---
title: 第10回 配信講義　計算科学技術特論A （2023）
tags:  #テンソルネットワーク #計算科学  
author: [R-CCS　計算科学研究推進室](https://www.docswell.com/user/2300203199)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/2EVL34VNJQ.jpg?width=480
description: 第10回6月22日 テンソルネットワークを用いた大規模計算  テンソルネットワーク形式による計算問題の記述法と、それを効率的に計算機で計算するための技術について紹介する。
published: June 20, 23
canonical: https://www.docswell.com/s/2300203199/5YWEP9-2023-06-20-101257
---
# Page. 1

![Page Image](https://bcdn.docswell.com/page/2EVL34VNJQ.jpg)

ςʫ
࠷খαΠζ
Computational
Science
Alliance
T h e Un ive r s i t y of To k yo
15mm
ςϯιϧωοτϫʔΫΛ༻͍ͨେ‫ن‬໛‫ࢉܭ‬
౦‫ژ‬େֶେֶӃཧֶ‫ڀݚܥ‬Պɹେ‫ٱ‬อ‫ؽ‬


# Page. 2

![Page Image](https://bcdn.docswell.com/page/57GX31L5EL.jpg)

ߨࢣͷഎ‫ܠ‬
େ‫ٱ‬อɹ‫ؽ‬ʢOKUBO Tsuyoshi)
౦‫ژ‬େֶཧֶ‫ڀݚܥ‬Պ
ྔࢠιϑτ΢ΣΞ‫࠲ߨ෇د‬
ಛ೚।‫ڭ‬त
‫ڀݚ‬෼໺ɿ
౷‫ܭ‬෺ཧɺ෺ੑ෺ཧɺ࣓ੑɺ‫ࢉܭ‬෺ཧ
•
•
•
߶ମԁ൫ͷϥϯμϜύοΩϯά
֊૚ࣾձ‫ܗ‬੒ͷฏ‫ۉ‬৔ղੳɾγϛϡϨʔγϣϯ
ʢ‫ݹ‬యʣϑϥετϨʔτ࣓ੑମͷ৽‫ن‬டং
•
•
•
Monte Carlo
multiple-Q ঢ়ଶ, Z2-vortex,…
୤ด͡ࠐΊྔࢠྟք‫ݱ‬৅
ςϯιϧωοτϫʔΫ
• ʢྔࢠʣεϐϯ‫ܥ‬ɺςϯιϧ‫܁‬ΓࠐΈ
•
ྔࢠଟମ‫ܥ‬ͷͨΊͷྔࢠΞϧΰϦζϜ
•
….
ʮ‫ژ‬ʯ
ʮ෋ַʯʁ


# Page. 3

![Page Image](https://bcdn.docswell.com/page/4EQ85DYLJP.jpg)

ࠓ೔ͷ࿩ͷྲྀΕ
•
͸͡Ίʹ
•
•
•
•
ςϯιϧͱςϯιϧωοτϫʔΫ
ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ‫ૅج‬
•
‫ج‬ຊతͳԋࢉ
•
ۙࣅతͳςϯιϧωοτϫʔΫॖ໿ɿςϯιϧ‫܁‬ΓࠐΈ‫܈‬
•
ม෼๏ʹΑΔ‫ݻ‬༗஋໰୊΁ͷԠ༻
ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷେ‫ن‬໛Խ
•
ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ࣮૷ɾಛ௃
•
ΞϧΰϦζϜʹಛԽͨ͠ฒྻԽɿHOTRG
•
൚༻తͳϥΠϒϥϦΛ࢖ͬͨฒྻԽɿTeNeS
·ͱΊ


# Page. 4

![Page Image](https://bcdn.docswell.com/page/KJ429ZW5E1.jpg)

͸͡ΊʹɿςϯιϧͱςϯιϧωοτϫʔΫ


# Page. 5

![Page Image](https://bcdn.docswell.com/page/LE1MKRY27G.jpg)

ςϯιϧʁ
•
ϕΫτϧ
ɿ
1࣍‫ݩ‬తͳ਺ࣈͷฒͼ
•
ߦྻ
ɿ
ɹ2࣍‫ݩ‬తͳ਺ࣈͷฒͼ
ҰൠԽ
•
(n֊ͷʣςϯιϧ
ɿ
ɹn࣍‫ݩ‬తͳ਺ࣈͷฒͼ
ʲ‫ج‬ຊతͳԋࢉʹॖ໿ʳ
ߦྻੵɿ
ॖ໿ɿ
&quot;଍&quot;͕ଟ͘ͳΔͱ
ද‫͕ه‬ෳࡶ...


# Page. 6

![Page Image](https://bcdn.docswell.com/page/GEW3D1G2E2.jpg)

μΠΞάϥϜΛ༻͍ͨςϯιϧද‫ه‬
•
ϕΫτϧ
ɿ
•
ߦྻ
ɿ
•
ςϯιϧ
ɿ
ςϯιϧͷੵʢॖ໿ʣͷද‫ݱ‬
ˎn֊ͷςϯιϧʹnຊͷ଍
C
ʹ
B
A
C
D
ʹ B
A


# Page. 7

![Page Image](https://bcdn.docswell.com/page/47ZVGPL4E3.jpg)

ॖ໿ͷ‫ྔࢉܭ‬
3ຊ
ߦྻੵɿ A, B =
C
ʹ
B
A
ͷ‫=ྔࢉܭ‬
ςϯιϧॖ໿ɿ
A=
B=
4ຊ
C
ʹ
A
ͷ‫=ྔࢉܭ‬
μΠΞάϥϜͱͷରԠ
•
•
ॖ໿ͷ‫ྔࢉܭ‬͸μΠΞάϥϜͷ଍ͷ਺Ͱ෼͔Δ
ʢϝϞϦ࢖༻ྔ΋෼͔Δʣ
4ຊ
B


# Page. 8

![Page Image](https://bcdn.docswell.com/page/YJ65YMWG7V.jpg)

ॖ໿ͷ‫ॱࢉܭͱྔࢉܭ‬
ςϯιϧॖ໿ɿ
A=
B=
C=
C
D
ʹ B
A
Case 1:
ͷ‫=ྔࢉܭ‬
Case 2:
ͷ‫=ྔࢉܭ‬
ॖ໿ͷධՁॱͰ‫͕ྔࢉܭ‬มΘΔʂ
C
B
A
AB
C
A
B
AC
ˎ࠷దॱংͷܾఆ͸NPࠔ೉ɻ࣮༻తͳΞϧΰϦζϜྫ
R.N.C. Pfeifer, et al., Phys. Rev. E 90, 033315 (2014).


# Page. 9

![Page Image](https://bcdn.docswell.com/page/GJ5YGZMXJ4.jpg)

ςϯιϧωοτϫʔΫ
ɿςϯιϧͷॖ໿Ͱߏ੒͞ΕͨωοτϫʔΫ
ςϯιϧωοτϫʔΫʢTNʣ
ʲʢͬ͘͟Γͨ͠ʣ෼ྨʳ
•
•
Openͳ଍ɿ͋Γ or ͳ͠
•
Openͳ଍͕͋ΓɿTNࣗ਎͕େ͖ͳςϯιϧ
•
Openͳ଍͕ͳ͠ɿTN͸਺ࣈ
ωοτϫʔΫߏ଄ɿ‫ن‬ଇత or ෆ‫ن‬ଇ
•
•
ωοτϫʔΫߏ଄͸໰୊ʹԠͯ͡มΘΔ
• ྫɿεϐϯ໛‫ܕ‬ͷ෼഑ؔ਺͸‫ن‬ଇత
• ྫɿ෼ࢠͷଟମిࢠঢ়ଶ͸ෆ‫ن‬ଇ
ωοτϫʔΫαΠζɿ༗‫ ݶ‬or ແ‫ݶ‬
•
‫ج‬ຊతʹ༗‫͕ͩݶ‬ɺ৔߹ʹΑͬͯ͸ແ‫ܥݶ‬΋
औΓѻ͑Δ


# Page. 10

![Page Image](https://bcdn.docswell.com/page/LE35R9W4J5.jpg)

ςϯιϧωοτϫʔΫͷྫ1ɿ౷‫ܭ‬෺ཧֶ
‫ݹ‬యΠδϯά໛‫ܕ‬ʢ࣓ੑମͷϞσϧʣ
Թ౓T Ͱͷ֬཰෼෍ɿϘϧπϚϯ෼෍
ঢ়ଶɿ
‫ٯ‬Թ౓ɿ
೤ྗֶࣗ༝ΤωϧΪʔ
෼഑ؔ਺ɿ
(2Nͷ࿨)
•
Openͳ଍͸&quot;ͳ͠&quot;
ʲ෼഑ؔ਺ͷςϯιϧωοτϫʔΫදࣔʳ• ‫ن‬ଇత
•
A
A
༗‫ݶ‬ʙແ‫ݶ‬
A
A
A
A
A
A
A
A


# Page. 11

![Page Image](https://bcdn.docswell.com/page/8EDDYGK5EG.jpg)

ςϯιϧωοτϫʔΫͷྫ2ɿྔࢠճ࿏
ྔࢠճ࿏ɿ
googleͷ&quot;ྔࢠ௒ӽ&quot; ճ࿏
F. Arute, et al., Nature 574, 505 (2019)
ྔࢠϏοτʹԋࢉ͢Δήʔτૢ࡞ͷճ࿏ਤ
Article
developed fast, high-fidelity gates that can be executed simultaneously
across a two-dimensional qubit array. We calibrated and benchmarked
the processor at both the component and system level using a powerful
new tool: cross-entropy benchmarking11. Finally, we used componentlevel fidelities to accurately predict the performance of the whole system, further showing that quantum information behaves as expected
when scaling to large systems.
2࣍‫ݩ‬ͷϕΫτϧɻ
ద౰ͳ‫ج‬ఈͷ‫Ͱݩ‬͸ɺϢχλϦ
ద౰ͳ‫ج‬ఈɺʢ | 0⟩, | 1⟩ʣͰ
ߦྻʢor &quot;ςϯιϧ&quot;ʣ
a
A suitable computational task
ද‫͢ݱ‬Δͱ2࣍‫ݩ‬ͷෳૉϕΫτϧ
To demonstrate quantum supremacy, we compare our quantum processor against state-of-the-art classical computers in the task of sampling
the output of a pseudo-random quantum circuit11,13,14. Random circuits
are a suitable choice for benchmarking because they do not possess
structure and therefore allow for limited guarantees of computational
hardness10–12. We design the circuits to entangle
b a set of quantum bits
Single-qubit
gate: logi(qubits) by repeated application of single-qubit and
two-qubit
25 ns
cal operations. Sampling the quantum circuit’s output produces a set
of bitstrings, for example {0000101, 1011100, …}.
Owing to quantum
Qubit
XYbitstrings
control
interference, the probability distribution of the
resembles
a speckled intensity pattern produced by light interference in laser
Two-qubit gate:
scatter, such that some bitstrings are much more likely
to occur than
12 ns
others. Classically computing this probability distribution
becomes
Qubit 1
Z control
exponentially more difficult as the number of qubits
(width) and number
of gate cycles (depth) grow.
Coupler
We verify that the quantum processor is working properly using a
2 compares how
benchmarking11,12,14,Qubit
which
A
B
CmethodDcalled cross-entropy
Z control
often each bitstring is observed experimentally with its corresponding
m
5
6
7
8
ideal probability computed via simulation on a classical computer. For
a given
circuit,into
we four
collect
the measured
bitstrings
i} and compute the
couplers
are divided
subsets
(ABCD), each
of which is{x
executed
11,13,14
•
linear
cross-entropy
benchmarking
fidelity
(see
also
Supplementary
simultaneously across the entire array corresponding to shaded colours.
Here
Information),
which
is
the
mean
of
the
simulated
probabilities
of the
we show an intractable sequence (repeat ABCDCDAB); we also use different
bitstrings
we
measured:
coupler subsets along with a simplifiable sequence (repeat EFGHEFGH, not
googleͷ&quot;ྔࢠ௒ӽ&quot;
ճ࿏ F. Arute, et al., Nature 574, 505 (2019)
Article
a
0
0
C
A
0
B
D
0
0
Row
Column
W
X
X
W
Y
Time
Cycle 1
A
B
2
D
C
3
4
Fig. 3 | Control operations for the quantum supremacy circuits. a, Example
quantum circuit instance used in our experiment. Every cycle includes a layer
each of single- and two-qubit gates. The single-qubit gates are chosen randomly
from { X , Y , W }, where W = (X + Y )/ 2 and gates do not repeat sequentially.
The sequence of two-qubit gates is chosen according to a tiling pattern,
coupling each qubit sequentially to its four nearest-neighbour qubits. The
single-qubit gates chosen randomly from
on all qubits,
followed by two-qubit gates on pairs of qubits. The sequences of gates
which form the ‘supremacy circuits’ are designed to minimize the circuit
depth required to create a highly entangled state, which is needed for
Adjustable coupler
b
Openͳ଍͸&quot;͋Γ&quot;
ྔࢠճ࿏=ςϯιϧωοτϫʔΫ
&quot;ͳ͠&quot;΋͋Δ
•
shown) that can be simulated on a classical computer. b, Waveform of control
FXEB = 2n&quot;P(xi)#i − 1
signals for single- and two-qubit gates.
(1)
ෆ‫ن‬ଇ
where nthese
is thecircuits
number
qubits,
P(x )processor
is the probability
ofthan
bitstring
of running
onofthe
quantum
is greater
at x
computed
for
the
ideal
quantum
circuit,
and
the
average
is
over
least 0.1%. We expect that the full
data for Fig. 4b should have similar the
• ༗‫ݶ‬
observed
bitstrings.
Intuitively,
is correlated
withtoo
how
often
F (red
fidelities, but since the simulation times
numbers) take
long
to we
ྔࢠίϯϐϡʔλͷ‫ݹ‬యγϛϡϨʔγϣϯ
{ X, Y, W}
ʹςϯιϧωοτϫʔΫͷॖ໿
Qubit
•
i
XEB
i
sample
high-probability
bitstrings.
When
there aresection).
no errors
in the
check,
we have
archived the data
(see ‘Data
availability’
The
10 mm
Fig. 1 | The Sycamore processor. a, Layout of processor, showing a rectangular
array of 54 qubits (grey), each connected to its four nearest neighbours with
couplers (blue). The inoperable qubit is outlined. b, Photograph of the
Sycamore chip.


# Page. 12

![Page Image](https://bcdn.docswell.com/page/V7PX63KDE8.jpg)

ςϯιϧωοτϫʔΫʹΑΔྔࢠճ࿏γϛϡϨʔγϣϯ
ྔࢠճ࿏ͷγϛϡϨʔγϣϯ=ςϯιϧωοτϫʔΫͷॖ໿
‫ݹ‬యίϯϐϡʔλͰͷ‫ࢉܭ‬ɿ
࣮ࡍͷճ࿏ͷ࣮ߦॱংʹΑΒͣɺ࠷దͳॱ൪Ͱςϯιϧͷॖ
໿‫ࢉܭ‬Λߦ͏͜ͱͰɺ‫ࢉܭ‬ίετɺϝϞϦίετ͕௿Լ
࠷ઌ୺ͷ‫ࢉܭ‬ɿ
Y. A. Liu, et al., Gordon bell Prize in SC21 (2021),
Google͕ྔࢠ௒ӽΛओுͨ͠ϥϯμϜྔࢠճ࿏ͷ‫ݹ‬యαϯϓϦϯά
10,000೥
ʢ࠷ॳͷ‫ੵݟ‬΋Γʣ
304ඵʂ ʢcf. ྔࢠίϯϐϡʔλ=200ඵʣ


# Page. 13

![Page Image](https://bcdn.docswell.com/page/2JVL34VG7Q.jpg)

ςϯιϧωοτϫʔΫͷྫ3ɿྔࢠଟମঢ়ଶ
ྔࢠଟମঢ়ଶɿ
‫ج‬ఈ
ྔࢠεϐϯɾbitɿ
ྔࢠԽֶɿ
i = ‫يࢠݪ‬ಓɾ෼ࢠ‫ي‬ಓͷ઎༗਺
܎਺͸ςϯιϧ
ྔࢠଟମঢ়ଶ
(a)
(b)
ςϯιϧωοτϫʔΫ෼ղ
(c)
PEPS (for 2d system)
&lt;latexit sha1_base64=&quot;XkC8uXv0uT1Qilx+79HHlBouX2k=&quot;&gt;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&lt;/latexit&gt;
ྔࢠ૬ؔͷ
খ͍͞ςϯιϧʹ෼ղ
ಛ௃Λར༻ͨۙ͠ࣅ
~eNͷಠཱཁૉ
~O(N)ͷಠཱཁૉ
ྔࢠଟମ‫ܥ‬ͷ௿ΤωϧΪʔঢ়ଶɿ
•
Ұൠͷঢ়ଶʢϥϯμϜϕΫτϧʣʹൺ΂ͯɺগͳ͍ྔࢠ૬ؔ
•
c.f. ΤϯλϯάϧϝϯτΤϯτϩϐʔͷ໘ੵଇ
ςϯιϧωοτϫʔΫʹΑΔߴਫ਼౓ͷۙࣅ
•
Openͳ଍͸&quot;͋Γ&quot;
‫ن‬ଇɾෆ‫ن‬ଇ
• ༗‫ݶ‬ɾແ‫ݶ‬
•


# Page. 14

![Page Image](https://bcdn.docswell.com/page/5EGX311DJL.jpg)

ςϯιϧωοτϫʔΫʹΑΔۙࣅγϛϡϨʔγϣϯ
ྔࢠճ࿏ͷۙࣅγϛϡϨʔγϣϯ
‫ݹ‬యίϯϐϡʔλͰͷ‫ࢉܭ‬ɿ
ྔࢠճ࿏ʹैͬͯҠΓมΘΔྔࢠঢ়ଶΛςϯιϧωοτϫʔ
ΫͰۙࣅతʹද‫͢ݱ‬Δ
•
ॳ‫ظ‬͸খ͍͞ςϯιϧͰද‫ݱ‬Մೳ
•
•
ඇৗʹଟ͘ͷqubitΛ‫ݹ‬యίϯϐϡʔλͰऔΓѻ͑Δ
ճ࿏͕ਂ͘ͳΔͱɺҰൠʹςϯιϧ͕େ͖͘ͳΔ
•
•
‫ࢉܭ‬ΛਐΊΔʹ͸ʢςϯιϧΛখ͘͞อͭʣۙࣅ͕ඞཁ
ਂ͘ͳΕ͹ͳΔ΄Ͳɺۙࣅਫ਼౓͕௿Լ


# Page. 15

![Page Image](https://bcdn.docswell.com/page/4JQ85DDX7P.jpg)

10
TABLE 2
a f2 (x) with error bound setting to 10 3 . For the 10th-order tensor, all 9
k r̄ as well as the number of total parameters Np are compared.
9
4
5
5
7
9
ςϯιϧωοτϫʔΫͷྫ4ɿςϯιϧ‫ܕ‬σʔλ
1
2
3
4
Np
5
6
7
1512 1944 2084 2144 2732 2328 3088
೚ҙͷςϯιϧ‫ܕ‬σʔλ
1512 1944 2064 2144 4804 4224 9424
1360
1828
1384
1360
2064
1324
1360
1788
1384
1360
1544
1348
1360
1556
1348
1360
1864
1384
1360
2832
1384
8
9
10376
7728
1360
1600
1272
3312
2080
1360
1324
1324
ɿྔࢠଟମঢ়ଶͱಉ༷ʹͯ͠෼ղՄೳ
ςϯιϧωοτϫʔΫ෼ղ
ςϯιϧ‫ܕ‬σʔλ
ce, sive results that are similar to that in noise free cases. TRpt ALSAR slightly overestimates the TR-ranks. It should be
As noted that TR-BALS can estimate the true rank correctly
m- and obtain the best compression ratio as TR-ALS given
৘ใͷ૬ؔߏ଄ͷ
In true rank. In addition, TR-BALS is more computationally
ral efficient than TR-ALSAR. In summary, TT-SVD and TRಛ௃Λར༻ͨۙ͠ࣅ
are SVD have limitations for representing the tensor data with
be symmetric ranks, and this problem becomes more severe
TR when noise is considered. The ALS algorithm can avoid this
ral problem due to the flexibility on distribution of ranks. More
(Q. Zhao, et al arXiv:1606.05535)
ata detailed results can be found in Table 3.
s aCOIL-100 dataset = 32 x 32 x 3 x 7200 ςϯιϧ
her 6.2 COIL-100 dataset
ors
We
are
10.
mal
der
ng
ks.
ed
go3,
en
•
Openͳ଍͸&quot;͋Γ&quot;
•
‫ن‬ଇɾෆ‫ن‬ଇ
•
༗‫ݶ‬
T
ྫ1ɿը૾σʔληοτ
ϐΫηϧ
৭ ը૾਺
M1
M2
r
ςϯιϧϦϯά෼ղ
M4
M3
ྫ̎ɿχϡʔϥϧωοτϫʔΫͷॏΈߦྻ
(Z.-F. Gao et al, Phys. Rev. Research 2, 023300 (2020).)
xi: input neuron (pixel)
yi: output neuron
matrix
x and y
Wij: weightZE-FENG
GAO etconnecting
al.
since t
grows
out th
of the
ment e
the sys
The ap
quantu
[40,41
In t
neural


# Page. 16

![Page Image](https://bcdn.docswell.com/page/K7429ZZ2J1.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ‫ૅج‬


# Page. 17

![Page Image](https://bcdn.docswell.com/page/LJ1MKRRKEG.jpg)

ςϯιϧωοτϫʔΫͷ਺஋‫ࢉܭ‬
ςϯιϧωοτϫʔΫΛ༻͍ͨԠ༻ͷ‫ج‬ຊ‫ࢉܭ‬ཁૉ
•
•
•
ςϯιϧͷॖ໿
•
‫ج‬ຊతʹɺ2ͭͣͭॖ໿‫ࢉܭ‬Λ͢Δ
•
ςϯιϧΛߦྻʹม‫͠ܗ‬ɺBLASͳͲΛ༻͍Δ
ςϯιϧͷ௿ϥϯΫۙࣅ
• ಛҟ஋෼ղʹΑΔ௿ϥϯΫۙࣅͷ֦ு
• ۙࣅతͳॖ໿Λߦ͏໨తͳͲʹ༻͍ΒΕΔ
• ଟ͘ͷ৔߹ɺςϯιϧΛߦྻʹม‫͠ܗ‬ɺߦྻͷಛҟ஋෼ղΛ༻͍Δ
ςϯιϧͷઢ‫ܗ‬໰୊
• ςϯιϧ͔Βߏ੒͞ΕΔߦྻͷʢҰൠԽʣ‫ݻ‬༗஋໰୊
•
ྔࢠଟମ໰୊ɺςϯιϧ෼ղͳͲͷ&quot;࠷దԽ&quot;Ͱ࢖༻
ςϯιϧͷ‫ج‬ຊԋࢉ͸ɺʢ‫ݱ‬ঢ়͸ʣߦྻʹม‫ߦͯ͠ܗ‬ΘΕΔ
cf. TBLIS ςϯιϧ޲͚ͷBLASʢBLIS= BLAS-like Library Instantiation Software)
https://github.com/devinamatthews/tblis


# Page. 18

![Page Image](https://bcdn.docswell.com/page/GJW3D11PJ2.jpg)

ςϯιϧͷߦྻ΁ͷม‫ܗ‬
ςϯιϧͷ଍Λ·ͱΊͯߦྻͱΈͳ͢
l
μΠΞάϥϜ
j
l
j
k
i
i
i
j
l
k
k
(0,0) → 0
(0,1) → 1
i, l = 0, 1ͷͱ͖ (1,0) → 2
(1,1) → 3
ςϯιϧ
‫ܗ‬ঢ়
•
ςϯιϧ༻ͷϥΠϒϥϦͰ؆୯ʹߦ͑Δɻʢྫɿnumpy.reshape)
•
ߦྻ΁ͷม‫ܗ‬͸ҰൠʹɺҰҙͰ͸ͳ͍
• Ͳͷ༷ʹߦྻԽ͢Δ͔͸ɺ໨తʹ߹ΘͤΔ


# Page. 19

![Page Image](https://bcdn.docswell.com/page/4EZVGPP6J3.jpg)

ςϯιϧωοτϫʔΫͷॖ໿
ςϯιϧωοτϫʔΫॖ໿ͷ‫ྔࢉܭ‬
ϧʔϓͷͳ͍πϦʔ‫ܕ‬ͷߏ଄Ҏ֎Ͱ͸ɺ
‫ྔࢉܭ‬͸ςϯιϧ਺ʹؔͯ͠ɺࢦ਺ؔ਺తʹ૿େ͢Δ
L×Lͷsquare lattice
௕͞Nͷchain
‫ॴہ‬ςϯιϧɿ
‫ॴہ‬ςϯιϧɿ
୺͔Βॱʹॖ໿ɿ
୺͔Βॱʹॖ໿ɿ
େ‫ن‬໛ͳςϯιϧωοτϫʔΫॖ໿͸ۙࣅతʹධՁ
2d ‫ن‬ଇTNʹର͢Δ൚༻తΞϓϩʔνɿ
•
•
•
ςϯιϧ‫܁‬ΓࠐΈ
ߦྻੵঢ়ଶ๏
֯సૹ‫܁‬ΓࠐΈ‫܈‬
ˎෆ‫ن‬ଇͰ΋ಉछͷۙࣅ͸Մೳ


# Page. 20

![Page Image](https://bcdn.docswell.com/page/Y765YMMLEV.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ‫ૅج‬ɿۙࣅॖ໿


# Page. 21

![Page Image](https://bcdn.docswell.com/page/G75YGZZM74.jpg)

ςϯιϧωοτϫʔΫ‫܁‬ΓࠐΈ‫܈‬
•
M. Levin and C. P. Nave, PRL (2007)ʹΑΔ Tensor network
Renormalization Group (TRG)͔Β࢝·ͬͨൺֱత৽͍͠ྲྀΕ
•
ςϯιϧωοτϫʔΫΛૈࢹԽ͍ͯ͘͜͠ͱͰɺۙࣅతʹॖ
໿Λ‫͢ࢉܭ‬Δ
•
•
ૈࢹԽ⟷࣮ۭؒ‫܁‬ΓࠐΈ‫܈‬
छʑͷʢ‫ن‬ଇ֨ࢠʣTNʹద༻Մೳ
•
෼഑ؔ਺ͷ‫ʹࢉܭ‬ద༻͢Δ͜ͱͰɺ෺ੑɺૉཻࢠɺ‫ࢠݪ‬
֩෼໺ͷ෺ཧ‫ʹڀݚ‬Ԡ༻͞Ε͍ͯΔ


# Page. 22

![Page Image](https://bcdn.docswell.com/page/9J2GYRRRJR.jpg)

TRGͰ΍Γ͍ͨ͜ͱ
‫܁‬ΓࠐΈ
ʢ௕͞εέʔϧ͕√̎ഒʣ
:
:
ʢۙࣅʣ
L×L ͷਖ਼ํ֨ࢠ
(L×L)/2 ͷਖ਼ํ֨ࢠ
ςϯιϧͷେ͖͞Λม͑ͣʹ
ςϯιϧͷ਺Λ‫ݮ‬Β͢


# Page. 23

![Page Image](https://bcdn.docswell.com/page/DEYVGDD5EM.jpg)

TRGͷ४උɿಛҟ஋෼ղ
ಛҟ஋෼ղ
೚ҙͷߦྻN×MߦྻA͸ҎԼͷ‫ʹܗ‬Ұҙʹ෼ղͰ͖Δ
A
=
͸ඇෛͷ࣮਺ɻ
rank(A) = ඇθϩͷಛҟ஋ͷ਺
ͱฒ΂Δͱศར
ର֯੒෼͕λͷ
ର֯ߦྻ
ҰൠԽϢχλϦߦྻ
Aͷ࠷దͳRϥϯΫۙࣅɿ ಛҟ஋Λେ͖͍ํ͔ΒR‫͠࢒͚ͩݸ‬ɺ
࢒ΓΛθϩͰஔ͖‫͑׵‬Δ


# Page. 24

![Page Image](https://bcdn.docswell.com/page/VJNMG66478.jpg)

TRGͷ४උɿಛҟ஋෼ղʹΑΔۙࣅ
Aͷ࠷దͳRϥϯΫۙࣅɿ ಛҟ஋Λେ͖͍ํ͔ΒR‫͠࢒͚ͩݸ‬ɺ
࢒ΓΛθϩͰஔ͖‫͑׵‬Δ
A
=
:M×N
(M ≦ N)
≃
:M×M
:(M, N)×M
ۙࣅ
:R×R
:(M, N)×R
͞Βʹ
=
=
ɿର֯੒෼͕
ͷର֯ߦྻ
X
SVDΛ࢖͏ͱ
Y
:M×R
:R×N
AΛখ͍͞ߦྻͷੵ
ʹ෼ղͰ͖Δ


# Page. 25

![Page Image](https://bcdn.docswell.com/page/YE9NWLL4E3.jpg)

ςϯιϧωοτϫʔΫ‫܁‬ΓࠐΈͷϨγϐ
̍ɽ෼ղ
ߦྻͱΈͳ͢
i
l
i
i
j
l
j
j
k
SVD
ʹΑΔۙࣅ
l
k
k
rank(A)=χ ʹۙࣅ
:
:
ʢۙࣅʣ


# Page. 26

![Page Image](https://bcdn.docswell.com/page/GE8KNXXV7D.jpg)

ςϯιϧωοτϫʔΫ‫܁‬ΓࠐΈͷϨγϐ
̎ɽૈࢹԽ
಺ଆͷ଍Λॖ໿
‫ݩ‬ͷςϯιϧ͕̎ͭ
৽͍͠ςϯιϧ̍ͭʹ
ૈࢹԽ͞Εͨ
:


# Page. 27

![Page Image](https://bcdn.docswell.com/page/LELZP88RJR.jpg)

ςϯιϧωοτϫʔΫ‫܁‬ΓࠐΈͷ‫ྔࢉܭ‬
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)
Z.-C. Gu, M. Levin and X.-G. Wen, Phys. Rev. B 78, 205116 (2008)
࣮͸O(χ5)·Ͱ‫ݮ‬ΒͤΔ
ʢ‫Ͱޙ‬ٞ࿦͠·͢ʣ
‫ྔࢉܭ‬ɿ
SVD= O(χ6)
ॖ໿= O(χ6)
ʢˎςϯιϧ͋ͨΓʣ
ˎ1 TRG εςοϓͰ ςϯιϧ਺͸1/2ʹͳΔ
ςϯιϧωοτϫʔΫॖ໿͕ςϯιϧ਺ʹରͯ͠ଟ߲ࣜͰ‫ࢉܭ‬Մೳ


# Page. 28

![Page Image](https://bcdn.docswell.com/page/4JMV466PJW.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ‫ૅج‬ɿ‫ݻ‬༗஋໰୊


# Page. 29

![Page Image](https://bcdn.docswell.com/page/PJR2WPPYE9.jpg)

‫ݻ‬༗஋໰୊ͷม෼๏
ྫɿ࠷௿ΤωϧΪʔঢ়ଶ
ίετؔ਺ɿ
Fͷ࠷খ஋
ͦͷ࣌ͷ
ม෼๏
•
•
‫ݻ‬༗஋໰୊ͷۙࣅղΛಘΔํ๏ͷҰͭ
Fͷ࠷খ஋Λ੍‫͞ݶ‬ΕۭͨؒͷൣғͰ୳͢
ͷ‫ܗ‬ΛԾఆ͢Δʹࢼߦؔ਺ɺม෼೾ಈؔ਺
ྫɿฏ‫ۉ‬৔ۙࣅɺςϯιϧωοτϫʔΫঢ়ଶɺχϡʔϥϧωοτϫʔΫ, ...
•
ྑ͍ࢼߦؔ਺ˠߴਫ਼౓ͷ࠷௿ΤωϧΪʔ
•
ෳࡶͳࢼߦؔ਺ˠίετؔ਺ͷ‫૿͕ྔࢉܭ‬େ


# Page. 30

![Page Image](https://bcdn.docswell.com/page/PEXDP3347X.jpg)

ςϯιϧωοτϫʔΫʹΑΔม෼๏
ม෼๏ͷࢼߦؔ਺ͱͯ͠ςϯιϧωοτϫʔΫΛ༻͍Δ
ςϯιϧੵঢ়ଶʢTPS,
PEPSʣ
ྫɿ ߦྻੵঢ়ଶʢMPSʣ
(b)
(c)
খ͍͞ςϯιϧʹ෼ղ
ίετؔ਺ɿ
FΛ࠷খʹ͢Δ
Λ୳͢ʂ
ϝϦοτɿ ‫ݩ‬ͷϕΫτϧۭؒͷ࣍‫͕ݩ‬aNͷ࣌ʹɺO(N)ͷίετͰ‫ࢉܭ‬Մೳ
ˎͨͩ͠ɺߦྻ͸ʢTNද‫ݱ‬Մೳͳʣૄߦྻ


# Page. 31

![Page Image](https://bcdn.docswell.com/page/3EKXDYYP7D.jpg)

Iterative optimization
(F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004))
i ൪໨ͷςϯιϧʹண໨ͨ͠‫࠷ॴہ‬దԽ
Minimize
খ͍͞αΠζͷҰൠԽ‫ݻ‬༗஋໰୊
(࠷খ‫ݻ‬༗஋ͷ‫ݻ‬༗ঢ়ଶΛ୳͢ʣ
i
஫: ߦྻαΠζ=
AiΛऔΓআ͘


# Page. 32

![Page Image](https://bcdn.docswell.com/page/L735R994E5.jpg)

Iterative optimization
(F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004))
AiΛi =1 ͔Β N·Ͱsweepͯ͠update
…
i =N ͔Β 1·Ͱ‫ʹ޲ํٯ‬sweep
…
ऩଋ͢Δ·Ͱ‫܁‬Γฦ͢


# Page. 33

![Page Image](https://bcdn.docswell.com/page/87DDYGG57G.jpg)

ߦྻͷίϯύΫτͳද‫ݱ‬
஫ʂ
͜ͷΞϧΰϦζϜ͸ɺߦྻࣗମ͕ςϯιϧωοτϫʔΫͰ
ޮ཰తʹද‫͖Ͱݱ‬Δ৔߹ʹͷΈ༗ޮʹ࢖͑Δ
͜ͷߦྻ͸ྫ͑͹ Matrix Product Operator (MPO) ͱ‫ݺ‬͹ΕΔ‫Ͱࣜܗ‬
ද͢͜ͱ͕Ͱ͖Δʢ৔߹͕͋Δʣɻ
=
ྫɿ 1࣍‫ࢠྔݩ‬S=1/2ϋΠθϯϕϧά໛‫ܕ‬ͷϋϛϧτχΞϯ
ͷMPOͰද‫͖Ͱݱ‬Δʢχ͸Nʹґଘ͠ͳ͍ʣ


# Page. 34

![Page Image](https://bcdn.docswell.com/page/VJPX633D78.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷେ‫ن‬໛Խ


# Page. 35

![Page Image](https://bcdn.docswell.com/page/2EVL344GJQ.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ࣮૷
ςϯιϧ‫܁‬ΓࠐΈɺม෼๏ͳͲͷςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ࣮૷
தɾখ‫ن‬໛ͷ‫ࢉܭ‬͸ɺ‫ط‬ଘͷϥΠϒϥϦͳͲͰ؆୯ʹ࣮૷Ͱ͖Δ
ྫɿ
•
•
python + numpy, scipy
Julia
MATLAB
‫ػ‬ցֶशϑϨʔϜϫʔΫ
•
FortranͰ΋ʢେม͚ͩͲʣͰ͖Δ
•
•
େ‫ن‬໛‫Ͱࢉܭ‬͸ɺ෼ࢄϝϞϦʹΑΔฒྻԽ͕ෆՄආ cf. TRGͷ‫ࢉܭ‬ίετ~O(χ6),
PEPSͩͱO(χ9)~O(χ12)
2ͭͷΞϓϩʔν
• ΞϧΰϦζϜʹಛԽͨ͠ฒྻ࣮૷Λߦ͏
• ൚༻తͳฒྻϥΠϒϥϦΛ༻͍Δ
ྫɿmptensor (https://github.com/smorita/mptensor)
෺ੑ‫ݚ‬ͷ৿ా͞Μ͕։ൃ
ˎલऀͷํ͕ੑೳ͕ߴ͘ͳΓ΍͍͕͢ɺଟ͘ͷTN‫ࢉܭ‬͸ʢͱͯ΋ʣෳࡶͰ͋Γɺ
൚༻తͳϥΠϒϥϦͷํ͕࣮૷ίετ͕େ෯ʹ௿͍


# Page. 36

![Page Image](https://bcdn.docswell.com/page/57GX31KDEL.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷಛ௃
•
ςϯιϧωοτϫʔΫͷ‫ج‬ຊతͳ‫ࢉܭ‬͸ߦྻͷઢ‫ܗ‬୅਺ʹͳΔ
•
ߦྻʹม‫ޙͨ͠ܗ‬͸ɺ‫ط‬ଘͷߦྻ༻ϥΠϒϥϦ͕࢖͑Δ
•
•
•
BLAS, LAPACK, ScaLAPACK, ....
ςϯιϧͱߦྻͷ૬‫ޓ‬ม‫׵‬ɺ&quot;transpose&quot;͕Կ౓΋‫ݱ‬ΕΔ
•
෼ࢄϝϞϦฒྻԽͰ͸ɺ௨৴͕ଟ਺ൃੜ͢ΔՄೳੑ
•
‫ڞ‬༗ϝϞϦͰ΋ɺϝϞϦॻ͖‫͑׵‬ίετ͕͋Δ
•
ʢม‫׵‬ͷͨΊͷindex‫ࢉܭ‬΋৔߹ʹΑͬͯ͸ॏ͍ʣ
ߦྻͷαΠζ͸ɺਖ਼ํߦྻΑΓ&quot;௕ํ‫ܗ‬ͷߦྻ&quot;͕ଟ਺Ͱͯ͘Δ
•
ϝϞϦͱ‫ྔࢉܭ‬ͷόϥϯε͕ѱ͘ͳΓ΍͍͢
•
‫͕཰ޮࢉܭ‬ग़ʹ͍͘৔߹΋ଟ͍


# Page. 37

![Page Image](https://bcdn.docswell.com/page/4EQ85DNXJP.jpg)

ޮ཰తͳTN‫ࢉܭ‬ͷͨΊͷ஫ҙ఺
ςϯιϧॖ໿ͷॱং
ॖ໿ͷධՁॱংͰ‫ࢉܭ‬ίετ͕มΘΔͨΊɺ‫ॱࢉܭ‬ংͷ࠷దԽ͕ॏཁ
•
؆୯ͳωοτϫʔΫͰ͋Ε͹ʢ‫׳‬Εͨʣਓ͕ؒ࠷దԽ͢Δ
•
NCONͳͲͷΞϧΰϦζϜɺϥΠϒϥϦΛ࢖ͬͯ࠷దԽ͢Δ
•
࠷ۙͷϥΠϒϥϦͩͱɺ࠷దԽͨ͠ॱংͰTNॖ໿Λ΍ͬͯ͘ΕΔ΋ͷ΋͋Δ
•
pythonͷopt_einsum
•
࠷దԽͷΞϧΰϦζϜʹ΋஫ҙ͕ඞཁʢਅ໘໨ʹ΍Δͱ஗͍ʣ
Transposeͷ஗Ԇ
ϓϩάϥϜͷ͋ΔߦͰςϯιϧͷindexΛฒͼସ͑ͯ΋ɺ
࣮ࡍͷ‫Ͱࢉܭ‬ඞཁʹͳΔ·Ͱ͸ɺϝϞϦ্ͷฒͼସ͑͸͠ͳ͍
ˎnumpyͳͲͷଟ͘ͷϥΠϒϥϦͰ‫ج‬ຊతʹ࣮૷͞Ε͍ͯΔ
ʢ஫ʣnumpyͷtransposeʢreshapeʣ͸εϨουฒྻʹରԠ͍ͯ͠ͳ͍ͨΊɺ
େ‫ن‬໛‫͕͜͜Ͱࢉܭ‬ϘτϧωοΫʹͳΔ͜ͱ͕͋Δ
→ࣗ෼ͰεϨουฒྻ൛ͷ࣮૷Λॻ͘ͳͲͯ͠ରԠ͢Δ


# Page. 38

![Page Image](https://bcdn.docswell.com/page/KJ429ZG2E1.jpg)

ޮ཰తͳTN‫ࢉܭ‬ͷͨΊͷ޻෉
ૄߦྻͷಛҟ஋෼ղ
TN‫Ͱࢉܭ‬͸ɺಛҟ஋෼ղΛʢେ෯ͳʣ௿ϥϯΫۙࣅʹ༻͍ΔͨΊɺ
શಛҟ஋Λ‫ٻ‬ΊΔඞཁ͕ͳ͍͜ͱ͕ଟ͍
Full SVD Ͱ͸ͳ͘ɺpartial SVD ʢtruncated SVD) Λ༻͍Δ
(ີߦྻιϧόʔ)
(ૄߦྻιϧόʔ)
ྫɿTRGͰͷಛҟ஋෼ղ
SVDʹΑΔۙࣅ
i
l
j
k
rank(A)=χ ʹۙࣅ
ີߦྻιϧόʔɿ
(શಛҟ஋Λ‫ٻ‬ΊΔʣ
ૄߦྻιϧόʔɿ
(্Ґχ‫ݸ‬ͷಛҟ஋Λ‫ٻ‬ΊΔʣ


# Page. 39

![Page Image](https://bcdn.docswell.com/page/LE1MKRDK7G.jpg)

ޮ཰తͳTN‫ࢉܭ‬ͷͨΊͷ޻෉
ૄߦྻͷಛҟ஋෼ղʢ͖ͭͮʣ
ૄߦྻιϧόʔͰ͸ɺߦྻϕΫτϧੵ͕‫͖Ͱࢉܭ‬Ε͹ྑ͍ͨΊɺ
ߦྻΛཅʹ࡞Δඞཁ΋ͳ͘ͳΔ
ྫɿTRGͰͷಛҟ஋෼ղ
SVDʹΑΔۙࣅ
i
l
j
k
rank(A)=χ ʹۙࣅ
ςϯιϧͷ
಺෦ߏ଄ɿ
͜ͷॖ໿͸
͜ͷॖ໿͸
χ‫ݸ‬ͷಛҟ஋


# Page. 40

![Page Image](https://bcdn.docswell.com/page/GEW3D1YPE2.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷେ‫ن‬໛ԽɿHOTRG
ʢΞϧΰϦζϜʹಛԽͨ͠ฒྻԽͷྫʣ


# Page. 41

![Page Image](https://bcdn.docswell.com/page/47ZVGPX6E3.jpg)

෼ࢄϝϞϦฒྻԽͷྫɿ3D-HOTRG๏ʹಛԽ
HOTRG: SVDΛςϯιϧʹ֦ுͨ͠HOSVDʹΑΔTRG
Z. Y. Xie et al, Phys. Rev. B 86, 045139 (2012)
⌘
&lt;latexit sha1_base64=&quot;6fFfiOA5K65pfNgux7h1VXJdczM=&quot;&gt;AAACaXichVFNLwNBGH66vqq+Wi4NF7GpODWzJSEO0sTFkVZLQtPsrsGwX/ajSTX+gJOb4EQiIn6Giz/g4CeII4mLg7fbTYQG72Rmnnnmfd55ZkZzDOH5jD3FpI7Oru6eeG+ir39gcCiZGi57duDqvKTbhu2ua6rHDWHxki98g687LldNzeBr2v5ic3+txl1P2NaqX3d4xVR3LLEtdNUnqrzJDwJRqyZllmVhjLcDJQIyoli2kzfYxBZs6AhggsOCT9iACo/aBhQwOMRV0CDOJSTCfY4jJEgbUBanDJXYfRp3aLURsRatmzW9UK3TKQZ1l5TjyLBHdste2QO7Y8/s49dajbBG00udZq2l5U516DhdfP9XZdLsY/dL9adnH9uYC70K8u6ETPMWektfOzx9Lc4XMo1JdsVeyP8le2L3dAOr9qZfr/DCBRL0AcrP524H5VxWmc7mVmbk/EL0FXGMYQJT9N6zyGMJyyjRuXs4wRnOYy9SSkpLo61UKRZpRvAtJPkT06uMOA==&lt;/latexit&gt;
೚ҙ࣍‫ݩ‬ͷ௒ཱํ֨ࢠʹద༻Մೳ
3࣍‫ݩ‬ͷ৔߹
ͷ‫ࢉܭ‬ίετ
‫ࢉܭ‬ͷϘτϧωοΫ
⌘
&lt;latexit sha1_base64=&quot;6fFfiOA5K65pfNgux7h1VXJdczM=&quot;&gt;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&lt;/latexit&gt;
ͷϝϞϦίετ


# Page. 42

![Page Image](https://bcdn.docswell.com/page/YJ65YM4L7V.jpg)

෼ࢄϝϞϦฒྻԽͷྫɿ3D-HOTRG๏ಛԽ
ฒྻԽͷࢦ਑ʢஜ೾େͷࢁԼɾᓎҪΒͷΞΠσΞΛվྑʣ
z
T. Yamashita and T. Sakurai, Comp. Phys. Comm. 278, 108423 (2022)
•
શ෦Ͱχ2ͷMPIϓϩηε
•
֤ϓϩηε͸(z,z&#039;)ͷ֤indexʹରԠ͢ΔμΠΞάϥϜΛ‫ࢉܭ‬
•
֤ϓϩηε͸(z,z&#039;)Λ‫ݻ‬ఆͨ͠ɺO(χ5)ͷςϯιϧΛϝϞϦʹอ࣋
•
‫ࢉܭ‬Λํ޲Λม͑ͯ‫܁‬Γฦͨ͢Ίʹɺχ4ͷσʔλΛχ2ͷ૬खʹbroadcst
•
z&#039;
Broadcast͢Δσʔλ͸χ4ʹ‫ݮ‬ΒͤΔ
ͷ‫ࢉܭ‬ίετ
/ϓϩηεͷ‫ࢉܭ‬ίετ
ͷϝϞϦίετ
/ϓϩηεͷϝϞϦίετ
MPIฒྻԽ


# Page. 43

![Page Image](https://bcdn.docswell.com/page/GJ5YGZQMJ4.jpg)

HOTRG 1stepͷฒྻԽੑೳ
ISSP sekirei
•
࣮ߦ࣌ؒ͸༧૝௨ΓͷO(χ9)εέʔϦϯά
•
εϨουฒྻͷੑೳ΋֓Ͷग़͍ͯΔ
‫ژ‬


# Page. 44

![Page Image](https://bcdn.docswell.com/page/9E2GYRPRER.jpg)

‫ࢉܭ‬ཁૉͷॏΈ
= 24
Sekirei
NBLF@6
#DBTU@6
NBLF@5
K
#DBTU@5
SFTIBQF@5
NBLF@6
&amp; 
&amp; 
&amp; 
&amp; 
&amp; 
&amp; 
&amp; 
&amp; 
&amp; 
DIJ PNQ
DIJ PNQ
DIJ PNQ
DIJ PNQ
&amp; 
#DBTU@6
NBLF@5
#DBTU@5
SFTIBQF@5
DIJ PNQ DIJ PNQ DIJ PNQ DIJ PNQ
•
make_U ʢHOSVDʹΑΔprojectorͷ‫ࢉܭ‬ʣ͸make_Tʢॖ໿ʣʹൺ΂ͯແࢹͰ͖Δ
•
Bcast_Tʢσʔλͷbroadcastʣ͸࣮ߦ࣌ؒͷ 5 ~10 %
֓Ͷຬ଍Ͱ͖Δੑೳ


# Page. 45

![Page Image](https://bcdn.docswell.com/page/D7YVGD557M.jpg)

ςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷେ‫ن‬໛ԽɿTeNeS
ʢ൚༻ϥΠϒϥϦΛ༻͍ͨฒྻԽͷྫɻ঺հͷΈʣ


# Page. 46

![Page Image](https://bcdn.docswell.com/page/VENMG6N4J8.jpg)

෼ࢄϝϞϦฒྻԽͷྫɿ2࣍‫ࢠྔݩ‬ଟମ໰୊
TPS (Tensor Product State)
(AKLT, T. Nishino, K. Okunishi, …)
PEPS (Projected Entangled-Pair State)
(F. Verstraete and J. Cirac, arXiv:cond-mat/0407066)
ྫɿ2࣍‫ݩ‬ਖ਼ํ֨ࢠͷTPS
4+1 ֊ͷςϯιϧ͕ෑ͖٧ΊΒΕͨωοτϫʔΫ
‫ࣗॴہ‬༝౓ɿs
Virtualࣗ༝౓ɿi, j, k, l
֤ΠϯσοΫεͷ࣍‫ʹݩ‬Ϙϯυ࣍‫ݩ‬ʢDʣ
ม෼೾ಈؔ਺ͱͯ͠ͷਫ਼౓ʹؔ܎͢Δύϥϝλ ʢD→∞Ͱ‫ʹີݫ‬ʣ
TPSΛม෼೾ಈؔ਺ͱ͢Δม෼๏
•
•
໘ੵଇΛຬͨͨ͢Ίɺ༗‫ݶ‬DͰ΋ਫ਼౓ͷྑ͍ۙࣅ
•
ແ‫ܥݶ‬΋௚઀ɺ༗‫ݶ‬ͷDͰ‫͖Ͱࢉܭ‬ΔɿiTPS
ςϯιϧωοτϫʔΫͷΈΛԾఆͨ͠ɺόΠΞεͷগͳ͍ม෼೾ಈؔ਺
•
Ϙϯυ࣍‫ݩ‬ͷ૿େʹΑΓɺ‫ܥ‬౷తʹਫ਼౓ΛվળͰ͖Δ


# Page. 47

![Page Image](https://bcdn.docswell.com/page/Y79NWLR473.jpg)

iTPS๏ͷద༻ྫ
ྫɿʢϞϯςΧϧϩ๏ͷͰ͖ͳ͍ʣϑϥετϨʔτ࣓ੑମ
H. YAMAGUCHI et al.
RE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09063-7
ϑϥετϨʔτਖ਼ํ֨ࢠ໛‫ܕ‬
5/9
0.4
3/9
Spin
1
0
0
1
site 1
0.1
3
B/J
g. 1 Calculated magnetization process for the spin-1/2 KAFM with the
earest-neighbor
interactionD.
J. Nakamura,
The tensor network
method
with the
R. Okuma,
T. Okubo
et al,
rojected entangledNat.
pair Commun.
state (PEPS)10,
is used.
vertical and horizontal
1229The
(2019).
xes represent magnetization M divided by saturated magnetization Ms and
0
site 2
1.0
0.5
1.5
(c)
2.0
site 1
0.3 site 2
0.2
2
site 1
H
0.4
0.3
2
Magnon
1/9
0
(b)
&lt;Sz &gt;
M /MS
7/9
(a)
Intensity
M/Msat
1
1.0
J6 J1
0.9
site 1
0.8
J2 J5
0.7
J4 J3 site 2
0.6
0.5
0.4
0.3
H=0
0.2
0.1
0
0
0.5
0.5
(&lt; Sx&gt;2+&lt;Sy&gt;2+&lt;Sz&gt;2 )1/2
Χΰϝ֨ࢠϋΠθϯϕϧά໛‫ܕ‬ͷ࣓Խ‫ۂ‬ઢ
0.2
site 2
0
0.5
1.0
1.5
0.1
H/|J1|
0
0.5
1.0
1.5
FIG. 4. The ground states under magnetic fields obtained from
H. Yamaguchi,
Y. Sasaki, T.
Okubo,
D = 6 iTPS calculation.
(a) Normalized
magnetization
curve, (b)
Phys.
Rev.
B
98,
094402
(2018).
average local magnetization, (c) average local moment at T = 0
calculated using the tensor network method assuming the ratios of
var
P(1
den


# Page. 48

![Page Image](https://bcdn.docswell.com/page/G78KNXWVJD.jpg)

Tensor Netwok Solver (TeNeS)
Y. Motoyama, T. Okubo, et al., Comput. Phys. Commun. 279, 108437 (2022).
https://github.com/issp-center-dev/TeNeS
ແ‫ܥݶ‬ͷTPSʢiTPSʣΛ༻͍ͨม෼๏ʹΑΔ‫ج‬ఈঢ়ଶ‫ࢉܭ‬
‫ൃؒ࣌ڏ‬ల๏ʹΑΔςϯιϧͷ࠷దԽ
MPI/OpenMPʹΑΔେ‫ن‬໛ฒྻ‫ʹࢉܭ‬ରԠ
mptensorʢ৿ాʣ ʹΑΔςϯιϧԋࢉͷฒྻԽ
Χΰϝ֨ࢠ໛‫ܕ‬ͷ࣓Խ‫ۂ‬ઢ
1
D = 10
0.8
@sekirei 72node
~4࣌ؒऑ
ೋ࣍‫ݩ‬ͷྔࢠεϐϯ‫ܥ‬΍Ϙκϯ‫͕ܥ‬؆୯ʹ‫ࢉܭ‬Մೳ
0.6
mVMC΍HPhiͱྨࣅͷinput le
ඪ४తͳೋ࣍‫ʹࢠ֨ݩ‬σϑΥϧτͰରԠ
‫ݪ‬ཧతʹ͸೚ҙͷೋ࣍‫ʹࢠ֨ݩ‬ରԠՄೳ
։ൃνʔϜ
•
•
•
•
•
େ‫ٱ‬อ‫ؽ‬ʢ౦େཧʣɿΞϧΰϦζϜ෦෼ͷ࣮૷
৿ా‫࢙ޛ‬ʢ෺ੑ‫ݚ‬ʣɿؔ࿈ϥΠϒϥϦɾπʔϧ࡞੒
ຊࢁ༟Ұʢ෺ੑ‫ݚ‬ʣɿϝΠϯϓϩάϥϜ౳ͷઃ‫ܭ‬ɾ࣮૷
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
٢‫ݟ‬Ұ‫ܚ‬ʢ෺ੑ‫ݚ‬ʣɿϢʔβʔςετɾαϯϓϧͷ࡞੒ɺϚωʔδϝϯτ
Ճ౻ַੜʢ෺ੑ‫ݚ‬ʣɿϢʔβʔςετɾαϯϓϧͷ࡞੒
ʲ෺ੑ‫౓ߴݚ‬ԽϓϩδΣΫτʳ
઒ౡ௚ًʢ෺ੑ‫ݚ‬ʣɿϓϩδΣΫτϦʔμʔ
fi
•
3-site unit cell


# Page. 49

![Page Image](https://bcdn.docswell.com/page/L7LZP8NRER.jpg)

·ͱΊ
•
ςϯιϧωοτϫʔΫʢTNʣ͸‫ࢉܭ‬Պֶͷ͍Ζ͍Ζͳ৔໘ʹ‫ݱ‬ΕΔ
•
•
໰୊ࣗମ͕TN‫Ͱࣜܗ‬ද‫͞ݱ‬ΕΔɻۙࣅͱͯ͠TN‫ݱ͕ࣜܗ‬ΕΔɻ
TNͷ‫ج‬ຊతͳ‫ࢉܭ‬͸ɺॖ໿ʢߦྻੵʣɺ௿ϥϯΫۙࣅʢSVDʣɺ͓Αͼɺ‫ݻ‬༗஋
໰୊ͰɺߦྻԋࢉϥΠϒϥϦ͕‫͖Ͱ༻׆‬Δ
•
•
খɾத‫ن‬໛ͷ‫ࢉܭ‬͸ɺ൚༻తͳϥΠϒϥϦͰ؆୯ʹ࣮૷Ͱ͖Δ
•
•
ॏཁͳԠ༻ྫɿςϯιϧ‫܁‬ΓࠐΈ‫܈‬ɺ‫ݻ‬༗஋໰୊ͷม෼๏
Transpose ΍ॖ໿ॱংͳͲɺςϯιϧಠࣗͷ໰୊΋͋Δ
େ‫ن‬໛‫ʹࢉܭ‬͸ɺ෼ࢄϝϞϦͰͷฒྻԽ͕ඞཁʹͳΔ
•
ΞϧΰϦζϜʹಛԽͨ͠ฒྻԽɿHOTRGͳͲ
•
൚༻ϥΠϒϥϦΛ༻͍ͨฒྻԽɿTeNeSͳͲ


# Page. 50

![Page Image](https://bcdn.docswell.com/page/4EMV46XPEW.jpg)

ࢀߟจ‫ݙ‬
•
೔ຊ‫ޠ‬ͷจ‫ݙ‬
•
ʮςϯιϧωοτϫʔΫ‫ࣜܗ‬ͷਐలͱԠ༻ʯ੢໺༑೥ɺେ‫ٱ‬อ‫ؽ‬ɺ೔ຊ෺ཧֶձࢽ Vol 72, No. 10, 2017
•
ʮςϯιϧωοτϫʔΫʹΑΔ৘ใѹॖͱϑϥετϨʔτ࣓ੑମ΁ͷԠ༻ʯେ‫ٱ‬อ‫ؽ‬ɺୈ63ճ෺ੑएख
ՆͷֶߍςΩετɺ෺ੑ‫ ڀݚ‬Vol. 7, No.2 (2018)
•
ʮςϯιϧωοτϫʔΫͷ‫ͱૅج‬Ԡ༻ɹ౷‫ܭ‬෺ཧɾྔࢠ৘ใɾ‫ػ‬ցֶशʯ੢໺༑೥ɺSGCϥΠϒϥϦɺ
αΠΤϯεࣾ
•
•
਺ཧՊֶ 2022೥2݄߸ɹಛूʮςϯιϧωοτϫʔΫͷਐలʯ
English
•
&quot;A practical introduction to tensor networks: Matrix product states and projected entangled pair states&quot;, R.
Orús, Annals. of Physics 349, 117 (2014).
•
&quot;Tensor networks for complex quantum systems&quot;, R. Orús, Nature Review Physics 1, 538 (2019).
•
&quot;Tensor Network Contractions&quot;, Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo,
Gang Su, Maciej Lewenstein, Lecture Note in Physics, vol. 964, Springer, (2020).
(Open access: https://link.springer.com/book/10.1007%2F978-3-030-34489-4)


