ςϯιϧωοτϫʔΫ๏ೖ໳౦େཧɹେ‫ٱ‬อ‫ؽ‬Introduction: େ‫ٱ‬อɹ‫ؽ‬ʢ͓͓͘΅ɹͭΑ͠ʣ‫ྺܦ‬•

1999-2008• ‫۝‬भେֶཧֶ෦෺ཧֶՊɺେֶӃཧֶ෎‫ڽ‬ॖ‫ܥ‬Պֶઐ߈•

2002-2008• ෺ੑཧ࿦‫ॴࣨڀݚ‬ଐʢࢦಋ‫ڭ‬һɿখా֞޹ઌੜʣ• ‫ڀݚ‬ςʔϚɿϥϯμϜύοΩϯάɺ‫ڥ‬քઁಈ໰୊ɺࣾձ෺ཧֶ

•2008-2012• େࡕେֶେֶӃཧֶ‫ڀݚܥ‬Պɹಛ೚‫ڀݚ‬һʢSupervisor: ઒ଜޫઌੜʣ
••‫ڀݚ‬ςʔϚɿϑϥετϨʔτ࣓ੑମʢ‫ݹ‬యεϐϯ໛‫ܕ‬ͷடংԽɾμΠφϛΫεʣ

2012-2017• ౦‫ژ‬େֶ෺ੑ‫ॴڀݚ‬ɹಛ೚‫ڀݚ‬һʢSupervisor: ઒ౡ௚ًઌੜʣ
•‫ڀݚ‬ςʔϚɿ୤ด͡ࠐΊྔࢠ૬సҠɺςϯιϧωοτϫʔΫ๏ͷྔࢠεϐϯ໛‫΁ܕ‬ͷద༻౳•

2017-2021.10• ౦‫ژ‬େֶେֶӃཧֶ‫ڀݚܥ‬Պɹಛ೚ߨࢣʢ౦‫ژ‬େֶ‫ࢉܭ‬ՊֶΞϥΠΞϯεʣ• ‫ڀݚ‬ςʔϚɿςϯιϧωοτϫʔΫΛத৺ͱͨ͠छʑͷτϐοΫɺϑϥετϨʔτ࣓ੑମ•

2019• JST ͖͕͚͞ʮྔࢠ৘ใॲཧʯྖҬɹ͖͕͚͞‫ऀڀݚ‬ʢ݉೚ʣ••‫ڀݚ‬ςʔϚɿςϯιϧωοτϫʔΫΛ‫͠༻׆‬ɺྔࢠίϯϐϡʔλͰྔࢠଟମ໰୊Λղ͖͍ͨ

2021.7• ౦‫ژ‬େֶେֶӃཧֶ‫ڀݚܥ‬Պɹྔࢠιϑτ΢ΣΞ‫ ࠲ߨ෇د‬ಛ೚।‫ڭ‬त‫ڀݚ‬ͷ‫ڵ‬ຯɿଟମ‫ܥ‬ͷ‫ݱྗڠ‬৅Ұൠɺ౷‫ܭ‬෺ཧֶͳͲɻ
Ωʔϫʔυɿ૬సҠɺ৽‫ح‬டংɺඇฏߧμΠφϛΫεɺςϯιϧωοτϫʔΫɺʜίϯςϯπ•͸͡ΊʹɿςϯιϧͱςϯιϧωοτϫʔΫ•֨ࢠ໛‫͑࢖Ͱܕ‬ΔςϯιϧωοτϫʔΫ๏••ςϯιϧ‫܁‬ΓࠐΈ‫܈‬•ߦྻੵঢ়ଶΛ༻͍ͨॖ໿

•ʢ֯సૹ‫܁‬ΓࠐΈ‫܈‬ʣ·ͱΊςϯιϧʁ•ϕΫτϧɿ

1࣍‫ݩ‬తͳ਺ࣈͷฒͼ

•

ߦྻ

ɿ

ɹ2࣍‫ݩ‬తͳ਺ࣈͷฒͼҰൠԽ•(n֊ͷʣςϯιϧ

ɿɹn࣍‫ݩ‬తͳ਺ࣈͷฒͼʲ‫ج‬ຊతͳԋࢉʹॖ໿ʳ
ߦྻੵɿ
ॖ໿ɿ

"଍"͕ଟ͘ͳΔͱද‫͕ه‬ෳࡶ...

μΠΞάϥϜΛ༻͍ͨςϯιϧද‫ه‬
•

ϕΫτϧ

ɿ

•

ߦྻ

ɿ

•

ςϯιϧ

ɿ

ςϯιϧͷੵʢॖ໿ʣͷද‫ݱ‬ˎn֊ͷςϯιϧʹnຊͷ଍

C

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B

A
C

D

ʹ B

A

ॖ໿ͷ‫ྔࢉܭ‬3ຊߦྻੵɿ A, B =
C

ʹ

B

A

ͷ‫=ྔࢉܭ‬ςϯιϧॖ໿ɿ

A=
B=

4ຊ

C

ʹ

A

ͷ‫=ྔࢉܭ‬μΠΞάϥϜͱͷରԠ••ॖ໿ͷ‫ྔࢉܭ‬͸μΠΞάϥϜͷ଍ͷ਺Ͱ෼͔Δ
ʢϝϞϦ࢖༻ྔ΋෼͔Δʣ

4ຊ

Bॖ໿ͷ‫ॱࢉܭͱྔࢉܭ‬ςϯιϧॖ໿ɿ

A=
B=
C=

C
D

ʹ B

A

Case 1:ͷ‫=ྔࢉܭ‬Case 2:ͷ‫=ྔࢉܭ‬ॖ໿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).

ςϯιϧωοτϫʔΫɿςϯιϧͷॖ໿Ͱߏ੒͞ΕͨωοτϫʔΫςϯιϧωοτϫʔΫʢTNʣ

ʲʢͬ͘͟Γͨ͠ʣ෼ྨʳ
••

Openͳ଍ɿ͋Γ or ͳ͠•Openͳ଍͕͋ΓɿTNࣗ਎͕େ͖ͳςϯιϧ•

Openͳ଍͕ͳ͠ɿTN͸਺ࣈωοτϫʔΫߏ଄ɿ‫ن‬ଇత or ෆ‫ن‬ଇ••ωοτϫʔΫߏ଄͸໰୊ʹԠͯ͡มΘΔ• ྫɿεϐϯ໛‫ܕ‬ͷ෼഑ؔ਺͸‫ن‬ଇత• ྫɿ෼ࢠͷଟମిࢠঢ়ଶ͸ෆ‫ن‬ଇωοτϫʔΫαΠζɿ༗‫ ݶ‬or ແ‫ݶ‬•‫ج‬ຊతʹ༗‫͕ͩݶ‬ɺ৔߹ʹΑͬͯ͸ແ‫ܥݶ‬΋औΓѻ͑Δʢࠓճ͸͜ͷྫΛѻ͍·͢ʣςϯιϧωοτϫʔΫͷྫ1ɿ౷‫ܭ‬෺ཧֶ‫ݹ‬యΠδϯά໛‫ܕ‬ʢ࣓ੑମͷϞσϧʣԹ౓T Ͱͷ֬཰෼෍ɿϘϧπϚϯ෼෍ঢ়ଶɿ‫ٯ‬Թ౓ɿ
೤ྗֶࣗ༝ΤωϧΪʔ෼഑ؔ਺ɿ
(2Nͷ࿨)

̍࣍‫ݩ‬Πδϯά໛‫ܕ‬ͷసૹߦྻ

෼഑ؔ਺
సૹߦྻ

+1

෼഑ؔ਺͸సૹߦྻͷੵͰ͔͚Δ

-1

+1
-1

෼഑ؔ਺ͷμΠΞάϥϜ
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సૹߦྻද‫ݱ‬ͷ֦ு̎࣍‫ݩ‬ਖ਼ํ֨ࢠ໛‫ܕ‬ͷ෼഑ؔ਺

ਖ਼ํ֨ࢠ

Aͷ഑ஔ

Ұͭͷਖ਼ํ‫ʹܗ‬஫໨֤ลͷϘϧπϚϯॏΈͷੵɿ̐֊ͷ“ςϯιϧ”Πδϯά໛‫ܕ‬A͸2×2×2×2ͷςϯιϧ

෼഑ؔ਺ʹςϯιϧͷ"ֻ͚ࢉ"෼഑ؔ਺ͷςϯιϧωοτϫʔΫද‫ݱ‬
A
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A

෼഑ؔ਺ʹςϯιϧAͷੵͷωοτϫʔΫςϯιϧωοτϫʔΫਖ਼ํ֨ࢠΠδϯά໛‫ܕ‬ˠ45౓܏͍ͨਖ਼ํ֨ࢠωοτϫʔΫˎ‫ݩ‬ͷ֨ࢠ఺ͱಉ͡৔ॴʹςϯιϧ͕഑ஔ͞ΕΔਖ਼ํ֨
ࢠωοτϫʔΫͰ෼഑ؔ਺Λද‫͢ݱ‬Δ͜ͱ΋Ͱ͖Δɻ

ςϯιϧωοτϫʔΫͷྫ2ɿྔࢠճ࿏
ྔࢠճ࿏ɿgoogleͷ"ྔࢠ௒ӽ" ճ࿏
F. Arute, et al., Nature 574, 505 (2019)ྔࢠϏοτʹԋࢉ͢Δήʔτૢ࡞ͷճ࿏ਤ
Articledeveloped 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.

2࣍‫ݩ‬ͷϕΫτϧɻద౰ͳ‫ج‬ఈͷ‫Ͱݩ‬͸ɺϢχλϦద౰ͳ‫ج‬ఈɺʢ | 0⟩, | 1⟩ʣͰߦྻʢor "ςϯιϧ"ʣa

A suitable computational taskද‫͢ݱ‬Δͱ2࣍‫ݩ‬ͷෳૉϕΫτϧTo 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 entangleb a set of quantum bits
Single-qubit
gate: logi(qubits) by repeated application of single-qubit andtwo-qubit
25 ns
cal operations. Sampling the quantum circuit’s output produces a setof bitstrings, for example {0000101, 1011100, …}.
Owing to quantum
Qubit
XYbitstrings
controlinterference, the probability distribution of theresembles
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 nsothers. Classically computing this probability distribution
becomes
Qubit 1
Z control
exponentially more difficult as the number of qubits(width) and numberof 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 givencircuit,into
we fourcollectthe measured
bitstringsi} and compute thecouplers
are divided
subsets(ABCD), eachof which is{xexecuted
11,13,14•
linearcross-entropy
benchmarking
fidelity(see
also
Supplementary
simultaneously across the entire array corresponding to shaded colours.
Here
Information),
which
is
the
mean
of
the
simulatedprobabilitiesof thewe show an intractable sequence (repeat ABCDCDAB); we also use different
bitstrings
wemeasured:
coupler subsets along with a simplifiable sequence (repeat EFGHEFGH, notgoogleͷ"ྔࢠ௒ӽ"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

ྔࢠճ࿏=ςϯιϧωοτϫʔΫsingle-qubit gates chosen randomly fromon 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 couplerbOpenͳ଍͸"͋Γ"• "ͳ͠"΋͋Δ

shown) that can be simulated on a classical computer. b, Waveform of control
FXEB = 2n"P(xi)#i − 1signals for single- and two-qubit gates.

ྔࢠίϯϐϡʔλͷ‫ݹ‬యγϛϡϨʔγϣϯ{ X, Y, W}ʹςϯιϧωοτϫʔΫͷॖ໿Qubit• ෆ‫ن‬ଇ(1)

where ntheseis thecircuits
numberqubits,P(xi)processor
is the probability
ofthanbitstringof runningonofthequantumis greater
at xi
computed
for
the
ideal
quantum
circuit,andtheaverageisoverleast 0.1%. We expect that the full• data for Fig. 4b should have similar theobserved
bitstrings.
Intuitively,
is correlated
withtoo
how
often
FXEB
fidelities, but since the simulation times(rednumbers) take
long
to we
samplehigh-probability
bitstrings.
When
there
are
no
errorsin thecheck, we have archived the data (see ‘Data availability’ section). The

༗‫ݶ‬

10 mm
Fig. 1 | The Sycamore processor. a, Layout of processor, showing a rectangulararray of 54 qubits (grey), each connected to its four nearest neighbours withcouplers (blue). The inoperable qubit is outlined. b, Photograph of the
Sycamore chip.

ʢ͜ͷྫ΋গؔ͠࿈͠·͢ʣςϯιϧωοτϫʔΫͷྫ3ɿྔࢠଟମঢ়ଶ
ྔࢠଟମঢ়ଶɿ

‫ج‬ఈྔࢠεϐϯɾbitɿྔࢠԽֶɿ
i = ‫يࢠݪ‬ಓɾ෼ࢠ‫ي‬ಓͷ઎༗਺܎਺͸ςϯιϧ
ྔࢠଟମঢ়ଶ

(a)(b)ςϯιϧωοτϫʔΫ෼ղ(c)

PEPS (for 2d system)<latexit sha1_base64="XkC8uXv0uT1Qilx+79HHlBouX2k=">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</latexit>ྔࢠ૬ؔͷখ͍͞ςϯιϧʹ෼ղಛ௃Λར༻ͨۙ͠ࣅ

~eNͷಠཱཁૉ

~O(N)ͷಠཱཁૉྔࢠଟମ‫ܥ‬ͷ௿ΤωϧΪʔঢ়ଶɿ•Ұൠͷঢ়ଶʢϥϯμϜϕΫτϧʣʹൺ΂ͯɺগͳ͍ྔࢠ૬ؔ
•c.f. ΤϯλϯάϧϝϯτΤϯτϩϐʔͷ໘ੵଇςϯιϧωοτϫʔΫʹΑΔߴਫ਼౓ͷۙࣅ• Openͳ଍͸"͋Γ"• ‫ن‬ଇɾෆ‫ن‬ଇ• ༗‫ݶ‬ɾແ‫ݶ‬

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

8

9

1512 1944 2084 2144 2732 2328 3088 10376 3312
೚ҙͷςϯιϧ‫ܕ‬σʔλ1512 1944 2064 2144 4804 4224 9424
7728
2080
1360
1828
1384

1360
2064
1324

1360
1788
1384

1360
1544
1348

1360
1556
1348

1360
1864
1384

1360
2832
1384

1360
1600
1272

ɿྔࢠଟମঢ়ଶͱಉ༷ʹͯ͠෼ղՄೳ
ςϯιϧωοτϫʔΫ෼ղςϯιϧ‫ܕ‬σʔλce, 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଄ͷIn true rank. In addition, TR-BALS is more computationallyral efficient than TR-ALSAR. In summary, TT-SVD and TRಛ௃Λར༻ͨۙ͠ࣅ
are SVD have limitations for representing the tensor data withbe symmetric ranks, and this problem becomes more severe
TR when noise is considered. The ALS algorithm can avoid thisral 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 ςϯιϧher 6.2 COIL-100 dataset

ors
We
are
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• Openͳ଍͸"͋Γ"

1360
1324
1324

• ‫ن‬ଇɾෆ‫ن‬ଇ• ༗‫ݶ‬

Tྫ1ɿը૾σʔληοτ
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
matrixx and yWij: weightZE-FENG
GAO etconnecting
al.

since tgrows
out thof thement ethe sys
The apquantu[40,41In tneuralςϯιϧωοτϫʔΫ‫ࢉܭ‬ͷ‫ૅج‬ςϯιϧωοτϫʔΫͷ਺஋‫ࢉܭ‬ςϯιϧωοτϫʔΫΛ༻͍ͨԠ༻ͷ‫ج‬ຊ‫ࢉܭ‬ཁૉ•••ςϯιϧͷॖ໿
•‫ج‬ຊతʹɺ2ͭͣͭॖ໿‫ࢉܭ‬Λ͢Δ•ςϯιϧΛߦྻʹม‫͠ܗ‬ɺBLASͳͲΛ༻͍Δςϯιϧͷ௿ϥϯΫۙࣅ• ಛҟ஋෼ղʹΑΔ௿ϥϯΫۙࣅͷ֦ு• ۙࣅతͳॖ໿Λߦ͏໨తͳͲʹ༻͍ΒΕΔ• ଟ͘ͷ৔߹ɺςϯιϧΛߦྻʹม‫͠ܗ‬ɺߦྻͷಛҟ஋෼ղΛ༻͍Δςϯιϧͷઢ‫ܗ‬໰୊• ςϯιϧ͔Βߏ੒͞ΕΔߦྻͷʢҰൠԽʣ‫ݻ‬༗஋໰୊•ྔࢠଟମ໰୊ɺςϯιϧ෼ղͳͲͷ"࠷దԽ"Ͱ࢖༻ςϯιϧͷ‫ج‬ຊԋࢉ͸ɺʢ‫ݱ‬ঢ়͸ʣߦྻʹม‫ߦͯ͠ܗ‬ΘΕΔςϯιϧͷߦྻ΁ͷม‫ܗ‬ςϯιϧͷ଍Λ·ͱΊͯߦྻͱΈͳ͢lμΠΞάϥϜ

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ςϯιϧωοτϫʔΫͷॖ໿ςϯιϧωοτϫʔΫॖ໿ͷ‫ྔࢉܭ‬ϧʔϓͷͳ͍πϦʔ‫ܕ‬ͷߏ଄Ҏ֎Ͱ͸ɺ‫ྔࢉܭ‬͸ςϯιϧ਺ʹؔͯ͠ɺࢦ਺ؔ਺తʹ૿େ͢Δ
௕͞Nͷchain

L×Lͷsquare lattice‫ॴہ‬ςϯιϧɿ‫ॴہ‬ςϯιϧɿ୺͔Βॱʹॖ໿ɿ

୺͔Βॱʹॖ໿ɿ

େ‫ن‬໛ͳςϯιϧωοτϫʔΫॖ໿͸ۙࣅతʹධՁ
2d ‫ن‬ଇTNʹର͢Δ൚༻తΞϓϩʔνɿ • ςϯιϧ‫܁‬ΓࠐΈ• ߦྻੵঢ়ଶ๏
• ֯సૹ‫܁‬ΓࠐΈ‫܈‬ˎෆ‫ن‬ଇͰ΋ಉछͷۙࣅ͸Մೳςϯιϧ‫܁‬ΓࠐΈ‫܈‬ςϯιϧ‫܁‬ΓࠐΈ‫܈‬•

M. Levin and C. P. Nave, PRL (2007)ʹΑΔ Tensor network
Renormalization Group (TRG)͔Β࢝·ͬͨൺֱత৽͍͠ྲྀΕ•෼഑ؔ਺ͷςϯιϧωοτϫʔΫද‫ݱ‬ΛૈࢹԽ͍ͯ͘͜͠ͱͰɺۙ
ࣅతʹ෼഑ؔ਺Λ‫͢ࢉܭ‬Δ••

ૈࢹԽ⟷࣮ۭؒ‫܁‬ΓࠐΈ‫܈‬छʑͷ֨ࢠ໛‫ʹܕ‬ద༻Մೳ•෺ੑ෼໺͚ͩͰͳ͘ɺૉཻࢠɾ‫֩ࢠݪ‬෼໺Ͱ΋ۙ೥‫͕ڀݚ‬ਐΜͰ͍Δ•‫ܦ‬࿏ੵ෼දࣔʹΑΓɺD࣍‫ݩ‬ͷྔࢠ‫ͱܥ‬D+1࣍‫ݩ‬ͷ‫ݹ‬య‫͕ܥ‬ରԠ

TRGͰ΍Γ͍ͨ͜ͱ෼഑ؔ਺‫܁‬ΓࠐΈ
ʢ௕͞εέʔϧ͕√̎ഒʣ: D×D×D×D
L×L ͷਖ਼ํ֨ࢠ

ʢۙࣅʣ: D×D×D×D(L×L)/2 ͷਖ਼ํ֨ࢠςϯιϧͷେ͖͞Λม͑ͣʹςϯιϧͷ਺Λ‫ݮ‬Β͢

TRGͷ४උɿߦྻͷ௿ϥϯΫۙࣅߦྻͷ֊਺ʢrankʣɿߦྻͷߦʢor ྻʣϕΫτϧͷ͏ͪઢ‫ܗ‬ಠཱͳ΋ͷͷ਺
A: N×Mߦྻ௿ϥϯΫۙࣅɿ
ߦྻAͷ௿ϥϯΫߦྻͰͷۙࣅ͍Βͳ͍৘ใΛͦ͗མͱͯ͠ɺॏཁͳ৘ใ͚ͩΛ࢒ۙ͢ࣅͷਫ਼౓Λຬͨ͢࠷దͳ௿ϥϯΫۙࣅ͸ಛҟ஋෼ղʢSVDʣ͔ΒಘΒΕΔ

TRGͷ४උɿಛҟ஋෼ղಛҟ஋෼ղ೚ҙͷߦྻN×MߦྻA͸ҎԼͷ‫ʹܗ‬Ұҙʹ෼ղͰ͖ΔA=͸ඇෛͷ࣮਺ɻ
rank(A) = ඇθϩͷಛҟ஋ͷ਺ͱฒ΂Δͱศར

ର֯੒෼͕λͷ
ର֯ߦྻҰൠԽϢχλϦߦྻAͷ࠷దͳRϥϯΫۙࣅɿ ಛҟ஋Λେ͖͍ํ͔ΒR‫͠࢒͚ͩݸ‬ɺ࢒ΓΛθϩͰஔ͖‫͑׵‬Δ

TRGͷ४උɿಛҟ஋෼ղʹΑΔۙࣅ
Aͷ࠷దͳRϥϯΫۙࣅɿ ಛҟ஋Λେ͖͍ํ͔ΒR‫͠࢒͚ͩݸ‬ɺ࢒ΓΛθϩͰஔ͖‫͑׵‬Δ
A

=

:M×N
(M ≦ N)≃:M×M:(M,N)×M

ۙࣅ:R×R:(M,N)×R͞Βʹ==ɿର֯੒෼͕ͷର֯ߦྻ

X

SVDΛ࢖͏ͱ

Y

:M×R
:R×N

AΛখ͍͞ߦྻͷੵʹ෼ղͰ͖Δςϯιϧ‫܁‬ΓࠐΈͷϨγϐ
̍ɽ෼ղߦྻͱΈͳ͢

i

l

i

i

j

l
j

j
k

SVD
ʹΑΔۙࣅ

l

k

k

rank(A)=Dʹۙࣅ: D×D×D×D: D2×D2

ʢۙࣅʣςϯιϧ‫܁‬ΓࠐΈͷϨγϐ
̎ɽૈࢹԽ಺ଆͷ଍Λॖ໿

‫ݩ‬ͷςϯιϧ͕̎ͭ৽͍͠ςϯιϧ̍ͭʹૈࢹԽ͞Εͨ

: D×D×D×D

ςϯιϧ‫܁‬ΓࠐΈͷϨγϐ
෼഑ؔ਺ʢप‫ڥظ‬ք৚݅ʣ̍ͭʹͳΔ·Ͱલड़ͷૢ࡞Λ‫܁‬Γฦ͢؆୯ʹ‫͖Ͱࢉܭ‬Δ෺ཧྔ
ࣗ༝ΤωϧΪʔɿΤωϧΪʔɿ

ʢඍ෼Λࠩ෼Ͱۙࣅʣ

ൺ೤ɿʢඍ෼Λࠩ෼Ͱۙࣅʣ

energy, the accuracy of the specific heat data is much less thanΠδϯά໛‫ܕ‬that of the internal energy, especially around the critical point.
This causes a big error in the determination of the exponent α
with the above formula. This problem can be solved by directly
evaluating this exponent from the temperature dependence of
the internal energy. From the temperature integration of thespecific heat, it is simple to show that the internal energy
shouldZ.exhibit
following
criticalbehavior:
Y. Xiethe
et al,
Phys. Rev.
B 86,
045139 (2012)ςϯιϧ‫܁‬ΓࠐΈͰͷ‫ྫࢉܭ‬ೋ࣍‫ݩ‬Πδϯά໛‫ܕ‬ͷࣗ༝ΤωϧΪʔ

XIE, CHEN, QIN, ZHU, YANG, AND XIANGU = Uc + at + bt 1−α ,̏࣍‫ݩ‬Πδϯά໛‫ܕ‬ͷΤωϧΪʔɾൺ೤
PHYSICAL REVIEW B 86, 045139 (2012)(a)*HOTRG
(n)

z

(b)

(n)

T y'1
x1
x'1
y'2
y1 i
x2
x'2
(n)
T
y2
z'

T

z

(n)

V

(n)

M

is already less than 10−7 even at the critical temperature,much more accurate than the TRG result.7,8 The HOSRG also

U x'

x U
y

(n)

(n)

V

y'
(n)

(n)

FIG. 4. (Color online) Comparison of the relative errors of freeenergy with respect to the exact results for the 2D Ising modelobtained by various√ methods with D = 24. The critical temperature
Tc = 2/ ln(1 + 2).

(17)

z'

M

FIG. 8. (Color online) The internal energy and the specific heat
for the 3D Ising model obtained by thez HOTRG with D = 14.
The Monte Carlo result (black curve) obtained
y' from an empirical(n+1)
x'
fit formula given in Ref.
T 27 is shownx for comparison.
(n+1)
T
y
z'
045

ςϯιϧ‫܁‬ΓࠐΈͱྟք‫ݱ‬৅ςϯιϧ‫܁‬ΓࠐΈͰ͸ɺྟք‫ݱ‬৅ʢྟքࢦ਺ͳͲʣ΋‫ࢉܭ‬Մೳςϯιϧͷ‫ݻ‬ఆ఺ɿʹ‫ݻ‬ఆ఺ςϯιϧͱྟքࢦ਺

‫܁‬ΓࠐΈม‫Ͱ׵‬ෆมͳςϯιϧɾసૹߦྻͷ‫ݻ‬༗஋͔Βྟքࢦ਺ΛಘΔ (Z. C. Gu and X. G. Wen, Phys. Rev. B 80, 155131 (2009).)ɾ‫܁‬ΓࠐΈૢ࡞͔Βྟքࢦ਺ΛಘΔ (cf. X. Lyu, R. G. Xu, and N. Kawashima, Phys. Rev. Research 3, 023048 (2021) )ʹͷ‫ݻ‬༗஋=εέʔϦϯάࢦ਺ߦྻੵঢ়ଶʹΑΔॖ໿

సૹߦྻʢ࠶‫ܝ‬ʣྫɿ̍࣍‫ݩ‬Πδϯά໛‫ܕ‬

෼഑ؔ਺
సૹߦྻ

+1

෼഑ؔ਺͸సૹߦྻͷੵͰ͔͚Δ

-1

+1
-1

సૹߦྻͷର֯Խసૹߦྻసૹߦྻ͸࣮ରশߦྻ
‫ݻ‬༗஋͸࣮Ͱɺ௚ߦߦྻͰର֯ԽՄೳ

෼഑ؔ਺෼഑ؔ਺ͷ‫ࢉܭ‬ˠసૹߦྻͷର֯Խ
ˎL ͕େ͖ͳ࣌͸ɺʢઈର஋ʣ࠷େͷ‫ݻ‬༗஋͕ࢧ഑త

̎࣍‫ܥݩ‬ͷసૹߦྻ
L×Mͷ̎࣍‫ܥݩ‬M‫ݸ‬ͷεϐϯΛ1ηοτͰߟ͑Δͱ̍࣍‫ͱܥݩ‬ಉ౳సૹߦྻͷେ͖͞

1࣍‫ܥݩ‬ɿ2×2
L×Mͷ̎࣍‫ܥݩ‬ɿ2M×2M (or 2L×2L)̎࣍‫ݩ‬Ҏ্Ͱ͸సૹߦྻ͕‫ܥ‬αΠζʹؔͯ͠ࢦ਺తʹେʂ‫ࢉܭͳີݫ‬͸͙͢ʹഁ୼͢Δ
̎࣍‫ݩ‬Πδϯά໛‫ͨͬͩܕ‬ΒɺM=50ఔ౓͕‫ݶ‬քʢૄߦྻͷର֯Խ໰୊ʣ

సૹߦྻͷԋࢉΛۙࣅతʹ‫ࢉܭ‬ʁTNͷॖ໿Ͱͷసૹߦྻͱ‫ݻ‬༗஋໰୊ೋ࣍‫ݩ‬ͷTNʢྫɿΠδϯά໛‫ܕ‬ͷ෼഑ؔ਺ʣ
1ߦΛऔΓग़͢

“సૹߦྻ”
ྻߦʢˎ؆୯ͷͨΊɺಉ͡సૹߦྻ͕‫܁‬Γฦ͞Ε͍ͯΔͱ͢Δʣઈର஋࠷େͷ‫ݻ‬༗஋λͱରԠ͢Δӈɾࠨ‫ݻ‬༗ϕΫτϧ‫ݻ‬༗ϕΫτϧͷ‫ࢉܭ‬ɿॳ‫ظ‬ϕΫτϧʹԿ౓΋సૹߦྻΛ͔͚Δɹ‫ݻ‬༗஋͸؆୯ͳԋࢉͰऔΓग़ͤΔ

ˎ‫֨ن‬Խɿ

ϕΫτϧͷσʔλѹॖɿςϯιϧωοτϫʔΫ෼ղసૹߦྻͷ‫ݻ‬༗ϕΫτϧͷ࣍‫ݩ‬ɿࢦ਺ؔ਺తʹେ͖͍ʂa: ଍ͷ࣍‫ݩ‬ʢΠδϯά໛‫Ͱܕ‬͸a=2ʣ

•ྔࢠଟମঢ়ଶͱಉ͡಺෦ߏ଄

•ಉ༷ʹςϯιϧωοτϫʔΫ෼ղ͕࢖͑Δʂ
(a)

(b)

ςϯιϧωοτϫʔΫ෼ղ(c)

PEPS (for 2d system)૬ؔͷখ͍͞ςϯιϧʹ෼ղಛ௃Λར༻ͨۙ͠ࣅ

~O(N)ͷಠཱཁૉςϯιϧωοτϫʔΫঢ়ଶΛ࢖ͬͨ‫ݻ‬༗ϕΫτϧ‫ࢉܭ‬Good reviews:

ߦྻੵঢ়ଶ (MPS)(U. Schollwöck, Annals. of Physics 326, 96 (2011))(R. Orús, Annals. of Physics 349, 117 (2014))

Nຊ଍ͷςϯιϧʢϕΫτϧʣΛߦྻͷੵͰද‫ݱ‬MPS

ɿindex i ͷߦྻ஫ɿ•MPS ͸ྔࢠଟମঢ়ଶͰ޿͘༻͍ΒΕ͖ͯͨ•

MPS ͸Ԡ༻਺ཧͰ͸"tensor train decomposition" ͱ΋‫ݺ‬͹Ε͍ͯΔ(I. V. Oseledets, SIAM J. Sci. Comput. 33, 2295 (2011))

ߦྻੵঢ়ଶ΁ͷ‫ͳີݫ‬ม‫׵‬೚ҙͷςϯιϧʢϕΫτϧʣ͸ಛҟ஋෼ղΛ‫܁‬Γฦ͢͜ͱͰ
‫ͳີݫ‬MPSද‫ʹݱ‬ৗʹม‫͖Ͱ׵‬Δ
SVD

SVD of<latexit sha1_base64="6m/gbzqlDKBTUkwq2Gwhy8OFn50=">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</latexit><latexit 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…

(column)(row)(column)͜ͷߏ੒Ͱ͸ߦྻͷ࣍‫ݩ‬͸৔ॴʹґଘ࠷େͷϘϯυ࣍‫ = ݩ‬aN/2
a:֤଍ͷ࣍‫ݩ‬ߦྻੵঢ়ଶʹ͓͚Δ௿ϥϯΫۙࣅ=࠷େϘϯυ࣍‫ = ݩ‬aN/2

࠷େϘϯυ࣍‫ = ݩ‬χ΋͠ɺ΋ͱͷςϯιϧ͕Ϙϯυ࣍‫ ݩ‬χ ͷߦྻੵঢ়ଶͰਫ਼౓ྑۙ͘ࣅͰ͖Ε͹ɺNͷࢦ਺ؔ਺ͷσʔλྔΛNͷଟ߲ࣜʹ·Ͱେ෯ʹ‫ݮ‬ΒͤΔʂ
•͜͜Ͱ͸ɺχ ͕ N ʹґଘ͠ͳ͍͜ͱΛ҉໧ͷ͏ͪʹԾఆͨ͠•1࣍‫ࢠྔݩ‬ଟମ‫ىྭͰܥ‬ΤωϧΪʔʹΪϟοϓ͕͋Δ‫ج‬ఈঢ়ଶͰ͸Ծఆ͸ਖ਼౰ԽͰ͖Δɻ•••͜Ε͸ɺ̎࣍‫ݩ‬Πδϯά໛‫ܕ‬ͷྟքԹ౓Ҏ֎ʹ૬౰ɻҰൠͷςϯιϧͰ͸ɺ͜ͷੑ࣭͕ඞͣ͠΋੒ཱ͸͠ͳ͍ɻʢcf. ໘ੵଇʣԾʹ χ ͕ N ͱ‫૿ʹڞ‬େ͢Δͱͯ͠΋ɺ࣮༻తʹ͸ɺMPS Λ࢖ͬͯςϯιϧΛۙࣅͰ͖Δɻ

ߦྻੵঢ়ଶ΁ͷ “ԋࢉࢠ” ͷద༻ͱۙࣅ
=U

ߦྻੵද‫ݱ‬ɿɿϘϯυ࣍‫ = ݩ‬χ

=
=
U
U

=

ɿϘϯυ࣍‫ = ݩ‬a 2χˎɹԋࢉࢠΛద༻͢ΔલͷMPSͱൺ΂ͯɺϘϯυ࣍‫ݩ‬͸Ұൠʹ͸૿େ‫܁‬Γฦ͠ԋࢉࢠΛద༻͢Δ৔߹ɺۙࣅʹΑΓϘϯυ࣍‫ݩ‬ΛԼ͛Δඞཁ‫ج‬ຊతͳۙࣅͷΞΠσΞ
ۙࣅͷྫɿ
=
U
‫ॴہ‬తͳςϯιϧͷΈΛมߋ•͜ͷۙࣅ͸ɺʢಉ͡Ϙϯυ࣍‫ݩ‬ͷൣғͰʣ࠷దͳۙࣅͰ͸ͳ͍͔΋͠Εͳ͍•Ұൠʹ͸ɺ‫ॴہ‬తͳԋࢉࢠͰ΋MPSશମͷςϯιϧʹӨ‫ڹ‬Λ༩͑Δ•

U ͕ϢχλϦԋࢉࢠͷ৔߹ʹ͸ɺ͜ͷۙࣅ͸΄΅࠷దपΓͷӨ‫ڹ‬Λແࢹͨ͠ૈ͍ۙࣅ
ߦ

ྻ

M

U

ߦྻͱΈͳ͢

ˎςϯιϧ‫܁‬ΓࠐΈͷखଓ͖ͱྨࣅˠSVDʂಛҟ஋෼ղʹΑΔۙࣅ

ߦ

=

U

M

=
SVD

ྻ

M

χ
2

2χ

χ

2χ

2

χ

େ͖ͳ χ‫ݸ‬ͷಛҟ஋Λ࢒͢

==χૈ͍ۙࣅͷҙຯ͜͜Ͱͷۙࣅ͸ɺ‫ॴہ‬తͳςϯιϧͷΈΛऔΓग़͍ͯ͠ΔҰൠతʹ͸ɺ͜ͷ‫ࣅۙॴہ‬͸ඞͣ͠΋શମΛ‫࣌ͨݟ‬ͷ࠷ળͳۙࣅͰ͸ͳ͍
Global approximation

Local approximation
U

U

MPSશମͷӨ‫ڹ‬Λߟ͑Δํ๏ͷྫɿTEBD๏
Time evolving block decimation (G. Vidal, Phys. Rev. Lett. 91, 147902 (2003))
Ωʔϫʔυɿ Schmidt coef cient, canonical form, entanglement, ...
ʮ‫ڥ؀‬ςϯιϧʯΛ‫ؚ‬ΊͨΫϥελʔʹର͢ΔSVD‫ͳີݫ‬MPSΛߏ੒ͨ͠ͱ͖ͷಛҟ஋ͱಉ͡fi

U

MPSͷ৔߹͸ɺglobalͳ࠷దԽͱ౳Ձʂ

ςϯιϧωοτϫʔΫද‫ݱ‬ͷར఺ɿiMPSର৅ͷςϯιϧ͕ฒਐର৅ੑΛ࣋ͭ৔߹ಉ͡ςϯιϧΛແ‫܁ʹݶ‬Γฦ͢͜ͱͰ
ແ‫ʹݶ‬௕͍ςϯιϧΛද‫ݱ‬Մೳ

iMPS
…

…

ˎಉ͡ԋࢉࢠΛશମʹ࡞༻ͤ͞Δ৔߹
…

…
U
•

U

U

ฒਐର৅ੑʹΑΓۙࣅͷࡍͷSVD͸શͯ౳Ձ

iTEBD • ‫ॴہ‬తͳ‫ࢉܭ‬1ճ͚ͩͰɺແ‫ܥݶ‬ͷ‫͖Ͱ͕ࢉܭ‬Δʂ
(G. Vidal, Phys. Rev. Lett. 98, 070201 (2007))(R. Orús and G. Vidal, Phys. Rev. B 78, 155117 (2008))

Ԡ༻ྫɿແ‫ʹݶ‬େ͖ͳ2࣍‫ݩ‬Πδϯά໛‫ܕ‬ͷ෼഑ؔ਺‫ࢉܭ‬ʢ͓·͚ʣ֯సૹ‫܁‬ΓࠐΈ‫܈‬֯సૹߦྻ‫܁‬ΓࠐΈ‫܈‬ʢCTMRGʣ
•Ԟ੢ɾ੢໺Β (1995)ʹΑΔஞ࣍తͳ“‫܁‬ΓࠐΈ”ʹΑΔςϯι
ϧωοτϫʔΫͷ‫ํࢉܭ‬๏••Corner Transfer Matrix Renormalization GroupʢCTMRGʣ

෼഑ؔ਺ͷςϯιϧωοτϫʔΫද‫ݱ‬ΛL→L+2ͷΑ͏ʹ਺αΠτͣͭେ͖͍ͯ͘͘͜͠ͱͰɺঃʑʹ‫͢ࢉܭ‬Δ•

ۙ೥ɺ̎࣍‫ࢠྔݩ‬ଟମ‫ܥ‬ͷ‫ج‬ఈঢ়ଶ‫ࢉܭ‬ΞϧΰϦζϜʢPEPS๏ɺTPS๏ʣͷҰ෦ʹ΋࢖ΘΕΔ

CTMRGͰ΍Γ͍ͨ͜ͱ
L×Lͷ෼഑ؔ਺͕

(L+2)×(L+2)ͷ෼഑ؔ਺Λ‫ࢉܭ‬ʢۙࣅతʹʣ‫͖ͨͰࢉܭ‬‫ܥ‬Λগ͠େ͖͘͢Δ

֯సૹߦྻද‫ݱ‬

C

C

C

C

: D×D

C

‫ܥ‬Λগ͠େ͖͘͢Δ

e e

e
e

e
e
C

C

e e

ʢۙࣅʣ

C

Cͷେ͖͞Λม͑ͣʹ‫ܥ‬Λେ͖͘͢Δ: D×D

֯సૹߦྻͷҙຯʢۙࣅͳ͠ʣ

4×4

6×6
C

=

=

=
e

֯సૹߦྻද‫ݱ‬

C

C

C

C

C

‫ܥ‬Λগ͠େ͖͘͢Δ

e e

C

C

e
e

e
e
e e

=

C

ʢۙࣅʣ

C
e

e

CTMRGͷϨγϐ
1. SVDʹΑΔ෼ղ
CߦྻͱࢥͬͯSVDe

M=
M͸࣮ରশʢΤϧϛʔτʣߦྻe
M : 2D×2D
C

e e

e
e

e
e
C

C

e e

=

=

C
ˎରশੑΛԾఆ

ಛҟ஋͕େ͖͍΋ͷD‫ݸ‬Λ࢒ͤ͹ྑ͍ۙࣅʂCTMRGͷϨγϐ: 2D×D

2. SVDΛ࢖ͬͯۙࣅ
C

e

=

e

C

e e

ର֯੒෼͕
ͷର֯ߦྻ

C

e
e

e
e
C

: D×D

e e

C

=

≃

ۙࣅ

CTMRGͷϨγϐ͘Γ͜Έม‫ͱ·׵‬Ί
1. LxL ͷ‫ܥ‬ͷ֯సૹߦྻΛSVD

C

e

e

3. (L+2)×(L+2)ͷ֯సૹߦྻΛ࡞੒

C

=
=

e

e

2. ProjectorΛ࡞Δಛҟ஋͕େ͖͍ํ͔ΒD‫͢࢒͚ͩݸ‬=େ͖͞Lͷ‫ܥ‬ͷ෼഑ؔ਺͕ஞ࣍‫·ٻ‬Δeࢀߟจ‫ݙ‬•ςϯιϧωοτϫʔΫ๏ղઆ‫ࣄه‬•਺ཧՊֶ 2022೥2݄߸ʮಛूɿςϯιϧωοτϫʔΫ๏ͷਐలʯɺαΠΤϯεࣾ•਺ཧՊֶ 2022೥11݄߸ͷҰ෦ʮྔࢠଟମ‫ͱܥ‬ςϯιϧωοτϫʔΫʯେ‫ٱ‬อ‫ؽ‬ɺαΠΤϯεࣾ•ʮςϯιϧωοτϫʔΫ‫ࣜܗ‬ͷਐలͱԠ༻ʯ੢໺༑೥ɼେ‫ٱ‬อ‫ؽ‬ɼ೔ຊ෺ཧֶձࢽ2017೥10݄߸
ʢhttps://www.jstage.jst.go.jp/article/butsuri/72/10/72_702/_article/-char/ja/ʣ•ʮςϯιϧωοτϫʔΫʹΑΔ৘ใѹॖͱϑϥετϨʔτ࣓ੑମ΁ͷԠ༻ʯେ‫ٱ‬อ‫ؽ‬ɺ෺ੑ‫ڀݚ‬Vol. 7, No. 2 ʢ෺ੑएखՆͷֶߍͷߨٛςΩετʣʢhttp://mercury.yukawa.kyoto-u.ac.jp/~bussei.kenkyu/archives/category/2018/vol07-2ʣ

•ςϯιϧωοτϫʔΫ๏ςΩετ•ʮςϯιϧωοτϫʔΫͷ‫ͱૅج‬Ԡ༻ɹ౷‫ܭ‬෺ཧɾྔࢠ৘ใɾ‫ػ‬ցֶशʯ੢໺༑೥ɺαΠΤϯε
ࣾ SGCϥΠϒϥϦ168 (2021).

•ςϯιϧωοτϫʔΫ๏ʹΑΔ਺஋‫ࢉܭ‬ͷʢ͓‫ק‬ΊʣReview•

R. Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled
pair states”, Annal. Phys. 349, 117 (2014).