Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains Hiroshi Shinaoka1 2Markus Wallerberger3Yuta Murakami4

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Multiscale space-time ansatz for correlation functions of quantum systems based on

quantics tensor trains

Hiroshi Shinaoka,1, 2 Markus Wallerberger,3Yuta Murakami,4

Kosuke Nogaki,5Rihito Sakurai,1Philipp Werner,6and Anna Kauch3

1Department of Physics, Saitama University, Saitama 338-8570, Japan

2JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

3Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria

4Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan

5Department of Physics, Kyoto University, Kyoto 606-8502, Japan

6Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland

Correlation functions of quantum systems—central objects in quantum ﬁeld theories—are deﬁned

in high-dimensional space-time domains. Their numerical treatment thus suﬀers from the curse

of dimensionality, which hinders the application of sophisticated many-body theories to interesting

problems. Here, we propose a multi-scale space-time ansatz for correlation functions of quantum

systems based on quantics tensor trains (QTT), “qubits” describing exponentially diﬀerent length

scales. The ansatz then assumes a separation of length scales by decomposing the resulting high-

dimensional tensors into tensor trains (known also as matrix product states). We numerically verify

the ansatz for various equilibrium and nonequilibrium systems and demonstrate compression rates

of several orders of magnitude for challenging cases. Essential building blocks of diagrammatic

equations, such as convolutions or Fourier transforms are formulated in the compressed form. We

numerically demonstrate the stability and eﬃciency of the proposed methods for the Dyson and

Bethe-Salpeter equations. The QTT representation provides a uniﬁed framework for implementing

eﬃcient computations of quantum ﬁeld theories.

I. INTRODUCTION

Correlation functions are central building blocks of

quantum ﬁeld theories for many-body and ﬁrst-principles

calculations [1]. A typical example is the Matsubara

or nonequilibrium Green’s functions. These correlation

functions are high-dimensional space-time objects, which

creates a severe challenge for numerical calculations. A

long-standing and fundamental problem of great practi-

cal importance is thus the search for compact represen-

tations of correlation functions.

Notable theoretical developments have been made in

the Matsubara-frequency domain for the one-particle

(1P) Green’s function, for which compact representa-

tions, such as Legendre [2,3] and Chebyshev [4] bases,

were constructed. The 1P Green’s function is related to

a spectral function through the ill-conditioned analytic

continuation kernel. This prior knowledge was recently

employed to construct the intermediate representation

(IR) [5,6] and the sparse sampling method [7,8], the

minimax method [9], and the discrete Lehmann represen-

tation (DLR) [10]. These methods are all based on the

same prior knowledge and allow to treat a wide range

of energy scales, from the bandwidth to low-temperature

phenomena, for one-dimensional (1D) objects in the Mat-

subara frequency domain. Their application enabled

ﬁrst-principles calculations of correlation functions for

unconventional [11] and phonon-mediated superconduc-

tors [12] with low Tc, as well as recent studies of transition

metal oxides [12–17].

Extending these developments to other space-time do-

mains, particularly to higher order correlation functions,

has been a central challenge in many diﬀerent ﬁelds of

computational physics. For the numerical renormaliza-

tion group (NRG) and diagrammatic calculations at the

two-particle (2P) level [18–21], compact representations

for the Matsubara-frequency dependence of 2P quanti-

ties have been proposed. Recently, the analytic structure

of arbitrary correlation functions has been clariﬁed [22];

this structure can be leveraged in NRG calculations [23],

compression of 2P quantities [24,25], and associated di-

agrammatic equations [26].

An eﬃcient description of the three-momentum de-

pendence of 2P quantities is another actively pur-

sued direction, relevant for diagrammatic calculations at

the 2P level and the functional renormalization group

(fRG) [27]. Examples of such eﬀorts include the trun-

cated unity approach based on a truncated form-factor

basis [28,29] and a machine-learning approach [30].

There is also an increasing demand for eﬃcient treat-

ment of 2P quantities in ab initio calculations for such

as the inclusion of vertex corrections in GW [31] and

the Migdal–Eliashberg theory [32]. For nonequilibrium

systems, a hierarchical low-rank data structure has been

proposed [33] for the real-time 1P Green’s function with

two time arguments.

Despite these extensive eﬀorts, a generic and eﬃcient

treatment of high-dimensional space-time objects has not

yet been established. The diﬃculty can be attributed to

the absence of a common and general ansatz for diﬀer-

ent space-time domains. A promising ansatz requires (1)

an accurate treatment of a wide range of length scales

in space-time, (2) systematic control over the truncation

error, (3) the possibility of eﬃcient computations in the

compressed form and (4) straightforward and robust im-

plementations as computer code.

arXiv:2210.12984v3 [cond-mat.str-el] 28 Apr 2023

In this paper, we propose the multi-scale space-time

ansatz based on quantics tensor trains (QTTs) [34,35]

as a universal solution. The space-time dependence is

described by auxiliary bits, which we call “qubits” in the

present study, representing exponentially diﬀerent length

scales in space-time. The resultant high-dimensional ob-

ject in the qubit space is decomposed into tensor trains

(TTs), to the physics community better known as ma-

trix product states (MPS), based on the assumption of

length scale separation. The QTT representation allows

us to describe the space-time dependence of correlation

functions in exponentially wide scales using memory and

computational resources which scale linearly, and thus es-

sentially removes a major bottleneck for numerical many-

body calculations. Basic operations such as the Fourier

transform can be formulated in compressed form and

the methods can be implemented straightforwardly us-

ing standard MPS libraries. We numerically verify the

ansatz for various equilibrium and nonequilibrium sys-

tems: from 1P and 2P Matsubara and real-time Green’s

functions. Compression rates of several orders of mag-

nitude are demonstrated for challenging cases. We also

numerically show the stability and eﬃciency of the pro-

posed methods for the Dyson and Bethe-Salpeter equa-

tions (BSEs).

Recently, related quantum-inspired algorithms using

the qubit mapping have been proposed for image com-

pression [36], and for solving Navier-Stokes equations for

turbulent ﬂows [37] or the Vlasov-Poisson equations for

collisionless plasmas [38]. A low-rank tensor train ap-

proximation has been applied to the numerical integra-

tion of high-order perturbation series of quantum sys-

tems without the multi-scale ansatz [39]. The quantics

represenation was used to represent spectral functions

in combination with a Boltzmann machine [40]. In this

paper, we clarify the fundamental question how such a

multi-scale ansatz performs in the context of quantum

ﬁeld theories. The QTT representation has the potential

not only to change the way in which numerical many-

body calculations will be performed, but also to bridge

the ﬁelds of quantum information theory and quantum

ﬁeld theory.

The paper is organized as follows: In Sec. II, we intro-

duce the QTT representation. We detail common opera-

tions performed with this ansatz in Sec. III. In Sec. IV, we

show the performance of the QTT representation in en-

coding the imaginary-time or Matsubara-frequency, mo-

mentum, and real-time dependence of correlation func-

tions in a variety of equilibrium and non-equilibrium sys-

tems. Sec. Vis devoted to the demonstration of the

computation of correlation functions. We summarize the

main results of the paper in Sec. VI. Appendices are

devoted to technical discussions on (A) matrix product

states, (B) matrix product operators, (C) Fourier trans-

forms, and (D) frequency meshes.

Note on nomenclature. QTT representation is based

on concepts already known in literature as quantics ten-

sor trains and we adopt this name also here. However, in

0π2π

101k1

F(1)

20101k2

F(2)

301010101k3

F(3)

F(4)

F(5)

F(6)

Figure 1. Multi-scale ansatz for momentum space. Each row,

numbered by the bond index b, corresponds to a diﬀerent

level of discretization of the 1D momentum k(diﬀerent length

scale). In this way, kcan be represented by a set of bits

k1,· · · , kR(see text). On the right, the QTT representation

of a momentum dependent function, Eq. (2), is shown.

the physics community tensor trains are better known as

matrix product states (MPS) and matrix product oper-

ators (MPO) and in the technical parts of the paper we

use these names.

II. MULTI-SCALE SPACE–TIME ANSATZ

In this section, we explain the multi-scale space-time

ansatz based on QTT. The essence of the ansatz is to in-

troduce multiple indices to describe diﬀerent space-time

length scales, and to assume low entanglement structures

between diﬀerent scales (see Fig. 1). We focus on the mo-

mentum space and the associated real space as the ﬁrst

examples.

A. Momentum space

Let us consider ﬁrst a function f(k) in momentum

space, where k∈[0,2π) is the (for now one-dimensional)

momentum. Usually, we discretize f(k) on an equidistant

grid of size, e.g., 2R. This technique is straight-forward to

implement, but comes with a series of drawbacks: many-

body propagators have sharp and intricate structures in

momentum space, which means that the precision of the

approximation only improves slowly with 2R.

Instead of considering a “ﬂat” discretization into a vec-

tor of 2Rmomenta, in the multi-scale ansatz, we ﬁrst

separate out Rdistinct scales k1, . . . , kR:

f(k)≈fk1π+k2

2+··· +kR

2π

2R=f(k1,··· , kR),

(1)

1 3 5 7 9 11 13 15

Bond b

100

101

102

Bond dimension

2 4 6 8 10 12 14

R= 16

2b−1

2R−1−b

I II III IV

Coarse Fine

(b)

Coarse Fine Coarse Fine

(a)

(c)

MPO

Plateau

Figure 2. (a) QTT representation in momentum space. The

rightmost bits (indices) represent ﬁne structures in momen-

tum space. Low entanglement structures are assumed be-

tween diﬀerent length scales. (b) Schematic illustration of the

bond dimensions along the chain representing the momentum

dependence. The dashed line indicates the maximum bond

dimensions in maximally entangled cases. (c) Fourier trans-

form from momentum space to real space by applying a ma-

trix product operator (MPO). The orange diamonds represent

the MPO tensors. The structure of the MPO is illustrated in

Fig. 27 in Appendix C.

where each kb, 1 ≤b≤R, now only takes two values:

zero or one. Put diﬀerently, k1, . . . , kRare the bits of

2Rk/(2π), i.e., k= 2π(k1···kR)2/2R. In this notation

k= 0 corresponds to (00 ···0)2,k= 2π/2Rcorresponds

to (00 ···1)2, and so forth.

We can interpret f(k1,··· , kR) in a couple of ways. In

terms of physics, we have separated out diﬀerent scales

of the problem, as illustrated in Fig. 1:k1partitions the

Brillouin zone into two coarse regions, [0, π) and [π, 2π),

and as we move towards kR, features on ﬁner and ﬁner

scales are captured. In terms of quantum information

theory, f(k1,··· , kR) can be regarded as an (unnormal-

ized) wavefunction in the Hilbert space of dimension 2R

spanned by S= 1/2 spins or qubits. In terms of linear

algebra, we have simply reinterpreted the 2R-vector of

momenta as 2 × ··· × 2 (R-way) tensor.

Since up to this point, we have merely reshaped our

data from a vector to a tensor, no information of the

original discretization is lost. The main idea of the

multi-scale ansatz is to express the single R-way tensor

fby a tensor train, a contraction of Rthree-way tensors

F(1),..., ˆ

F(R):

f(k1,··· , kR)≈

α1=1 ···

DR−1

αR−1=1

F(1)

k1,1α1··· ˆ

F(R)

kR,αR−11

≡ˆ

F(1)

k1·ˆ

F(2)

k2·. . . ·ˆ

F(R)

kR,(2)

where ˆ

F(b)is an auxiliary 2 ×Db−1×Dbtensor,

α1, . . . , αR−1forming bonds between neighbouring ten-

sors, and Dbis the bond dimension of the b-th bond.

D= maxbDbis the bond dimension of the whole MPS.

(We refer the reader to Appendix Afor more details.)

We illustrate Eq. (2) in Fig. 2(a).

The ansatz (2) is still exact if the bond dimension

is very large, D∼2R; it becomes approximate if the

bonds are truncated to the most important contribution.

The core insight is that for many functions, including,

as we shall show, the propagators in momentum space,

the bond dimension needed to approximate the original

tensor grows only modestly with the desired accuracy 

measured by the Frobenius norm [see Eq. (A5)], thus al-

lowing us to compress the function signiﬁcantly.

More speciﬁcally, Fig. 2(b) illustrates how the bond di-

mension typically varies along the chain when the MPS is

truncated with a certain cutoﬀ . First, the bond dimen-

sion increases exponentially in the region I, where coarse

global structures are not compressible. This is followed

by region II (plateau region), where diﬀerent length scales

are not strongly entangled (separation in length scale).

In region III, the bond dimension decreases but there is

still a ﬁnite entanglement that is important for a quan-

titative description of the kdependence within the given

. In region IV, the bond dimension is 1 and the tensor

train can be truncated without sacriﬁcing any accuracy.

The eﬃciency of the QTT representation relies on the

existence of the plateau.

B. Real space

We construct a similar representation for the real

space that is associated with the momentum space by

Fourier transform. In the case of a regular lattice, the

lattice points are labeled by natural numbers, r/a =

0,1,··· ,2R−1, where ais the lattice constant. As we

did for k, we map natural integers to binary numbers as

r/a = (r1···rR)2. Note that riand kR+1−icorrespond

to the same length scale. We represent the Fourier trans-

formed function f(r) in the rspace as an MPS:

f(r) = X

e−ikr f(k)

≡1

2RX

k1,··· ,kR

e−2πi(k1/2+ ··· +kR/2R)rf(k1,··· , kR)

≈

α1=1 ···

rR−1

αR−1=1

F(R)

rR,1α1···F(1)

r1,αR−11

=F(R)

rR·F(R−1)

rR−1·(. . .)·F(1)

r1.(3)

···

xR−1

yR−1

Figure 3. Matrix product states representing a 2D space

spanned by xand y. The expansion can be truncated at the

right edge.

C. Other spaces

The aforementioned representation can be applied to

other variables. However, special care is needed for

imaginary-time and Matsubara-frequency spaces. An

imaginary time τis represented as 2Rτ/β = (τ1···τR)2.

A Matsubara frequency is represented as ν= (2(n−

2R−1) + ξ)π/β using n= 0,1,2,··· ,2R−1 (ξ= 0,1 for

bosons and fermions, respectively). Note that the artiﬁ-

cial periodic boundary condition maps negative Matsub-

ara frequencies as −n= 2R−nto the higher half of the

positive part. This allows to treat momentum space and

frequency space consistently in the implementation. The

boundary eﬀects of this periodic boundary condition van-

ish exponentially with increasing Rand do not matter in

practice. A real time t(0 ≤t<tmax) is represented by

a natural number, 2Rt/tmax. In a similar manner, a real

frequency ω(−W≤ω < W ) is represented by a natural

number, 2R(ω+W)/(2W).

D. Higher dimensions

It is easy to construct QTT representations for higher

dimensional objects spanned by multiple space-time axes.

As an example, let us consider a 2D space spanned by

the two variables x= (x1···xR0)2and y= (y1···yR)2

(R0< R). We assume that we are going to truncate

the expansion at xR0and yR. In other words, the right

qubits correspond to ﬁne resolution for both xand y. In

the present study, we use the MPS structure shown in

Fig. 3. An important point is that two qubits/tensors

corresponding to the same length scale are next to each

other because they are expected to be strongly entangled.

III. OPERATIONS IN THE QTT

REPRESENTATION

A. Fourier transform

The discrete Fourier transform (DFT) in Eq. (3) can be

represented as a matrix product operator (MPO), with

a small bond dimension. (We refer the reader to Ap-

pendix Bfor more details.) This can be intuitively un-

derstood by the fact that two space-time indices rR+1−i

and kiat the same position (i= 1,··· , R) in Fig. 2(c)

correspond to the same length scale. The small bond di-

5 10 15 20

Bond dimension

= 10−25

Figure 4. Bond dimensions of the MPO for the discrete

Fourier transform recursively constructed with truncation

cutoﬀ = 10−25.

mension of the MPO was shown numerically in 2017 [41].

As detailed in Appendix C, one can construct MPOs

recursively for R= 1,2,3,···. Figure 4shows the bond

dimensions of the numerically constructed MPOs with

= 10−25. One can clearly see that the bond dimension

weakly depends on Rand becomes saturated for R > 10.

This crossover point shifts to larger Ras the cutoﬀ is

reduced. This result indicates that the Fourier transform

can be performed eﬃciently, with a computational time

O(R) for ﬁxed target accuracy.

B. Element-wise product

Solving the Dyson equation requires the computation

of the element-wise product of two MPSs, Aand B:

C(iν) = A(iν)B(iν).(4)

To be precise, for given MPSs Aand B, one needs to

compute an MPS for the product C. In the compressed

form, the product can be expressed as

C(νR,··· , ν1)

=A(νR,··· , ν1)B(νR,··· , ν1)

ν0

1,··· ,ν0

AνR,··· ,ν1

ν0

R,··· ,ν0

1B(ν0

R,··· , ν0

1),(5)

where

AνR,··· ,ν1

ν0

R,··· ,ν0

1≡A(νR,··· , ν1)δνR,ν0

R···δν1,ν0

1.(6)

An MPO for the auxiliary linear operator Acan be con-

structed from the MPS tensors of Aas

(AνR

1,a1δνR,ν0

R)···(Aν1

aR,1δν1,ν0

1).(7)

The MPO is illustrated in Fig. 5(a). This allows to use an

eﬃcient implementation of an MPO–MPS multiplication.

νR

ν1

νR

ν1

···

νR

ν1

···

νR

ν1

νR

ν1

νR

ν1

···

ν1

νR

(a)

(b)

νR

Figure 5. Tensor contraction for the element-wise prod-

uct [(a)] and matrix product [(b)] of two MPSs Aand B.

The ﬁlled circles in (a) denote a superdiagonal tensor whose

nonzero entries are one. The dashed squares denote the ten-

sors of the auxiliary MPOs.

C. Matrix multiplication for two-frequency objects

To solve the BSE or the Dyson equation for the non-

equilibrium Green’s function, one needs to multiply two-

frequency quantities:

C(iν, iν00) = X

ν0

A(iν, iν0)B(iν0,iν00),(8)

where we perform the summation on the mesh of size 2R.

This can be expressed as an MPO-MPS product:

C(νR, ν00

R,··· , ν1, ν00

(ν0

1ν000

1)··· ,(ν0

Rν000

A(νRν00

R),··· ,(ν1ν00

(ν0

Rν000

R),··· ,(ν0

1ν000

1)B((ν0

Rν000

R),··· ,(ν0

1ν000

1)).

(9)

Here, we introduced a combined index of dimension 4 (=

22), and an auxiliary MPO, A, which is illustrated in

Fig. 5(b).

D. Linear transformation of arguments of

multidimensional objects

Another typical operation required for solving a dia-

grammatic equation is the linear transformation of argu-

ments of multidimensional objects. As as an example,

we consider a function with two time arguments, f(t, t0).

We want to transform this to a function g(t1, t2) =

f((t−t0)/2,(t+t0)/2) which depends on the relative and

average times. This linear transformation can be repre-

sented by an MPO with a small bond dimension of O(1)

because the linear transformation can be performed al-

most independently at diﬀerent length scales. Indeed, the

MPOs can be constructed using adders or subtractors of

binary numbers.

IV. COMPRESSION

A. Imaginary-time/Matsubara-frequency Green’s

function

As the minimum example, we consider the imaginary-

time and Matsubara-frequency dependence of the

fermionic Green’s function generated by a few poles. The

Green’s function reads

G(iν) = Zdωρ(ω)

iν−ω=

i=1

iν−ωi

,(10)

G(τ) = −

i=1

cie−τωi

1 + e−βωi,(11)

with

ρ(ω) =

i=1

ciδ(ω−ωi),(12)

where ωiand ciare the positions of the poles and the

associated coeﬃcients, respectively. The G(iν) decay

asymptotically as O(1/iν) for large Matsubara frequen-

cies (high frequency tail). Since we know the normaliza-

tion factor PNP

i=1 cia priori from the commutation rela-

tion of the operators, this contribution can be subtracted

G(iν)≡G(iν)−PNP

i=1 ci

iν,(13)

where ˜

G(iν) decays faster than O(1/(iν)2). As we will

see later, this subtraction slightly suppresses the bond

dimension at high temperatures.

For NP= 1, G(τ) can be represented as an MPS of

bond dimension 1:

G(τ) = −c1

1 + e−βω1

t=1

e−τt2−tβω1,

=−c1

1 + e−βω1G(1) ·(···)·G(R)(14)

with the t-th TT tensor

G(t)

αt,αt+1 ≡e−τt2−tβω1δαt,αt+1 ,(15)

where τ/β = (0.τ1τ2···τR)2and t= 1,2,··· , R, while αt

and αt+1 are indices of the virtual bonds. The coeﬃcient

in Eq. (14) can be absorbed into one of the tensors.

For NP>1, the bond dimension of the natural MPS

of G(τ) is bounded from above: D≤NP. This explic-

itly constructed MPS is highly compressible as we will

demonstrate below.

We investigate the compactness of the representation

for a model with NP= 100 where the position and co-

eﬃcients of the poles are chosen randomly according to

the normal Gaussian distribution. We use the truncation

parameter = 10−20.

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Multiscalespace-timeansatzforcorrelationfunctionsofquantumsystemsbasedonquanticstensortrainsHiroshiShinaoka,1,2MarkusWallerberger,3YutaMurakami,4KosukeNogaki,5RihitoSakurai,1PhilippWerner,6andAnnaKauch31DepartmentofPhysics,SaitamaUniversity,Saitama338-8570,Japan2JST,PRESTO,4-1-8Honcho,Kawaguchi,Sait...

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Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains Hiroshi Shinaoka1 2Markus Wallerberger3Yuta Murakami4

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