 4. (a)

D TN representation of the Floquet dynam-

ics of an

-site kicked Ising chain, starting from a product state

Ψ

Þi¼j

⊗

. TNF contraction direction is either along the

spatial direction or inverse time directon (see inverse direction

results in Supplemental Material

[43]

). Conventional MPS-MPO

contraction is along the time direction. (b) Shows the dynamics

of the half-system entropy

of a 14-site chain at

J; h; g

Þ¼

;

. Inset of (b) presents the entanglement spectrum

for the 40 largest eigenvalues of the reduced density matrix

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The above can be seen as equating a TNF with a type of

circuit computational model, in this case without a memory.

Although the tensor network contraction is of polynomial

cost, it is not very concise with respect to the number of

tensors, due to representing floating point numbers in

binary. In principle, we can introduce new elements to

the computational model that improve the conciseness. In

the Supplemental Material

[43]

, we show how to use the

arithmetic circuit tensor network construction of Ref.

[63]

to compute with tensor networks with floating point entries,

and the tensor network contraction is efficient if we further

introduce the ability to store results from contractions of

tensor network subgraphs.

Conclusions

—

In summary, we have introduced tensor

network functions (TNFs), a new perspective on the tensor

network state, as a flexible representation for quantum

many-body systems. The broadened perspective removes

the conventional restrictions of tensor network computa-

tions on the form of the ansatz, and the computational need

to converge the auxiliary bond dimension in an approxi-

mate contraction. On the other hand, TNFs inherit the

conceptual advantages of tensor networks. Unlike generic

function approximators, such as neural networks, the

function architecture is directly suggested by the physics

at hand and one can reuse standard tensor network

algorithms. For example, in time evolution, the TNF

structure can follow the evolution circuit and choice of

contraction path; in variational optimization, existing TN

algorithms provide a deterministic guess for the TNF

tensors, which may then be further relaxed. TNFs derived

from fermionic tensor networks naturally encode fermionic

sign structure

[56]

, avoiding the need for more complicated

constructions

[64

–

67]

. Finally, TNFs can be further gen-

eralized into a wider class of states, where nonlinearity

other than the singular value decomposition is introduced

into the tensor network computational graph. These pos-

sibilities open up many new avenues to investigate and to

explore with tensor networks for the future.

Acknowledgments

—

This work was supported by the U.S.

NationalScienceFoundationunderGrantNo.CHE-2102505.

G. K. C. acknowledges additional support from the Simons

Investigator program and the Dreyfus Foundation under the

program Machine Learning in the Chemical Sciences and

Engineering.

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