ARTICLES

Sci. Rep. Kanazawa Univ.
Vol. 65, pp. 1–9, 2021

Cost reduction of the bottleneck contraction in thehigher-order tensor renormalization group
Hideaki O BA*
Graduate School of Natural Science and Technology, Kanazawa University
Kakuma, Kanazawa 920-1192, Japan(Received January 23, 2021 and accepted in revised form March 2, 2021)

Abstract We reduce the computational cost of the bottleneck contraction in the higherorder tensor renormalization group (HOTRG) by using the singular value decomposition.
To conﬁrm the effectiveness of the proposed technique, we examine the HOTRG with orwithout the proposed technique by using the square lattice Ising model at criticality. Then,
we ﬁnd that the proposed method provides the consistent results with the original HOTRG
while the computational cost and the elapsed time are reduced. Therefore, we conclude thatthe performance of the proposed method is better than that of the original one.
Keywords. Higher-order tensor renormalization group, tensor contraction, singular value
decomposition

1

Introduction

Tensor network (TN) methods are very useful tools to study quantum and classical many-bodysystems [1,2]. By using the methods, one can calculate a wave function and a partition function ofa lattice system. There are remarkable properties of TN methods. For example, they can handle alarge volume system easily, and avoid the sign problem which occurs in some lattice models whenusing the Monte Carlo methods. Therefore, we expect that we can explore the models which havethe sign problem by using the TN methods.
Tensor renormalization group (TRG) [3] is the one of the TN methods by which one can calculate the partition function of a two-dimensional classical lattice model. The concept of TRG
is that the tensor network of the partition function is transformed into one which has less numbers of tensors by using the singular value decomposition (SVD). By extending TRG, various TN
methods have been proposed since it was introduced [4–18], and there are some algorithms whichallow us to deal with arbitrary-dimensional systems. Higher-order tensor renormalization group(HOTRG) [6] is one of them and its important feature is to use the higher-order SVD. However, thecomputational cost of the HOTRG turns out to be worse when the dimension of the lattice model
becomes large. The computational cost of the HOTRG is O(χ 4d−1 ), where χ is the bond dimension and d is the dimension of the system. The most time-consuming calculation in the HOTRG
is the contraction for making a coarse-grained tensor since one has to deal with a lot of tensors’* Email:oba.h534@gmail.com

2

Cost reduction of the bottleneck contraction in the higher-order tensor renormalization groupbonds in this calculation. To solve this problem, anisotropic tensor renormalization group [16, 17]
and triad tensor renormalization group [18] have been proposed and they can signiﬁcantly reducethe computational cost for the high-dimensional system compared to the HOTRG, though the accuracies of the results obtained by using the algorithms are worse than that of the HOTRG withthe same bond dimension.
In this paper, we focus on the HOTRG and reduce the computational cost of the algorithm byapplying the SVD to the tensor in the bottleneck contraction. We compare the original HOTRG
to the HOTRG with the proposed technique by using the square lattice Ising model at criticality.
Then, we ﬁnd that the proposed method provides the consistent results with the original HOTRG
while the computational cost and the elapsed time are reduced. We conclude that the performanceof the proposed method is better than that of the original one especially when χ becomes larger.
The rest of this paper is organized as follows. In Sec. 2, we review the original HOTRG
algorithm for a square lattice system. In Sec. 3, the proposed contraction technique is explained.
In Sec. 4, we present numerical results of the square lattice Ising model for a comparison between
the original HOTRG and the HOTRG with the proposed technique, and discuss their performances.
A summary and outlooks are given in Sec. 5.

2 Higher-order tensor renormalization group
In this section, we review the HOTRG [6] for a square lattice model. First of all, we rewrite thepartition function of the system in terms of a tensor network as
Z = Tr ∏ Ti;x0 x1 y0 y1 ,(init)(2.1)

i(init)

where Ti
is an initial tensor, i runs all lattice sites, and Tr means summing up all the bondindices of the initial tensors. Then, we execute the coarse graining of the tensor network to reducethe degrees of freedom by using the HOTRG [6]. The whole ﬂow of the HOTRG is shown in
Figure 1. There are two steps in the HOTRG, making the isometries Us and inserting them inthe tensor network [Fig. 1(a)], and contracting two T s and two Us [Fig. 1(b)]. First, to obtain theisometry U, two T s are contracted in the following ways,χ

Kx1 y0 x̃1 ỹ0 =∑ Tx x y y Tx x̃ ỹ y ,(2.2)∑ Tx x y y Tx̃ x ỹ y ,(2.3)∑ Tx x y y Tx x̃ y ỹ ,(2.4)

∑ Tx x y y Tx̃ x y ỹ .

(2.5)

0 1 0 10 1 0 1

x0 ,y1 =1
χ

Lx1 y2 x̃1 ỹ2 =Kx′ 1 y1 x̃1 ỹ1 =Lx′ 1 y3 x̃1 ỹ3 =1 2 2 31 2 2 3x2 ,y3 =1
χ

0 1 0 10 1 0 1

x0 ,y0 =1
χ

1 2 2 3x2 ,y2 =1

1 2 2 3

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Hideaki O BA

x0

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Figure 1: The ﬂow of the HOTRG. (a) Making the isometries Us and inserting them in the tensor
network. (b) Contracting two T s and two Us to obtain T ′ .
where T is the χ × χ × χ × χ tensor 1 . Next, K and L are contracted to M, and then K ′ and L′ arecontracted to N as
χ

My0 y2 ỹ0 ỹ2 =∑ Kx y x̃ ỹ Lx y x̃ ỹ ,(2.6)

∑ Kx′ y x̃ ỹ Lx′ y x̃ ỹ .

(2.7)

1 0 1 01 2 1 2

x1 ,x̃1 =1
χ

Ny1 y3 ỹ1 ỹ3 =1 1 1 11 3 1 3x1 ,x̃1 =1

1 Even if the bond dimensions of the tensor T are smaller than χ , one can implement the contractions and decompositions of the tensors in HOTRG similarly.

4

Cost reduction of the bottleneck contraction in the higher-order tensor renormalization group

Then, one implements the SVD or the eigenvalue decomposition to M and N as,
χ2

My0 y2 ỹ0 ỹ2 = ∑ U{L}y0 y2 i Λ{L}iiU{L}ỹ0 ỹ2 i ,

(2.8)

i=1

χ2

Ny1 y3 ỹ1 ỹ3 = ∑ U{R}y1 y3 j Λ{R} j jU{R}ỹ1 ỹ3 j ,

(2.9)

j=1

where the singular values, Λ{·} s, are ordered such that Λ{·}kk ≥ Λ{·}ll ≥ 0 (k < l), and evaluates
ε1 , ε2 deﬁned by

ε1 = ∑ Λ{L}ii ,(2.10)

ε2 = ∑ Λ{R} j j .

(2.11)i>χ

j>χ

If ε1 < ε2 , then we set the isometry U = U{L} , which is truncated by keeping the index i corresponding to the largest χ singular values. Otherwise U = U{R} truncated in the same way as U{L} .
Then, the isometries Us are inserted in the tensor network [see Fig. 1(a)].
As second step, T and U are contracted in the following two ways to obtain A and B as,χ

Ax0 x1 y0 y3 y′1 = ∑ Tx0 x1 y0 y1 Uy1 y3 y′1 ,(2.12)

Bx1 x2 y0 y3 y′0 = ∑ Tx1 x2 y2 y3 Uy0 y2 y′0 .

(2.13)

y1 =1
χ
y2 =1

Then A and B are contracted to make a coarse-grained tensor, T ′ ,Tx′0 x2 y′ y′ =
0 1

χ

∑

x1 ,y0 ,y3 =1

Ax0 x1 y0 y3 y′1 Bx1 x2 y0 y3 y′0 .

(2.14)

The computational cost of making U is O(χ 6 ) and that of the contraction of four tensors isO(χ 7 ). Therefore, the latter part is the bottleneck in the HOTRG.

3 Cost reduction of the bottleneck contraction in the HOTRG
In this section, we explain how to apply the SVD to the tensor T and reduce the computational
cost of the bottleneck contraction part in the HOTRG, Eq. (2.14). The arrow named (a) of Figure 2
represents executing Eq. (2.14) and the ﬂow of the proposed technique can be seen from (b) to (c)of Figure 2. First, we decompose T by implementing the truncated SVD with the bond dimensionχp (χ ≤ χp ≤ χ 2 ) as,χp

Tx0 x1 y0 y1 ≈ ∑ Xx1 y0 n SnnYx0 y1 nn=1
χp

= ∑ Cx1 y0 n Dx0 y1 n ,

(3.1)

n=1

where

p
Snn Xx1 y0 n ,p
Dx0 y1 n = SnnYx0 y1 n .
Cx1 y0 n =

(3.2)
(3.3)

5

Hideaki O BA

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T

(a)

(c)

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T0

T 00

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Figure 2: The way to implement the SVD to the tensor T in the bottleneck contraction of the
HOTRG. (a) The contraction of two T s and two Us in the HOTRG. (b) T is decomposed to C andD. (c) The contractions shown in Eq. (3.4)–(3.6) are executed and T ′′ is obtained.

Then, we replace Eq. (2.12)–(2.14) with the following calculations,χ

A′x0 y3 y′ n = ∑ Dx0 y1 nUy1 y3 y′1 ,1

B′x2 y3 y′ n =
0

y1 =1
χ

∑

x1 ,y0 ,y2 =1
χ χp

(3.4)



Tx1 x2 y2 y3 Uy0 y2 y′0 Cx1 y0 n ,Tx′′0 x2 y′ y′ = ∑ ∑ A′x0 y3 y′ n B′x2 y3 y′ n ≈ Tx′0 x2 y′ y′ .
0 1

y3 =1 n=1

1

0

0 1

(3.5)
(3.6)

The computational cost of this alternative method to obtain T ′′ , which is the approximation of T ′in Eq. (2.14), is O(χp χ 5 ). If χp < χ 2 , then the computational cost of this part becomes less thanO(χ 7 ).

4 Numerical results
In this section, we show the numerical results of the 215 × 215 Ising model at criticality with theperiodic boundary condition in order to conﬁrm the effectiveness of the proposed technique 2 . The
initial tensor of the Ising model on the square lattice at the temperature T = 1/β is given as(init)

2

Tx0 x1 y0 y1 =∑δ pq δqr δrs δspWpx0 Wqx1 Wry0 Wsy1 ,(4.1)

p,q,r,s=1

where

W=

W11 W12
W21 W22" pcosh β= pcosh β

#
p
sinh
β
p
.
− sinh β(4.2)

2 The spectrum of the tensor T decays the most slowly and χ should be the largest at criticality. Therefore, it ispenough to investigate the performance of the proposed algorithm only at the critical point since the computational costis the maximum.

6

Cost reduction of the bottleneck contraction in the higher-order tensor renormalization group
-7

HOTRG
scut=1x10-3
scut=1x10-4-5
scut=1x10

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f

1x10

-9

1x10

30

35

40

45

50

55

60

65

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Figure 3: The relation between χ and δ of the original HOTRG and the proposed method whenscut changes. We use the 215 × 215 Ising model at criticality with the periodic boundary condition.
To see the accuracy of the free energy, the relative error of it is deﬁned byδ=f − fexact,
fexact(4.3)

where f is the free energy given by the HOTRG with or without the proposed technique, and fexact
is the exact value obtained by the Onsager’s solution [19]. The free energy of 215 ×215 Ising model
can be obtained by implementing the 30 HOTRG steps. The result of δ is shown in Figure 3. In
this ﬁgure, the bond dimension χ is set to be the integers in 32 ≤ χ ≤ 64. scut is a parameter which
represents the truncation magnitude of the singular values in Eq. (3.1), and deﬁned by 1.0 × 10−m ,where m is set to be some integers. For given scut , χp in Eq. (3.1) is deﬁned by the number ofsingular values, Snn , which satisfy Snn ≥ S11 × scut . From Fig. 3, it is enough to set scut ≤ 1 × 10−5
in 32 ≤ χ ≤ 64 since the results of the proposed method (scut = 1 × 10−5 ) are consistent with thoseof the original HOTRG.
Next, let us see the difference between the free energies obtained by the proposed method andthe original HOTRG in detail. For this purpose, we deﬁne another relative error of the free energy,
δ f , asδf =fproposal − fhotrg,
fhotrg(4.4)

where fproposal means the free energy given by the proposed method and fhotrg is the free energy
obtained by the original HOTRG. The relation between χ and δ f is shown in Figure 4. From thisﬁgure, we ﬁnd that the values of δ f depend on scut rather than χ , and δ f = 0 with double precision
for scut = 10−8 . Therefore, we conclude that fproposal is equal to fhotrg when scut = 10−8 .
Then, to see the performance of the proposed method, we introduce χpM , which is the largest
χp after the 7th coarse-graining of the proposed method. Figure 5 shows the relation between χand χ 2 /χpM . From this ﬁgure, we can see χ 2 /χpM is increasing when χ becomes larger. Therefore,
the power of χ in χpM is less than 2 as χpM is the functions of χ . Moreover, we can see the relation
between χ and χpM in Figure 6. To investigate the power of χ in χpM , we ﬁt the data with thefollowing function:

χpM = aχ c + b,

(4.5)

7

Hideaki O BA

1x10

-10

1x10

-11

1x10

-12

-5

scut=1x10
scut=1x10-6-7
scut=1x10-8
scut=1x10<latexit sha1_base64="WKAnB/nEtT052OKy3kG0yH0Cwk4=">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</latexit>f1x10-91x10-13
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<latexit sha1_base64="BBENFyPtyQcO1GVK6x5IeKQK2ns=">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</latexit>

Figure 4: The relation between χ and δ f when scut changes. We cannot see the results for scut =1 × 10−8 since δ f = 0 with double precision.
9-5

scut=1x10-6
scut=1x10-7
scut=1x10
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Figure 5: The relation between χ and χpM /χ 2 of the proposed method when scut changes.
Table 1: The ﬁtting results of the data in Figure 6 with Eq. (4.5).
scut

a

b

c

10−5

1.01(44) × 101−6.85(38) × 101

8.7(39)1.05(76) × 101
1.0(10) × 102−4.85(38) × 101−1.0(86) × 101−4.4(37) × 102

9.63(90) × 10−1
1.045(92)1.05(15)
6.6(19) × 10−1

10−6
10−7
10−8

and show the ﬁtting results in Table 1. The ﬁtting range is set to be 32 ≤ χ ≤ 64. From this table,
we ﬁndc≈1(4.6)

in Eq. (4.5) at the all scut . Therefore, the proposed method can reduce the computational cost fromO(χ 7 ) to O(χ 6 ).
Finally, we investigate the total elapsed times of the two algorithms and a performance ratio
which is deﬁned as the division of the elapsed time of the original HOTRG by that of the proposed
method with the same bond dimension. The results of the elapsed times and the performance ratio

8

Cost reduction of the bottleneck contraction in the higher-order tensor renormalization group<latexit sha1_base64="iccLv24AtFhD+NNa0VnDazLg5no=">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</latexit>

M
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s =1x10-51100 cuts =1x10-6
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Figure 6: The relation between χ and χpM of the proposed method when scut changes. The solidlines are the ﬁtting lines of the data with the function shown in Eq. (4.5).
2.6

10

2.2
2
1.8

4

10

103
30

1.6
HOTRG
scut=1x10-8
Performance ratio

40

50

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1.4

Performance ratio

Elapsed time [sec]

2.4
5

1.2
1

Figure 7: The elapsed times of the original HOTRG and the proposed method (scut = 10−8 ), andthe performance ratio between the two methods as the functions of χ .
are shown in Figure 7. From this ﬁgure, we ﬁnd that the elapsed time of the proposed method
is smaller than that of the original HOTRG in 32 ≤ χ ≤ 64 and the performance ratio becomeslarger as the bond dimension grows. Therefore, we conclude that the performance of the proposed
method is better than that of the original HOTRG. For measuring the elapsed times, we used themachine which has 24GB for the memory and Xeon X5670 (2.93 Ghz 6 core) for the two CPUs.
The programs of the algorithms are written with python 2.7.6rc1 and we used numpy.tensordot
in numpy 1.8.0 for the tensor contractions and scipy.linalg.svd in scipy 0.14.0 for the SVD of thetensors.

5

Summary and outlooks

We reduce the computational cost of the bottleneck contraction in the HOTRG by using the SVD.
The computational cost of the bottleneck part in the HOTRG becomes O(χp χ 5 ) from O(χ 7 ). To
conﬁrm the effectiveness of the proposed technique, we use the square lattice Ising model and ﬁndthat the free energy of the proposed method are consistent with the original HOTRG in 32 ≤ χ ≤ 64
if we set scut = 10−8 . We also ﬁnd that the proposed method can reduce the computational cost

Hideaki O BA

9from O(χ 7 ) to O(χ 6 ), and actually the elapsed time of the proposed method is smaller than that of
the original HOTRG. Therefore, we conclude that the proposed method shows better performance
than the original HOTRG.
For the future works, we consider extending this technique to the HOTRG which can be applied
to the high dimensional systems. However, the spectra of the tensors in the bottleneck calculation
seem to decay much more slowly. To solve this situation, we may take account of applying theenvironment of the tensor [4] and disentanglers [7] for the new algorithm.

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