Improved local truncation schemes for the higher-order tensor renormalization group method

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Improved local truncation schemes for the higher-order tensor

renormalization group method

Jacques Blocha, Robert Lohmayera,b, Maximilian Meistera, Michael Nunhofera

aInstitute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany

bLeibniz Institute for Immunotherapy (LIT), 93053 Regensburg, Germany

Abstract

The higher-order tensor renormalization group is a tensor-network method providing estimates for the par-

tition function and thermodynamical observables of classical and quantum systems in thermal equilibrium.

At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor

dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and

backward tensor modes are projected on the same lower dimensional subspaces. In this paper we present

two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or

even minimize the local approximation errors, while satisfying this constraint.

1. Introduction

Physical systems in thermal equilibrium are described by their partition function, whose complexity

grows exponentially in the volume. The standard method to simulate such statistical systems is the Markov

chain Monte Carlo method (MC), which eﬃciently samples the relevant states of the system to produce

reliable estimates of observables. A fundamental prerequisite for the MC method is the positivity of the

sampling weights. Models which do not satisfy this condition cause the infamous sign problem and require

alternative simulation methods. Quantum systems with complex actions are typical examples of systems

with a sign problem. An important topical application in high energy physics is the simulation of quantum

chromodynamics (QCD) at nonzero quark chemical potential, which allows for the investigation of the QCD

phase diagram as a function of temperature and baryon density.

There exist numerous methods to circumvent the sign problem, and some even solve it for particular

systems [1, 2, 3]. Very mild sign problems can be circumvented by reweighting, which uses the Monte

Carlo method on an auxiliary ensemble with positive weights, and reweights the observables to the target

ensemble. The main issue with this method is that the statistical error increases exponentially with the

volume such that it is hardly usable in any realistic situation, except for the validation of other methods in

regions where the sign problem is small. Other methods which have shown their merit on some models, but

are known to have fundamental problems for other ones, are the complex Langevin method, the thimbles,

the density of states method and the method of dual variables, where the simulations are usually performed

with the worm algorithm. Common to those methods is the stochastic sampling of the partition function.

An alternative approach that has recently drawn a lot of interest is that of tensor networks, see [4]

for a review. In these methods the partition function is ﬁrst rewritten as a full contraction of a tensor

network covering the entire lattice. The exact computation of the partition function and observables in

this formulation would have an exponential complexity. The tensor renormalization group (TRG) [5] and

higher order tensor renormalization group (HOTRG) [6] methods avoid this exponential cost by blocking

the lattice iteratively and truncating the inﬂated dimensions of the coarse grid tensor at each blocking step

Email addresses: jacques.bloch@ur.de (Jacques Bloch), robert.lohmayer@ur.de (Robert Lohmayer)

Preprint submitted to Elsevier October 5, 2022

arXiv:2210.02266v1 [hep-lat] 5 Oct 2022

→ → → →

Figure 1: Blocking procedure to reduce a two-dimensional 4 ×4 lattice to a single tensor using alternating contractions in the

horizontal and vertical directions.

using truncated higher order singular value decompositions (HOSVD) [7], which are based on the matrix

singular value decomposition (SVD).

We consider the partition function of a d-dimensional classical or quantum system in thermal equilibrium,

written as a fully contracted tensor network [8],

Z= tTr

x=1

T(x)

ix,−1ix,1...ix,−dix,d ,(1)

with a tensor T(x)at each site x= 1, . . . , V . In general the local tensor is the same on all sites, i.e., T(x)=T

for all x. For each lattice direction ν, the tensor has one mode for the forward and one mode for the backward

orientation, corresponding to the indices ix,ν and ix,−ν≡ix−ˆν,ν , respectively, where ˆνis a unit step in the

νdirection. We will often refer to these modes as backward and forward modes of the physical tensor. The

trace in the partition function stands for a full contraction over all tensor indices, where two adjacent tensors

share exactly one index.

Thermodynamical observables, which are deﬁned as derivatives of the partition function with respect to

one of its parameters, can be computed using either a ﬁnite-diﬀerence approximation or an impurity tensor

formulation involving the analytical derivative of T[9].

In the following we will restrict our discussion to HOTRG, because it can be applied to any number of

dimensions, whereas TRG is limited to two-dimensional systems. The HOTRG method uses an iterative

blocking procedure that reduces the size of the lattice by a factor of two during each blocking step by

contracting pairs of adjacent tensors. The procedure is illustrated for a two-dimensional 4 ×4 lattice in

Fig. 1. Its extension to higher dimensions is obvious, and below we will further discuss the HOTRG method

for the three-dimensional case.

When contracting two adjacent tensors Tover their shared link, a tensor Mof higher order is produced.

Such a contraction in the 1-direction is illustrated for the three-dimensional case in Fig. 2 and can be written

MjX,−1jX,1jX,−2jX,2jX,−3jX,3=X

ix,1

Tix,−1ix,1ix,−2ix,2ix,−3ix,3Tiy,−1iy,1iy,−2iy,2iy,−3yx,3(2)

where y=x+ˆ

1 and therefore iy,−1=ix,1, by deﬁnition, X= (x, y) labels sites on the coarse grid and

jX,−1=ix,−1, jX,1=iy,1

jX,−2= (ix,−2, iy,−2), jX,2= (ix,2, iy,2)

jX,−3= (ix,−3, iy,−3), jX,3= (ix,3, iy,3))fat indices.(3)

For any direction perpendicular to the direction of contraction, the tensor Mhas modes originating from

both contracted tensors. To keep the order of the tensor unchanged, we gather every such pair of modes in

a new fat mode corresponding to its direct product space. Assuming that the modes of the local tensor have

dimension D, then the fat modes will have dimension D2. In HOTRG these fat modes are truncated back

jX,−1ix,1=iy,−1jX,1

ix,−2iy,−2

jX,−2

ix,−3iy,−3

jX,−3

ix,3iy,3

jX,3

ix,2iy,2

jX,2

T T

Figure 2: Illustration of the contraction T ?1T=Malong the 1-direction in a three-dimensional system, as in (2). The square

nodes represent the fusion of the original tensor indices into the combined fat indices (3) of M.

to dimension Dusing a modiﬁed version of the HOSVD approximation method, such that the dimension of

the coarse grid tensor remains the same as that of the original local tensor throughout the entire blocking

procedure.

In general, step k+ 1 of the HOTRG procedure can be summarized as

T[k]?νT[k]=: Mtruncate

−→ T[k+1],(4)

where the ?ν-operation symbolically represents a forward-backward contraction in direction ν. The precise

construction of T[k+1] will be discussed in Sec. 3.

In the standard approximation procedure using HOSVD [7], referred to as HOSVD approximation in the

following, the dimension of each tensor mode gets reduced by projecting it on a lower dimensional subspace,

which is generically diﬀerent for each mode. This HOSVD approximation is modiﬁed when used as part of

the iterative blocking procedure in the standard HOTRG algorithm, as it is essential for the accuracy and

eﬀectiveness of the method that the backward and forward modes for every direction get projected on the

same subspace. Each of these subspaces will be characterized by a frame, which is a set of orthonormal basis

vectors spanning the subspace. Constructing appropriate frames will be the major subject of this paper.

The standard HOTRG procedure for the construction of frames [6] is not optimal, in particular when the

local tensor is not symmetric in its backward and forward modes. In this paper we present two improved

methods for the construction of common subspaces for pairs of backward and forward modes: the SuperQ

and the iterative SuperQ method (ISQ), which is an iterative improvement of the former in search of the

optimal subspaces. Note that the discussion in this paper solely focusses on the optimization of the rank

reduction of the local tensors at every blocking step, but does not take into account global eﬀects on the

full contraction of the tensor network.

Here is a brief outline of the paper. In Sec. 2 we review the standard HOSVD method to construct a

reduced rank approximation for an arbitrary tensor. In Sec. 3 we explain why the HOTRG method uses a

modiﬁcation of this rank reduction procedure such that the backward and forward modes are projected on

the same subspace. We then propose two methods to improve the standard HOTRG truncation: In Sec. 4

we present the SuperQ method, and in Sec. 5 we derive the more sophisticated ISQ method. Finally, we

summarize and conclude in Sec. 7.

2. Rank reduction and HOSVD approximation

Below we ﬁrst review the general idea of rank reduction for an arbitrary tensor, before describing the

HOSVD procedure [7] which can be used to generate a quasi-optimal rank-reduced approximation in an

eﬃcient way.

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Improvedlocaltruncationschemesforthehigher-ordertensorrenormalizationgroupmethodJacquesBlocha,RobertLohmayera,b,MaximilianMeistera,MichaelNunhoferaaInstituteforTheoreticalPhysics,UniversityofRegensburg,93040Regensburg,GermanybLeibnizInstituteforImmunotherapy(LIT),93053Regensburg,GermanyAbstractThehi...

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Improved local truncation schemes for the higher-order tensor renormalization group method

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