テンソルネットワーク法による情報圧縮
東京大学大学院理学系研究科

大久保毅

大久保の自己紹介
•

•

•

•

•

•

1999 •
九州大学理学部物理学科
2002 - 2008•
物性理論研究室（指導教員：小田垣先生）•
ランダムパッキング、社会物理学、境界摂動問題
2008 - 2012•
大阪大学 宇宙地球科学専攻 特任研究員（Supervisor：川村先生）•
フラストレート磁性体（古典スピン模型の秩序、ダイナミクス等）
2012 - 2017•東京大学 物性研究所 特任研究員（Supervisor：川島先生）•
脱とじ込め転移、量子スピン模型、テンソルネットワーク等
2017 •
東京大学大学院 物理学専攻 特任講師
2019.10 •JST さきがけ研究者（量子情報処理領域）•
テンソルネットワークを活用して、量子コンピュータで量子多体問題を解く

研究の興味：多体系の協力現象一般。相転移、新奇秩序、非平衡ダイナミクス…
研究手法：主に、計算機シミュレーション

この講義の進め方と成績評価•

講義はスライドを用いて行います•••

講義スライドのpdfは大久保のwebページからダウンロードできます•google検索：「大久保毅 東京大学」で検索•

https://exa.phys.s.u-tokyo.ac.jp/ja/members/okubo/lecture

スライドは日本語の部分と英語の部分が混在しています

成績評価•

出席点とレポート（最後の回に課題を提示）で評価しますTentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

現代物理学における巨大なデータと情報圧縮

Huge data in physics（イントロなのでふんわり分かればOK!）Huge data in physics
Many-body problems in physics• Celestial movement （天体運動）• Gases, Liquids• Molecules, Polymers (eg. Proteins), ...
• Electrons in molecules and solids• Elemental particles (Quantum Chromo Dynamics)（量子色力学）In these problems, "systems" contain huge degrees of freedoms:
6N-dimensional phase space for classical mechanics
O(eN)-dimensional Hilbert space for quantum system

Complex particle system
Eg. Poliovirus capsid in electrolyte solution（ポリオウイルス

カプシド）（電解質溶液）Y. Ando et al, J. Chem. Phys. 141, 165101(2014).

doh et al.
165101-4

Long-range coulomb interaction
Poliovirus
capsidAndohet al.

6.5 million atoms

J. Chem. Phys. 141, 165101 (2014)（クーロン相互作用）Dynamics of water molecules

FIG. 1. A snapshot of the equilibrated poliovirus capsid in electrolyte solution. VP1, VP2, VP3, and VP4 are shown in blue, red, green, and gray,

J. Chem. Phys. 141, 165101 (2014)

FIG. 3. The accumulated number of water molecules that passed throughthe capsid from the inside to the outside (red circle) and from the outside to
the inside (green square) defined by the number of water molecules that werelocated in one side of the capsid at t = 100 ns and found in the opposite side att = t (upper panel), and its increment for 10 ns at t = t (lower panel). The errorof the rate represents the standard deviation (SD) of nine measurements of
the rate from the increment in the lower panel excluding the initial increment

Localized electrons as quantum spin systems
Eg. Antiferromagnetic Mott insulator Na2IrO3e pellet of
etized ande for 72 h,uenched in
imensionsof a semigrow crysd 1150 ° C
ow belowThus, T
ature. The
and single
g powder（反強磁性）（モット絶縁体）
Y. Singh and P. Gegenwart, Physical Review B 82, 064412 (2010)

Si : spin operator

PHYSICAL REVIEW B 82, 064412 !2010"

Why we need information compression?
1. We can not understand huge information directly.
We try to characterize "systems" thorougha few parameters.
Examples:Thermodynamics:
Systems are characterized by thermodynamic quantities,
Internal energy, Entropy, Pressure, Volume, Particle number,...

Critical phenomena:
Critical systems are characterized by a few critical exponents.
（臨界指数）

Why we need information compression?
2. We can not treat entire data in the present computers.
Available memories in the present computers:
Personal computers: ~10 GB

Double precision real number= 8 Bite~109

Super computers: ~100 GB / node
K@RIKEN,
Oakforest-PACS@UTokyoand Tsukuba Univ,
Sekirei@ISSP, UTokyo
...

~1 PB
(whole system)

~1010
~1014

Notice: In quantum system, the size of Hilbert space is O(eN)Why we need information compression?
2. We can not treat entire data in the present computers.
Try to reduce the "effective" dimension of(Hilbert) space.
By taking proper basis set,
we can represent a quantum state efficiently.
••••

Krylov subspace
Matrix product state
Tensor network states...

Examples of information compression 1
Krylov subspace
linear subspace generated by a square matrix (M) and a vector (v) as

For quantum many body problems::Hamiltonian:wavevector
Solve the eigenvalue problem withina restricted space (Krylov subspace)
Lanczos method, Arnoldi method* In these method, we do not necessarily need explicit matrix.
It is enough to know the result of matrix vector multiplication.

Examples of information compression 2
Compression of an image
image = matrix
Original# of "singular values"

CompressedExamples of information compression 2
Compression of a color image

About 10% compressed

image = tensor
Original

SVD

HOSVD

Examples of information compression 3
Wave function:（波動関数）orN-rank tensor(or Vector)# of Elements＝2N

T
m1

m2

m3

m4

m5

m3 m4

m5

Approximation asa product of "matrices"

Matrix Product States (行列積状態)（Tensor train decomposition）：Matrix for state mm1 m2

Singular value decomposition （特異値分解）
Singular value decomposition (SVD):
For a K × L matrix M,(half) unitary
Diagonal

Singular values:
Singular vectors:

By taking only several larger singular values,
we can approximate M as a lower rank matrix.

Examples of information compression 3
Wave function:（波動関数）orN-rank tensor(or Vector)# of Elements＝2N

T
m1

m2

m3

m4

m5

m3 m4

m5

Approximation asa product of "matrices"

Matrix Product States (行列積状態)（Tensor train decomposition）：Matrix for state m

m1 m2

Example of MPS: AKLT state
S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian:…
<latexit sha1_base64="yVh46hI6V0gjzAzAgoi6BBoWKpQ=">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</latexit>

The ground state of AKLT model:

S=1 spin:
S=1/2 spin

χ=2 iMPS: (U. Schollwock, Annals. of Physics 326, 96 (2011))

Spin singlet

1360
1828
1384

1360
2064
1324

1360
1788
1384

1360
1544
1348

1360
1556
1348

1360
1864
1384

1360
2832
1384

1360
1600
1272

1360
1324
1324

Application of MPS to data sciencesive results that are similar to that in noise free cases. TRALSAR slightly overestimates the TR-ranks. It should be
noted that TR-BALS can estimate the true rank correctly
and obtain the best compression ratio as TR-ALS given
true rank. In addition, TR-BALS is more computationallyefficient than TR-ALSAR. In summary, TT-SVD and TRSVD have limitations for representing the tensor data withsymmetric ranks, and this problem becomes more severe
when noise is considered. The ALS algorithm can avoid thisproblem due to the flexibility on distribution of ranks. Moredetailed results can be found in Table 3.

Tensor train (TT) and tensor ring (TR) decompositions for real data.

COIL-100 dataset = 32 x 32 x 3 x 7200 tensor

6.2 COIL-100 dataset

Q. ZHAO et al. TENSOR RING DECOMPOSITION

Compression by tensor
ring decomposition.
TABLE 4The comparisons of different algorithms on Coil-100 dataset. ✏, rmax ,r̄(Q. Zhao,
et al
arXiv:1606.05535)
denote relative
error, themaximumrank and the average rank,
respectively. For classification task, the ratio of training samples⇢ = 50%, 10% were considered.

error

CP-ALS

TT-SVD

TR-SVD

TR-ALS

Rank

✏

rmax

r̄

0.20
0.30
0.39
0.47
0.19
0.28
0.37
0.46
0.19
0.28
0.36
0.43
0.20
0.30
0.40

70
17
5
2
67
23
8
3
23
10
5
3
23
10
5

70
17
5
2
47.3
16.3
6.3
2.7
12.0
6.0
3.5
2.3
12.0
6.0
3.5

Acc. (%)(⇢ = 50%)
97.46
97.56
90.40
45.05
99.05
98.99
96.29
47.78
99.14
99.19
98.51
83.43
95.10
97.32
95.77

Acc. (%)(⇢ = 10%)
80.03
83.38
77.70
39.10
89.11
88.45
86.02
44.00
89.29
89.89
88.10
73.20
70.38
80.71
79.92

Examples of tensor decompositions
Good for 1d gapped systems

MPS:(1d correlation in data)

MERA:

RG

PEPS, TPS:

For higher dimensional systems
Extension of MPS(higher correlation in data)

Scale invariant systems（スケール不変）Real space renormalization（実空間繰り込み）
Corse graining of a tensor network representing a scaler.

"Contraction" toa new tensor

Approximation
by SVDwhere t = |1 − T /Tc |. However, as the specific heat data are
obtained Ising
simply modelfrom the innumerical
derivative
infinite
size of the internalenergy, the accuracy of the specific heat data is much less thanthat 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
Partition function = tensor network
should exhibit Z.
theY.following
behavior:
Xie et al,critical
Phys. Rev.
B 86, 045139 (2012)

Example of calculation

Error of free energy for 2D Ising model

XIE, CHEN, QIN, ZHU, YANG, AND XIANGU = Uc + at + bt 1−α ,Energy and specific heat of 3D Ising model
PHYSICAL REVIEW B 86, 045139 (2012)

(a)
(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

−7

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'

Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

Examples of many body problems:
格子スピン模型と統計力学

相転移•

温度や圧力等の“パラメタ”を変えると

圧力

臨界点

自由エネルギーに異常（特異点）が現れる場合がある。→相転移

液体
固体
気体

•

相転移で区別された状態＝相•水だと、常圧で温度を下げると気体→液体→固体の３つの相が出現

物性物理の研究対象の一つ•

どんな相があるか？
••

長距離秩序、トポロジカル秩序、…

それらを分ける相転移の性質は？

三重点

温度

磁性体（スピン模型）の相
典型的には２つの相が存在
磁気秩序相

強磁性

反強磁性

無秩序相
相転移

実際の物質や複雑なスピン模型では、多種多様な磁気秩序が生じる

１次転移と２次転移•相転移には大きく分けて不連続転移と連続転移が存在•

不連続転移：
相転移で自由エネルギーの１階微分が不連続に変化＝１次転移••

例：液体⟷固体の相転移….

連続転移：
自由エネルギーの１階微分は連続に変化•

多くの場合、２階微分が不連続に変化する＝２次転移•

例：気体⟷液体の臨界点、イジング模型の相転移…

臨界現象
２次転移では臨界現象が生じる
相転移点（臨界点）では、特徴的な長さスケールが発散
スケール不変性

種々の物理量が非自明なべき関数の振る舞い
相関長：
比熱：

べき指数＝臨界指数

感受率：
ユニバーサリティ
臨界指数は相転移で“破れる”対称性と臨界現象は対称性に注目した空間次元で決まり系の詳細には依存しない

シンプルな模型で調べられる

統計力学とカノニカル分布
カノニカル分布：状態（例えば、粒子の位置・運動量）：Γが実現する確率：逆温度

分配関数＝カノニカル分布の規格化因子：ハミルトニアン

熱力学自由エネルギーとの関係

分配関数の対数＝自由エネルギー

カノニカル分布での物理量の期待値
物理量Oの期待値：
物理量の期待値⟷マクロな系で観測される物理量すべての状態の和が計算出来れば、熱力学量が分かる
現実：

はとてつもなく大きいので、手では計算できない（計算機を使っても厳密に計算するのは難しい）

計算科学の手法＝分配関数・期待値を数値的に計算する

格子スピン模型：格子
格子スピン模型：
格子上にスピン自由度が定義され相互作用する模型
格子
１次元

鎖

正方格子

カゴメ格子

２次元

３次元

立方格子，FCC格子，…

格子スピン模型：スピン自由度
スピン自由度＊量子スピンは角運動量演算子
スピン量子数Sで特徴付け
Sが小さいほど量子効果が強く、新規量子相が実現する可能性＊古典スピンの極限に対応する
イジングスピン：上か下かしか向かない

ハイゼンベルグスピン：
3成分の単位ベクトル：

普通の磁気秩序は古典スピンで十分理解できる

格子スピン模型：相互作用
典型的なハミルトニアン：スピン自由度の２体相互作用（強磁性相互作用）：同じ向きを向くと得：反対向きを向くとエネルギーが得（反強磁性相互作用）
現実の物質ではは大で十分に小さくなる

最近接格子点間のみの相互作用を考えるのが良い近似：最近接格子点ペアの和スピン模型と相転移
例：正方格子強磁性イジング模型

自由エネルギー：
低温：エネルギーが小さい方がFが小さい
高温：エントロピーが大きい方がFが小さい
高エントロピー

低エネルギー（磁化の和がほぼゼロになる状態はたくさん）

相転移！

イジング模型の分配関数
統計力学の処方箋：
分配関数を計算したい

しかし、和は、スピンがN個の時、で指数的に大きい！（N=100でも、1030の項がある！）
定義通りに計算することは困難
計算科学によるアプローチ

（古典）格子スピン模型の数値解法•

大きく分けて２種類のアプローチ••

乱数を使って物理量の期待値を統計誤差つきで求める•

乱択アルゴリズム、モンテカルロ法•

統計誤差の範囲内で得られる期待値は厳密分配関数を近似的に計算する•得られた分配関数には近似に基づく系統的な誤差•

誤差は計算の規模を増大することで減らせる•

転送行列法、テンソルネットワーク法

Quantum spin systems（テンソルネットワーク法の主要な対象）

Quantum systems
Quantum system: governed by Schrödinger equation:Hamiltonian:Wave function (state vector)（波動関数 or 状態ベクトル）

Inner product:

Nature: Elementary particles, e.g. electrons, obey quantum mechanics.
素粒子

Static problems: Time-independent Schrödinger equation

Energy= Eigenvalue problem

Quantum systems
Example of quantum system: Array of quantum bits
1 bit

A quantum bit is represented by two basis vectors.
or

2 bits<latexit sha1_base64="b0TSKdinjwk9T/dIa6/LeFDtppE=">AAACsnichVG7SgNBFD1Z3/EVtRFsgsEXSJiIYLASbCzVmCgaCbPrGBcnu8vsJhKjPyBYW1gpWIilnWhl4w9Y+AliqWBj4d3Ngi/UO8zMmTP33DnD1R1puh5jDxGtobGpuaW1Ldre0dnVHevpzbl2WRkia9jSVis6d4U0LZH1TE+KFUcJXtKlWNa3Z/375YpQrmlbS17VEeslXrTMTdPgHlGF2MjoXr7scKXsnbziVlGK+Hh8L79h71ifybFCLMGSLIj4T5AKQQJhzNuxK+SxARsGyihBwIJHWILDpbGGFBgc4tZRI04RMoN7gX1ESVumLEEZnNhtWot0WgtZi85+TTdQG/SKpKlIGccQu2fn7JndsQv2yN5+rVULavheqrTrda1wCt0H/ZnXf1Ul2j1sfaj+9OxhE+nAq0nenYDxf2HU9ZXdo+fM9OJQbZidsifyf8Ie2C39wKq8GGcLYvH4Dz86eQmq+IjalPrelJ8gN5FMEV6YTMykw4a1YgCDGKWuTGEGc5hHll45xCWucaNNaqsa14x6qhYJNX34Epp8B/j3oU8=</latexit>

The Hilbert space is spanned by four basis vectors.
ヒルベルト空間
Simple notation::complex number
The Hamiltonian for 2 bits system can be represented in these bases.

Matrix element:

Quantum systems
Example of quantum system: Array of quantum bits
N bits: Dimension of the Hilbert space = 2N
Hamiltonian is 2N × 2N matrix

Need to solve eigenvalue problem of huge matrix!
In physics,• We often interested in the "ground state" (smallest eigenvalue)
基底状態

We can concentrate to a special state.
•

Typical system only has "short range" interactions

Hamiltonian matrix becomes sparse.

(Quantum) spin system
Spin systems:
Spin degree of freedoms defined on a lattice and interact each other

Lattice

Chain
1D
Square lattice

Kagame lattice

2D

3D

simple cubic，FCC lattice, BCC lattice, …

Quantum spin
Spin operator:
Commutation relation（交換関係）

Spin quantum number operator:（スピン量子数）

Simultaneous eigenstate of Sz and S2:(Hereafter, we set)

Quantized spin number

Quantum spin: S=1/2
Matrix representation of the spin operators:

We can consider S=1/2 spin as a quantum bit :Spins on a chain:

1

2

3

L-1 L

"Transverse field Ising model" （横磁場イジング模型）

L=2

Quantum spin: S=1/2

1

2

3

L-1 L

"Transverse field Ising model"

Larger TFIM
Non-zero elements in the Hamiltonian(Figure from Yamaji-sensei)

commutative
Total matrix elements=22L

antum ﬂuctuations
Zero point motion

seelements

O(L)

able
archical matrix?

# of non-zero elements

Sparse!

Classical problems
Two types of classical many-body problems
1. Approximation of quantum problems
Nature: Elementary particles obey quantum mechanics.

Classical mechanics is an approximation

2. Pure classical problems
Classical problems not necessary based on quantum mechanics• Percolation, covering, packing, …
• Stochastic process, “dynamical” system, ..
• Critical phenomena• …

Classical problems as an approximation: magnetism
Electron Spin: “Quantum” degree of freedom
For accurate treatment, the spin quantum number S is important
S=1/2, 1, 3/2, …
However, we can approximate the system by taking
the limit of S→∞ .
“classical” spin model•••

Classical Heisenberg model
Anisotropy: Ising model, XY model….

Classical spin degree of freedom
1.
limit of quantum spin
Classical spin:
2. simple degree of freedom reflecting symmetry
1. Heisenberg spin
Three component unit vector:

A lot of magnetism can be understand through classical Heisenberg spin
2. Ising spin・Strong easy axis anisotropy
・Representing underlying Z2 symmetry

3. XY spin

Two component unit vector:

・Strong easy plane anisotropy
・Representing underlying U(1) symmetry

Classical Ising spin vs. quantum spin
S=1/2 quantum spin

Ising spin
"Ising model""Transverse field Ising model"

In the case of classical system, the Hamiltonian is "diagonal".
••

We do not need explicit diagonalization.
"State" can be represented by a product of local DOF.
(Degrees Of Freedom)•

Although, # of states isTentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

線形代数の復習（必要なければ飛ばします）

Outline•••

Vector space- Abstract vectors•

General vector space (with inner product)•

Basis and relation to coordinate vector space•

Vector subspace and spanned vector subspace

Matrix and linear map•

Relation between matrices and linear maps•

Important properties and operations for matrices•

Relation to simultaneous linear equations

Eigenvalue problem and diagonalization

Vector space -Abstract vectors-

Geometric vector
Geometric vector: Arrow on the plane (or the space) ,which has "Direction" and "Length"

We can express a vector by its component:

P

OProperties of vector
Properties of addition:

Commutative property（交換法則）
Associative property（結合法則）
zero vector
inverse vector

Multiplication of scaler<latexit sha1_base64="yHNU29cvgkODsaZzHW9ah/DR9Z4=">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</latexit>（実数）:
Distributive property（分配法則）

Inner product of vector
Inner product:

Example:Properties:<latexit sha1_base64="yHNU29cvgkODsaZzHW9ah/DR9Z4=">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</latexit>

Norm (length):

Vector space (linear space)
Vector space<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>: generalization of geometric vector

Set of elements (vectors) satisfying following axioms（公理）Properties of addition:

Commutative property（交換法則）
Associative property（結合法則）
Existence of unique zero vector
Existence of unique inverse vector
Multiplication of scaler:<latexit sha1_base64="yHNU29cvgkODsaZzHW9ah/DR9Z4=">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</latexit><latexit sha1_base64="DUfBQ52zcsm8Y4SKMxOfsVJMzpY=">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</latexit>: Real vector space: Complex vector space

Inner product space (metric vector space)
Inner product space:（計量空間）

Vector space + definition of inner product

Inner product:
Axiom:*If a norm defined from the inner product is "complete"（完備）,that space is called Hilbert space.

Examples of vector spaces(1) Coordinate space（数ベクトル空間）
Vector:<latexit sha1_base64="2uBzX+kUjh41NNMG3hWM6qkWNiM=">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</latexit>

Inner product:(2) Wave vectors in quantum physics
Vector:Inner product:<latexit sha1_base64="aRaFEjTTBPsLdrGEER8lQY2BAak=">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</latexit>Linearly independent or dependent（線形独立）（線形従属）

Linear combination:<latexit sha1_base64="A6OV28GvUn4Wqat5zeMuPGPZdPI=">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</latexit><latexit sha1_base64="ytLai4T+KKlyHaJIJn6AzL+OqHU=">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</latexit>

A set

is linearly independent whenis satisfied if and only if

A set

is linearly dependent when

it is not linearly independent.

Basis of vector space
（基底）
A set
when

is a basis（基底）of<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit><latexit sha1_base64="K8bmd3naQuKI33R2S+FyhhfCLDU=">AAACpXichVG7SiRBFD22uutr11ET0WRwcNdNhtuLoBiJJiaCrxkFR5ru3nIs7BfdNQPaDBj7AwZGCgpioLGpiT9g4CcsGyqYGHi7p0FUdG9RVadO3XPrFNcKHBkporsWrbWt/cvXjs6u7p5v33tzff3lyK+FtijZvuOHa5YZCUd6oqSkcsRaEArTtRyxam3PJverdRFG0vdW1E4gNlyz6slNaZuKKSM3NFapCzsWDUPmK9LLV1xTbVlWXG78MnIFKlIa+fdAz0ABWSz4uStU8Ac+bNTgQsCDYuzARMRjHToIAXMbiJkLGcn0XqCBLtbWOEtwhsnsNq9VPq1nrMfnpGaUqm1+xeEZsjKPUbqlM7qnGzqnv/T0Ya04rZF42eHdampFYPTuDy4//lfl8q6w9aL61LPCJiZTr5K9BymT/MJu6uu7B/fLU0uj8Q86pn/s/4ju6Jp/4NUf7JNFsXT4iR+LvaRVEsRt0t825T0o/y7qVNQXxwvTM1nDOjCMEYxxVyYwjTksoMSv7OEUF7jUfmrz2opWbqZqLZlmAK9CM54Bu2mbaA==</latexit>

is linearly independent.
andAny vectors in<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>

are represented by its linear combination.

: basis vector# of basis vectors (n) is called dimension （次元）of<latexit sha1_base64="EWJVbb/BF0zxhkH5YYfnLhWcicc=">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</latexit><latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>.

Relation (map) to coordinate vector space,

By using a basis

is uniquely represented as<latexit sha1_base64="9W3YAzhE8MVwAZe4hY+IiU+41Nw=">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</latexit>(* From linear independency)

We can represent

as a coordinate vector<latexit sha1_base64="D135yu9VEsQWRVUkdShX5gLdgz4=">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</latexit>

By selecting a basis, we obtain a "concrete" coordinate vector
for an "abstract" vector

Orthonormal basis （正規直交基底）
When a vector space has an inner product,is orthogonal （直交）if.

Orthonormal basis
A basisis an orthonormal basis when

*A basis can be transformed into an orthonormal basis.
cf. Gram-Schmidt orthonormalization

Example: wave vector
2 qbits:

We can choose following four vectors as the (orthonormal) basis.

Simple notation::complex number<latexit sha1_base64="VhBou3LUrisOmUa1VZpbxz2W9jo=">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</latexit>

Many qbits:<latexit sha1_base64="3+vO9KwHHJFoOc9rv8rkjB5vCNw=">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</latexit>

basis:<latexit sha1_base64="xezPqh1OiyL5Cra4QEZ18CP9ieo=">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</latexit><latexit sha1_base64="sH+yUz5zCglxrI1LAPIX89I7T4g=">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</latexit><latexit sha1_base64="e7rTq7dG+v7M/OFwKI8/sxAjrpo=">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</latexit><latexit sha1_base64="AkmuMLOQKO9bsro6Y1Qq0Kbtr0o=">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</latexit>

Vector subspace (linear subspace)
Vector subspace（ベクトル部分空間）:
A subset
when<latexit sha1_base64="t2cn6g+Juk8RDUuVygS3pMXrPtY=">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</latexit>of a vector space<latexit sha1_base64="t2cn6g+Juk8RDUuVygS3pMXrPtY=">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</latexit><latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>

is a vector subspace ofsatisfies the same axioms of vector space with<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>.

The following conditions are necessary and sufficient.
<latexit sha1_base64="A2ko25QtC+MQ5vhTq0DorANy3W0=">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</latexit><latexit sha1_base64="DEC3SjDDW7oMJGte1TMl8dd39aI=">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</latexit><latexit sha1_base64="9MAla4eXZS7KPRLHLs/Q1dWsaQs=">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</latexit><latexit sha1_base64="Xl47a3ygrRABm1uIkCDxi4N790U=">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</latexit>(In the case of complex vector space)<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>

Spanned vector subspace
Spanned subspace:
For a subset<latexit sha1_base64="VQJcofYrlZQmSyk1h/mzgs5+bb8=">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</latexit>of a vector space<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>, a set of linear combinations<latexit sha1_base64="Ol/aAsKqQ3b+H9HoIkE2yNgYOu0=">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</latexit>

becomes a vector subspace of<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit>.

We often use

to represents a vector subspace spanned by a set of vectors

Matrix and linear map

Matrix（行列）Matrix: "Table" of (complex) numbers in a rectangular form
M × N matrix

Product of matrices:<latexit sha1_base64="gtiQE1989ZCrX0bSyAx4+aXiRUc=">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</latexit>

In general:*We also know addition, multiplication of scalar.

Identity matrix（単位行列）
Identity matrix:
N × N matrix(Square matrix)
Product:* Element of the identity matrix:(Kronecker delta)Transpose, complex conjugate and adjoint
Transpose:（転置）Complex conjugate:（複素共役）

Adjoint:（随伴）
orHermitian conjugate:（エルミート共役）("Dagger" is convention in physics)

Multiplication to coordinate vector<latexit sha1_base64="ySrnqsxLNKhSyK5f3jGTCHhoYuQ=">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</latexit><latexit sha1_base64="1Lvgp2u7m2lKraoOV0QIu+gm+Kk=">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</latexit>

M × N matrix transforms a N-dimensional coordinate vector
to a M-dimensional coordinate vector.
M × N matrix

Linear map:
1 to 1 （線形写像）<latexit sha1_base64="uKwQ2fMCrJaBL3wa3nRvtFa897Q=">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</latexit>

General linear map
Map:<latexit sha1_base64="eTVqPfp++mXtKjsSldyAruwt9p8=">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</latexit><latexit sha1_base64="703f5XcrtLE5xgFRihsCw8LoA40=">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</latexit>

Linear map:<latexit sha1_base64="bNf2LJZpkJuWfKZcSRV+cFVavGM=">AAACvnichVHLShxBFD220UxGoxPdBLJpMigGZLhtAglZDc4mS18zCo4M3ZUaLewX3TWDk6Z/ID8QQlYJZBHyB9m4cGM+wIWfELI04MaFt3saREVzm6576tQ9t05xndBVsSY6HTFGH4yNPyw9Kk9MPp6arjyZacVBLxKyKQI3iLYcO5au8mVTK+3KrTCStue4ctPZb2Tnm30ZxSrwN/QglDueveurrhK2ZqpTebnQ7kuRHKSLeR6kZlv5Ztuz9Z7jJK100RTXmEb6otypVKlGeZi3gVWAKopYCSq/0MZ7BBDowYOED83YhY2Yv21YIITM7SBhLmKk8nOJFGXW9rhKcoXN7D6vu7zbLlif91nPOFcLvsXlP2KliTk6oR90Rsf0k/7QxZ29krxH5mXA2RlqZdiZ/vh0/fy/Ko+zxt6V6l7PGl28yb0q9h7mTPYKMdT3P3w6W3+7NpfM0zf6y/6/0ikd8Qv8/j/xfVWufbnHj8Ne8i4Z4jFZN4dyG7SWahbVrNVX1fpyMbASnuE5Fngqr1HHO6ygybd8xiGO8duoG13DM4JhqTFSaGZxLYyDS4RGpRU=</latexit>

Examples:Rotation (e.g. θ rotation around z-axis)<latexit sha1_base64="9Y6ya019z2wSMEzYQQliuzRpdEI=">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</latexit>

Hamiltonian operator<latexit sha1_base64="GcXULGD1+aC+KfAWumVndEDHUGg=">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</latexit>

Matrix representation of linear map
By using a basis, we can represent a linear map in a matrix.
<latexit sha1_base64="eTVqPfp++mXtKjsSldyAruwt9p8=">AAACqHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvIplkpxOQmlmQkJVWH1SrElOQjcdXjBZQN9AzAQAGTYQhlKDNAQUC+wC6GGIYUhnyGZIZShlyGVIY8hhIgO4chkaEYCKMZDBkMGAqAYrEM1UCxIiArEyyfylDLwAXUWwpUlQpUkQgUzQaS6UBeNFQ0D8gHmVkM1p0MtCUHiIuAOhUYVA2uGqw0+GxwwmC1wUuDPzjNqgabAXJLJZBOguhNLYjn75II/k5QVy6QLmHIQOjC6+YShjQGC7BbM4FuLwCLgHyRDNFfVjX9c7BVkGq1msEig9dA9y80uGlwGOiDvLIvyUsDU4Nm43FPEtAtYFNALGA0GaJHCiYjzEjP0EDPMNBE2cEJGmEcDNIMSgwawFgxZ3Bg8GAIYAgF2tLMsJJhE8NmJi2mAKZwpkiIUiZGqB5hBhTAlAQAYlacgQ==</latexit>

Vector space<latexit sha1_base64="nH9dqbr2SMg7S+MxFrNoQASMGtQ=">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</latexit><latexit sha1_base64="dP747jHS7MgB3CvxbHQEhh9CsAg=">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</latexit>

Basis

Transformation of basis vectors:<latexit sha1_base64="eTVqPfp++mXtKjsSldyAruwt9p8=">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</latexit>

1 to 1（if we fix basis)

Examples of matrix
Rotation (e.g. θ rotation around z-axis)<latexit sha1_base64="9Y6ya019z2wSMEzYQQliuzRpdEI=">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</latexit>

Hamiltonian operator<latexit sha1_base64="GcXULGD1+aC+KfAWumVndEDHUGg=">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</latexit>

Matrix element:（行列要素）* In this notation, basis should be orthonormal.

Image of a map<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit><latexit sha1_base64="Xu834rVCmnDGdEEJmV0djTOUqCo=">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</latexit><latexit sha1_base64="eTVqPfp++mXtKjsSldyAruwt9p8=">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</latexit>

Image of f :（像）
Vector subspace whose elements are
mapped from by .
<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit><latexit sha1_base64="RTRxGZ9VG3gQWmp1pdBxxkznC38=">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</latexit>

ImageKernel of a map<latexit sha1_base64="ENpFY4z+3c58x+mVTPzrrhEezw8=">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</latexit><latexit sha1_base64="Xu834rVCmnDGdEEJmV0djTOUqCo=">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</latexit><latexit sha1_base64="eTVqPfp++mXtKjsSldyAruwt9p8=">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</latexit>

Kernel of f :（核）
KernelVector subspace whose elements are
mapped into zero vector by .
<latexit sha1_base64="rIaSUFoF7I3i+2D4X0RqPzhWWKc=">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</latexit>

Theorem:Rank of matrix
Rank （ランク or 階数）of a matrix A:

Rank is identical with
Maximum # of linearly independent column vectors （列ベクトル）in A
Maximum # of linearly independent row vectors （行ベクトル）in A

for a N×M matrix A.

Regular matrix and its inverse matrix
A square matrix A is a regular matrix （正則）if a matrix X satisfying
exists. The matrix X is called inverse matrix（逆行列） of A and
it is written as.

Properties:

is unique.

A is a regular matrix
Can we consider an "inverse matrix" of a non-regular matrix (including a
rectangular matrix) ?

Eigenvalue problems and diagonalization

Eigenvalue and Eigenvector
For a square matrix A:eigenvector（固有ベクトル）<latexit sha1_base64="5XMHbWb5ZfcGwCz/mOCpMFZiseM=">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</latexit>:eigenvalue（固有値）

Properties:
If

is an eigenvector,

is also an eigenvector.

Eigenspace（固有空間）:
The set of eigenvectors corresponds an eigenvalue λ.

Eigenvectors corresponding to different eigenvalues arelinearly independent.

Right and left eigenvectors
In general, left eigenvectors can be different from the right eigenvectors.

:Right eigenvector:Left eigenvector
Properties:
Set of eigenvalues are identical between the right and the left eigenvectors.
A left eigenvector and a right eigenvector are orthogonal when theycorrespond to different eigenvalues.

Diagonalization
Diagonalizaiton（対角化）：

A has N linearly independent
eigenvectors.
necessary

A can be diagonalized.

and
suﬃcient

Normalization:<latexit sha1_base64="DXBoFBdp0/FFqfQ+D7lbzglf9qQ=">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</latexit>Meaning of diagonalization
General transform using a regular matrix:
It is a transform of the basis:
Diagonalization:
By using eigenvectors as a basis,
we can obtain a simple linear map
represented by a diagonal matrix.
* The determinant of A is invariant under this transformation:(This relation is true even if A cannot be diagonalized)

Unitary matrix
Unitary matrix（ユニタリ行列）：
Real Orthogonal matrix（実直交行列）：<latexit sha1_base64="MMZu34D2z0e8232G3KW8GN0F8hI=">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</latexit>

When we consider a unitary matrix as a set of vectors:

it is a orthonormal basis:
The linear map represented by a unitary matrix(unitary transformation) does not change• the norm of a vector•"distance" between two vectors

Normal matrix
Normal matrix（正規行列）：
We can always diagonalize it by a unitary matrix
as<latexit sha1_base64="3wRXTwh31ELD4HsWs65QjiS9Wqc=">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</latexit>

Its eigenvalues could be complex.
(even if A is a real matrix)

Hermitian matrix and its eigenvalue
Hermitian matrix（エルミート行列）：
Real symmetric matrix（実対称行列）：<latexit sha1_base64="HaDGtz6BpcCR57zqz/NHO23wg1Q=">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</latexit>

It is a special normal matrix.
Its eigenvalues are real.
We can always diagonalize it by a unitary matrix<latexit sha1_base64="F/l2wc4m2QILAddmQcJ9vbmUmQ8=">AAACo3ichVHLLgRBFD3a+z3YSGwmJiNWk9siIVbCRmLDMEjMZNLdCpXpV7prJmHCB/gBCyvCQoQPsLXxAxY+QSxJbCzc7ulEENxKVZ06dc+tU7mmb8tQET02ac0trW3tHZ1d3T29ff2pgcG10KsGlihYnu0FG6YRClu6oqCkssWGHwjDMW2xblbmo/v1mghC6bmras8XJcfYceW2tAzFVDk1XLQ5ecsoy3RRuumiY6hd06znD8qpDOUojvRPoCcggySWvNQtitiCBwtVOBBwoRjbMBDy2IQOgs9cCXXmAkYyvhc4QBdrq5wlOMNgtsLrDp82E9blc1QzjNUWv2LzDFiZRpYe6JJe6J6u6Inef61Vj2tEXvZ4Nxta4Zf7j4ZX3v5VObwr7H6q/vSssI3p2Ktk737MRL+wGvra/vHLykw+Wx+jM3pm/6f0SHf8A7f2al0si/zJH35M9hJXiRC3Sf/elJ9gbSKnU05fnszMziUN68AIRjHOXZnCLBawhAK/cohzXONGy2qLWl5bbaRqTYlmCF9CK30AKxGaxQ==</latexit>

Hermitian (or real symmetric) matrices often appear in physics.

Generalization of diagonalization••

Eigenvalue problems and diagonalizations aredefined for a square matrix.
Even if A is a square matrix, it may not be diagonalized.
••

Is it possible to transform all square matrixes into
diagonal forms by generalizing the diagonalization?
Is it possible to generalize it to a rectangular matrices?

Yes. The singular value decomposition(特異値分解) is an generalization of the diagonalization.
(We can also consider a decomposition of a tensor.)Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

特異値分解と低ランク近似

Outline•

Singular value decomposition (SVD)•

Low rank approximation••

Low rank approximation by SVD•

Low "rank" approximation for tensor

Application of the low rank approximations to images

Singular value decomposition

Diagonalization
Diagonalizaiton（対角化）：(Square matrix)••

Eigenvalue problems and diagonalizations aredefined for a square matrix.
Even if A is a square matrix, it may not be diagonalized.
•

Normal or Hermitian matrices are always diagonalized
by a unitary matrix

Spectral decomposition(For a normal matrix A,)
Spectral decomposition（スペクトル分解）

Note:(
Matrix decomposition into a sum of
projectors onto its eigen subspaces.

)
Projector:

Singular value decomposition (SVD)
Singular value decomposition（特異値分解）Unitary

Unitary

Diagonal matrix with
non-negative real elements
Singular values

Properties of SVD 1
1. Any matrices can be decomposed as SVD:*

is a Hermitian matrix.

It can be diagonalized bya unitary matrix .

eigenvector*

is a positive semi-definite matrix.
（半正定値、準正定値）
Its eigenvalues are
non-negative

Properties of SVD 1
1. Any matrices can be decomposed as SVD:

Suppose(There are r positive eigenvalues.)
Make new orthonormal basis

in<latexit sha1_base64="OOe3ax2yoWPG5moeiGwkvxBoAgY=">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</latexit>

For
For

Any orthonormal basis orthogonal to(For simplicity, we set

for

.)

Properties of SVD 1
1. Any matrices can be decomposed as SVD:
We can perform same "proof" by using.
is the unitary matrixaswhich diagonalize

In summary,• A matrix A can be decomposed as SVD:•

Singular values are related to the eigenvalues of

and

.
•

and

are eigenvectors ofand,respectively.

asProperties of SVD 2
2. # of positive singular values is identical with the rank.
<latexit sha1_base64="hOjFD4jefF3+7fx3h3i24n/X/+g=">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</latexit><latexit sha1_base64="ACSRWZSnEG1/OrFEzKR2WKj/VNo=">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</latexit><latexit sha1_base64="MZuAfsA7opqxv/oBf+Sd2I4vpHQ=">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</latexit><latexit sha1_base64="V+bD9Kv4YW8KAHflxPUPzVbe4cI=">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</latexit>

Remember
The orthonormal basis

satisfies

Here,

and

<latexit sha1_base64="d0UabPWEqAW1/uMIjpe+XbXCmjQ=">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</latexit><latexit sha1_base64="eoQ/+fy4otrdwITFbGQ+He8V0CY=">AAACsnichVFNS9xQFD3Gr+nUj1E3hW5CB62LIi8y4CAMCG66Ej86jujYkLy+GR/miyQT1DB/oNB1F64UXEiX7kRXbvoHuvAniMspdNOFN5mAqKg35L3zzr3nvvO4pmfJIGTsukfp7esfGMy9yb8dGh4ZLYyNrwduy+eiyl3L9TdMIxCWdEQ1lKElNjxfGLZpiZq5u5jka5HwA+k6X8J9T2zbRtORDcmNkCi98PGTWo8Ej/faakWtBy1bj2VFa39dUhd12U1FbV3m9UKRzbA01KdAy0ARWSy7hXPU8Q0uOFqwIeAgJGzBQEDfFjQweMRtIybOJyTTvEAbedK2qEpQhUHsLq1NOm1lrEPnpGeQqjndYtHvk1LFJPvDTlmH/Wa/2A37/2yvOO2ReNmn3exqhaePfn+39u9VlU17iJ171YueQzRQTr1K8u6lTPIK3tVHBz87a/Ork/EUO2a35P+IXbMreoET/eUnK2L18AU/JnlJuySIxqQ9HspTsD47oxFeKRUXytnAcniPD5imqcxhAZ+xjCrd8gNnuMClUlI2FUPh3VKlJ9NM4EEo1h2P0J/I</latexit><latexit sha1_base64="2RQDmhA+0p7E2w9j5S84fVS5NWY=">AAAC5nichVFNS91AFD2m1vrts24EN8GHohuZiFARFIsbV+LXU+E9G5I46mC+SOYFbcgfcNFdceFGBRfiP3AnbvoHuvAXiLhpUXDjwpu8QLEPdUJmzpx7z71nuKZvi1Aydt2gfGj82PSpuaW1rb2js6vQ/Xkl9KqBxUuWZ3vBmmmE3BYuL0khbb7mB9xwTJuvmjszaXw14kEoPHdZ7vl83TG2XLEpLEMSpRfKXysRt+LdRJ1UK2HV0WMxqSXf5tQZXahDtWCU6GL4ZTyuSL4rs/ZxwDeSOEiSOoleKLIRli21Hmg5KCJf817hAhVswIOFKhxwuJCEbRgI6StDA4NP3Dpi4gJCIotzJGglbZWyOGUYxO7QvkW3cs66dE9rhpnaoi42/QEpVQyw3+yM3bNf7JzdsqdXa8VZjdTLHp1mTct9vWu/d+nxXZVDp8T2P9WbniU2MZ55FeTdz5j0FVZNH30/uF+aWByIB9kJuyP/x+yaXdEL3OjBOl3gi4dv+DHJS1YlRTQm7f+h1IOV0RGN8MJYcXoqH1gz+tCPIZrKF0xjFvMoUZdL3OAP/irbyg/lp3JQS1Uack0PXizl6BlvsrWP</latexit>＝# of positive singular values

ImageProperties of SVD 3
3. Singular vectors,
For,right singular vector
left singular vector
Relation to image and kernel:

cf. Hermitian matrix

Properties of SVD 4 (optional)
4. Min-max theorem (Courant-Fischer theorem), Hermitian matrix
Suppose its eigenvalues are.
We can prove thisby considering vector
subspace spanned byeigenvectors.
(see references)

Orthonormal complement（直交補空間）
Intuitive examples:

Maximum appears for the eigenvector.

Properties of SVD 4 (optional)
4. Min-max theorem (Courant-Fischer theorem)
Suppose its singular values are

By setting k=1,

which meansfor.

We can easily prove thisby using
HermitianProperties of SVD 5 (optional)
5. Singular values for multiplication and addition
singular value of matrix A)(for i > rank(A), we set
Multiplication:

Addition:

If,*Following properties can be provenby using min-max theorem.

Libraries for SVD*Linear Algebra PACKage

There are LAPACK routines for SVD.
DGESDD, ZGESDDDGESVD, ZGESVD(For dense matrices)

At netlib.org (reference implementations)+
A lot of vender implementations• Intel MKL• Apple Accelerate Framework• Fujitsu SSLII• ...

numpy and scipy modules in python have routines for SVD.
numpy.linalg.svd
scipy.linalg.svd
scipy.sparse.linalg.svds(For dense matrices)(For sparse matrices or
calculation of partial singular values)

Computation cost
For a M×N matrix (M≦N):

Full SVD: O(NM2)
Partial SVD: O(NMk)k : # of singular values
to be calculated

Low rank approximation

Amount of data in SVD representation,neglect zero
singular values

If rank(A) is much smaller than M and N,
we can reduce the data to represent A.
(At this stage, no data loss)

Low rank approximation
Low rank approximation（低ランク近似）
Find an approximate matrix
with lower rank:
Through the low rank approximation,
we can reduce amount of the data.
An example of information compressions.
Notice! In order to quantify accuracy of the approximation,
we need a measure of distance between matrices.

Low rank approximation by SVD
Consider a matrix obtained by neglecting smaller singular values

Keep the largest k singular values(and corresponding singular vectors).

This approximation is one of the low rank approximation.
*

For this approximation, we need O(MNk) calculationsfor SVD of a M×N matrix.

Norm of matrices
There are two popular norms:(1) Frobenius norm （フロベニウス

ノルム）*Trace（対角和）(2) Operator norm（作用素ノルム）*We define the norm
for a vector as*inf =infimum（下限）
By using these norms, we define the distance between matrices:Accuracy of low rank approximation by SVD
TheoremFor

Because the k-rank approximation by SVD gives

it is an "optimal" approximation with rank k.

Short proof of the theorem: Frobenius norm (optional)*This proof is based on"システム制御のための数学（１）"太田快人 著

From the inequality of singular values for matrix addition (property 5),
for j=1,... ,min(M,N)

Property 5
By taking a square and summing up them*Note

Short proof of the theorem: operator norm (optional)*This proof is based on"システム制御のための数学（１）"太田快人 著

From the min-max theorem of singular values (property 4),

Property4 with

Expand thevector space

Definition of the operator norm

Relation to principal component analysis （主成分分析）
Data set {Xij} : X: N × M matrix
i = index for data, j = data type (coordinates, momentum, ...)* Suppose the mean of data is 0:<latexit sha1_base64="Yuh0BxnHtww1JENQQnWOHAr5c+Y=">AAACn3ichVHLSiNBFD2243N8xMdCcNNjUFzJzTCgCErAha4k6sREojTdbaml/aK7E0ia/IA/4MKNDswi+gduZzM/MAs/YZilA25ceLvTICrqLarq1Kl7bp3iGp4lg5DotkPp/NTV3dPb1/95YHBoODMyuh24Vd8URdO1XL9s6IGwpCOKoQwtUfZ8oduGJUrGyUp8X6oJP5Cu8z2se2LP1g8deSBNPWRKy4zvBlVbk2pU1iJ53Gyq6pJKWiZLc5SE+hrkUpBFGgU3c4Nd7MOFiSpsCDgIGVvQEfCoIAeCx9weIuZ8RjK5F2iin7VVzhKcoTN7wushnyop6/A5rhkkapNfsXj6rFQxTX+oRXf0m67pLz28WStKasRe6rwbba3wtOHTia37D1U27yGOnlTveg5xgIXEq2TvXsLEvzDb+lrj7G5rcXM6mqEf9I/9X9It/eIfOLX/5s8NsXn+jh+DvSRVYsRtyr1symuw/XUux3jjWza/nDasF5OYwix3ZR55rKGAIr/SwAVauFK+KKvKulJopyodqWYMz0LZeQTKoZht</latexit>

Covariance matrix（共分散行列）:<latexit sha1_base64="NLdQauCvRuKVtNo4J80F8f9bv68=">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</latexit>

Principal component analysis (PCA):
Data compression through the spectrum decomposition of C.
<latexit sha1_base64="k/MZWH5TL3QJ7k4398KweELzCAQ=">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</latexit><latexit sha1_base64="HHjusBinogfxC/stv7VxJZSx1u4=">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</latexit>: diagonal matrix,<latexit sha1_base64="qyPbNrTWJpqHtCnGeSKeaI5F/jo=">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</latexit>

corresponding to large λi contains important information.

<latexit sha1_base64="KYQK9o8Av2mv/y4bVW7ZSfsZ+PM=">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</latexit>

By construction, λ and V are related to SVD of X !
PCA can be regarded as
the low rank approximation X.
<latexit sha1_base64="0ed0dY+FBbdCXn0AQl13Z25nPXY=">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</latexit><latexit sha1_base64="xJtbrlcnisgdbhEY1jZU6wqkypc=">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</latexit>

Generalization to tensors

Scalar, Vector, Matrix, Tensor,...
Scalar:<latexit sha1_base64="xjTtQZt/S06bAHzsr4SBpd4jRhs=">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</latexit>

Vector:<latexit sha1_base64="wlWQUpOEA9Mmb6ql6UkCS7qzJwA=">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</latexit>

Matrix:i

NumberOne dimensional array of numbers
Two dimensional array of numbers<latexit sha1_base64="PUy+WJIK1cOmBYjLGbwuDqULCAk=">AAACj3ichVG7SgNBFD2u72cSbQSbYIhYyUSEiIUEbbQQkmiiECXsrqNOsi92JwFd/AErO1ErBQvxD2xt/AELP0EsFWwsvLtZEBX1DjNz5sw9d85wNccQnmTssU1p7+js6u7p7esfGByKxRPDZc9uuDov6bZhuxua6nFDWLwkhTT4huNy1dQMvq7VF4P79SZ3PWFba3Lf4VumumuJHaGrkqjyStUXtcNqPMWmWBjJnyATgRSiyNvxW2xiGzZ0NGCCw4IkbECFR6OCDBgc4rbgE+cSEuE9xyH6SNugLE4ZKrF1WnfpVIlYi85BTS9U6/SKQdMlZRJp9sCu2Qu7Zzfsib3/WssPawRe9mnXWlruVGNHo6tv/6pM2iX2PlV/epbYwWzoVZB3J2SCX+gtffPg5GV1rpj2J9gleyb/F+yR3dEPrOarflXgxfM//GjkJawSIGpT5ntTfoLy9FSGcGEmlVuIGtaDMYxjkrqSRQ5LyKNEr9RwjFOcKQklq8wruVaq0hZpRvAllOUPc6eTSw==</latexit>

Tensor:Higher dimensional array of numbersi

Scalar: 0-dim. tensor
Vector: 1-dim. tensor
Matrix: 2-dim. tensorjkGraphical representations for tensor network•

Vector：•Matrix：Tensor：•* n-rank tensor = n-leg object

When indices are not presented in a graph, it represent a tensor itself.

Graphical representations for tensor network
Matrix product
C

＝

A

B

C

＝

A

B

Generalization to tensors
C
B

A

Contraction of a network = Calculation of a lot of multiplications（縮約）Low rank approximation: generalization to tensor
Tensor:ij

Naive application of SVD:
Make a matrix by dividing indices into two parts.

kThen apply SVD (and low rank approximation).
i

j

i

l

k

l

j

i

=

T

T

i

j
k

l

k

k
Note: The result depends on the initial mapping to a matrix.

TjCP decomposition

Review: T. G. Kolda et al, SIAM Review 51, 455 (2009)

CP (Canonical Polyadic) decomposition: Hitchcock (1927)
i

i

j
T

j

min (r) = tensor rank

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is NP-hard problem.

<latexit sha1_base64="1fLBlzjRgpe/u+S0r7W+zNnKUHM=">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</latexit>

Low "rank" approximation<latexit sha1_base64="qWKliztd2gb1TbpeGIkGIvYurBE=">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</latexit><latexit sha1_base64="R8pMI9SEqjFP8rz0DPu1oxuWiT0=">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</latexit>

rank-r' approximation

Tucker decomposition: generalization of SVD
Tucker decomposition:(Tucker (1963))
i

U(2)

U(1)

j

=

T
l

Review: T. G. Kolda et al, SIAM Review 51, 455 (2009)k

Low "rank" approximation

S
U(4)

U(3)

U(i) : Factor matrix(usually unitary)
S :Core tensor*If S is "diagonal", Tucker decomposition
becomes CP decomposition.

<latexit sha1_base64="0b5vuDqbmGkatQ8ZNzpv9EM32JA=">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</latexit>

rank-(I',J', K', L') approximation

Higher order SVD (HOSVD)
L. De Lathauwer et al, SIAM J. Matrix Anal. & Appl., 21, 1253 (2000)

Define a factor matrix from matrix SVD:
i

j
T

l

i

j

j
k

T
l

k

=

k

U(1)
l

Core tensor is calculated as

Properties of the core tensor

Dot product

Generalization of the diagonal matrix Σ in matrix SVD.
* Low-rank approximation based on HOSVD is not optimal.

Application of low rank approximation
Sample codes will be uploaded on the website

SVD_sample.zip(python3 (or python2.7) + numpy + PIL)
(python and jupyter notebook codes)

Image compression: grayscale image
Image: 1024 × 768 pixels

768 x1024 matrix A
Amount of data=786,432
Perform SVD of A:
approximation
Amount of data=(768 +1024 + 1)×χ

Image compression:
compression grayscale image

Rank:
Data: 786,432(Original)

179,300

179,30Image compression: color image
Image: 1024 × 768 pixels
768 x1024x3 tensor T
Amount of data=2,359,296* Sub matrices for RGB colors

Two image compressions:
Perform SVD for R, G, B

approximation for RGB matrices
Amount of data=3×(768 +1024 + 1)×χ

Perform HOSVD for T

approximation
Amount of data= (768 +1024 + 3χ)×χImage compression:
compression color image1
SVD

~10%
Compression<latexit sha1_base64="b4XVXZ3MrmVrQbCxgQ6mbAPMPgo=">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</latexit>

268,950

HOSVD<latexit sha1_base64="AtcCApO1+if/lZgDTu2jfkyFiQo=">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</latexit>

Original
Data: 2,359,296

209,200

Image compression:
compression color image2
SVD

~10%
Compression<latexit sha1_base64="b4XVXZ3MrmVrQbCxgQ6mbAPMPgo=">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</latexit>

268,950

HOSVD<latexit sha1_base64="AtcCApO1+if/lZgDTu2jfkyFiQo=">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</latexit>

Original
Data: 2,359,296

209,200

Image compression:
compression multi images

92×122 pixels 10 images

Images were taken from ORL Database of Faces,
AT&T Laboratories Cambridgeby HOSVD

92 x122 x 10 tensor T
Amount of data=112,240

approximation
Amount of data=(92 +122 )×χ + 10×χ' + χ2χ'Image compression:
compression multi images
Data
Original

112,240

15,52014,61010,970SVD of wave function?
Wave function:<latexit sha1_base64="sH+yUz5zCglxrI1LAPIX89I7T4g=">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</latexit>

T:N-leg tensor (or Vector)
m1

m2

m3

m4

We can consider it as a matrix by making two groups:

T{m1 ,m2 ,··· ,mM },{mM +1 ,··· ,mN }<latexit sha1_base64="xU9rERi5ezQJsoBrgDMy8DMa3kU=">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</latexit>

We can perform the low rank approximation of T.
<latexit sha1_base64="BBENFyPtyQcO1GVK6x5IeKQK2ns=">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</latexit>

T
m1

m2

m3

'
<latexit sha1_base64="pBZwnagoKpoajh8xjQf7DXhtIW0=">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</latexit>

m4

m5

m1

m2

What does it mean?

m3

m4

m5

m5

References:•

齋藤正彦、「線形代数入門」東京大学出版会•

太田快人、「システム制御のための数学（１）ー線形代数編ー」、コロナ社•

T. G. Kolda et al, SIAM Review 51, 455 (2006).

Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

テンソルネットワーク繰り込みによる情報圧縮（古典スピン模型の実空間繰り込み群）

Outline•Tensor network representation of a scalar•••Tensor network renormalization•

Tensor Renormalization Group (TRG) in two dimension•

Generalization to higher dimensions

Tensor network renormalization around critical point••

Partition functions in statistical mechanics

Fixed point of TRG: Corner double line tensors

Report problems

Tensor network representation of a scalarイジング模型の分配関数
統計力学の処方箋：
分配関数を計算したい

しかし、和は、スピンがN個の時、で指数的に大きい！（N=100でも、1030の項がある！）
定義通りに計算することは困難
計算科学によるアプローチ

転送行列
例：１次元イジング模型

分配関数
転送行列

+1

分配関数は転送行列の積でかける

-1

+1
-1

転送行列の対角化
転送行列

転送行列は実対称行列
固有値は実で、直行行列で対角化可能

分配関数

分配関数の計算→転送行列の対角化

２次元系の転送行列
L×Mの２次元系
M個のスピンを1セットで考えると１次元系と同等
転送行列の大きさ

1次元系：2×2
LxMの２次元系：2M×2M (or 2L×2L)
２次元以上では転送行列が系サイズに関して指数的に大！
厳密な計算はすぐに破綻する
２次元イジング模型だったら、M=40程度が限界（疎行列の対角化問題）

転送行列の積を近似的に計算転送行列表現の拡張
２次元正方格子模型の分配関数

正方格子

Aの配置

一つの正方形に注目

各辺のボルツマン重みの積：４階の“テンソル”
イジング模型
Aは2×2×2×2のテンソル

分配関数＝テンソルの"掛け算"ダイアグラムを用いたテンソル表記
•

ベクトル

：

•

行列

：

•

テンソル

：

テンソルの積（縮約）の表現＊n階のテンソル＝n本の足

C

＝

A

B

C
B

A

分配関数のダイアグラム
例：１次元イジング模型
T<latexit sha1_base64="pjn+Vxlv/8kO2wMR3fuPydf06lo=">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</latexit><latexit sha1_base64="ylsTIBa3eHai+z8oexIJCTjS+Oc=">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</latexit>

T<latexit sha1_base64="Br4u+JQD+E5HkV+WYqVBGKr43T8=">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</latexit>

T

T

T

T

T

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T

T

T

T

T

T

分配関数のテンソルネットワーク表現
A
A
A

A

A
A

A
A

A
A

分配関数＝テンソルAの積のネットワーク
テンソルネットワーク
正方格子イジング模型→45度傾いた正方格子ネットワーク

(Real space) Renormalization group（補足資料）Example: Ising model
Ising model

Canonical ensemble:
Magnetization at T:

Coarse graining （粗視化）
Block spin transformation（ブロックスピン変換）

:1

:-1

"Length scale"changes
2×2 system
6×6 systemcoarse grained spin::=

Example of block spin transformation
Figure taken from a book "Scaling and Renormalization in Statistical Physics", John Cardy

T=Tc (critical point)(befor)•(after)T > Tc(befor)At the critical point, the block spin transformation does not change"image" qualitatively.
"Scale invariance"•(after)At T > Tc, the block spin transformation changes typical "cluster size"

Partition function after coarse graining
Partition function after a block spin transformation:(for simplicity, we set(d-dimensional system with length L))(d-dimensional system with length L/b)

By block spin transformation, the partition function isrepresented by smaller # of spins with a modified Hamiltonian
Information compression by "tracing out"
short range fluctuations

Coarse grained Hamiltonian
Partition function after a block spin transformation:
Sum over spin configurations
corresponds to {S'}Suppose H' has the same form with the original Hamiltonian,which characterized only one parameter K:

By repeating the procedure, we can draw a flow of "K""renormalization group"（繰り込み群）Rb : transformation with scale b

Renormalization flow
Renormalization group:
Fixed point（固定点）:
Unchanged under renormalization
Typically, we have three fixed points for a phase transition:
Corresponding T=0, T=∞, and T=Tc

General caseIn general, H' contains many body interaction such as SiSjSkSl.
We need more than one parameter:

Renormalization group:
RG characterize a flow in parameter spaceCritical exponents and eigenvalues
Linearization around Kc::Matrix applied in parameter space:Eigenvalue of:Eigenvector"Relevant"increases along renormalization"Irrelevant"decreases along renormalization

Critical exponents are related to relevant eigenvalues!

Tensor network renormalization

Tensor network renormalization（テンソルネットワーク繰り込み）••

Approximate calculation of a tensor network contraction
by using "coarse graining" （粗視化）of the network•

Coarse graining ⟷ Real space renormalization•（粗視化） ⟷（実空間繰り込み）

It can be applicable to (basically) any lattices, and the idea(algorithm) is independent on "models" represented bytensor networks.
•

Potential application to wide range of the science.

Outline of tensor network renormalization
Scaler represented
by L×L tensors(L×L)/2 tensors

Corse graining (Renormalization)
into √２times longer scale.

: D×D×D×D

Approximation

Reduce the number of tensors
keeping their size constant: D×D×D×D

Key technique: low rank approximation by SVD
Best low-rank approximation of a matrix = SVD
A

=

:M×N
(M ≦ N)

≃
:R×R

:M×M

(Diagonal matrix):(M,N)×M(Keeping the R largest
singular values):(M,N)×R

In addition,==:Diagonal matrix
those elements are

X

Y

:M×R
:R×M

By SVD, we candecompose a matrix
into a product of"small" matrices.

Recipe of Tensor Renormalization Group (TRG)
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)

1. Decomposition
Regard a tensor as a matrix
l

i

i

i

j

l
j

j
k

: D×D×D×D

D-rank approximation
by SVDlkk: D2×D2

Approximation

Recipe of Tensor Renormalization Group (TRG)
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)

2. Coarse graining
Contraction of
inner indices

In total, two originaltensors are coarse
grained into a new tensor.

: D×D×D×D

Recipe of Tensor Renormalization Group (TRG)
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)

Calculation cost:

SVD= O(D6)Contraction= O(D6)(per tensor)*By one TRG step, # of tensors is reduced by 1/2.

We can calculate the contraction in polynomial cost!

Tensor renormalization group for higher dimensions
Simple generalization of TRG to cubic lattice (three dimension)

6-rank8-rankTensor renormalization group by using HOSVD
Anisotropic coarse graining by using HOSVD instead of SVD
Z. Y. Xie et al, Phys. Rev. B 86, 045139 (2012)

Basic idea of HOTRG algorithm:(For details, see the original paper.)==Low-rank approximation
by HOSVD

U(1)

U(3)

S

U(4)

U(2)

(Reference of HOSVD)

Tucker decomposition: generalization of SVD
Tucker decomposition:(Tucker (1963))
i

U(2)

U(1)

j

=

T
l

Review: T. G. Kolda et al, SIAM Review 51, 455 (2009)k

Low "rank" approximation

S
U(4)

U(3)

U(i) : Factor matrix(usually unitary)
S :Core tensor*If S is "diagonal", Tucker decomposition
becomes CP decomposition.

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rank-(I',J', K', L') approximation

(Reference of HOSVD)

Higher order SVD (HOSVD)
L. De Lathauwer et al, SIAM J. Matrix Anal. & Appl., 21, 1253 (2000)

Define a factor matrix from matrix SVD:
i

j
T

l

i

j

j
k

T
l

k

=

k

U(1)
l

Core tensor is calculated as

Properties of the core tensor

Dot product

Generalization of the diagonal matrix Σ in matrix SVD.
* Low-rank approximation based on HOSVD is not optimal.

Power of the HOTRG
Advantage:•

HOTRG does not change the network structure.
••

We can easily generalize it to higher dimensions.

Low-rank approximation is based on the cluster of two tensors.
•

At the approximation, we take into account more information.

•

More efficient than TRG where SVD is done for a single tensor.

Disadvantage:•

HOTRG needs higher cost than TRG.
•

O(D7) in HOTRG ⟷ O(D6) in TRG

Application to a classical partition function
Partition function(Periodic boundary condition)
Repeat TRG step
until only a few
tensors remain.

We can easily calculate physical quantities from Z.
Free energy:
Energy:(Use difference approximation)

Specific heat:(Use difference approximation)

Example of calculation
Error of free energy for 2D Ising model

XIE, CHEN, QIN, ZHU, YANG, AND XIANG

energy, the accuracy of the specific heat data is much less thanIsing model in infinite sizethat 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)
U = Uc + at + bt 1−α ,Energy and specific heat of 3D Ising model
PHYSICAL REVIEW B 86, 045139 (2012)

(a)
(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

Tensor network renormalization at the critical point— When the accuracy of TRG becomes worse?
（少し高度な内容）

Correlation (entanglement) within a tensor
Eg. Correlation in (i,j) and (k,j)

General tensor

Diagonalize M and N matrices.

i

i
l

l

λ

l

M

j

j

N

k

k

New rule for representation of the correlation:ij =(+ we neglect eigenvalues in the graph.)λkFixed point of TRG: Corner Double Line tensor(固定点）
Corner Double Line (CDL) tensor:
There are correlations
among the nearest legs.

SVD
Original bond dimension =Single line: bond dimension

Degree of freedoms
connecting two tensors.
Two lines =No truncation error at SVD(Original rank = D)Fixed point of TRG: Corner Double Line tensor
Contraction of four tensors in TRG:

×=
Constant

proportional to a CDL tensor!

CDL tensor is a fixed point of TRG (and also HOTRG).
CDL tensor remains as CDL tensor along TRG.

Problems in TRG: accumulation of correlations
Correlations remains after TRG.

Correlation in several scales

TRG

Ideal renormalization:
TRG：

Correlation in shorter scales should be removed.
Only the correlation in the present scale exists.
Correlations in all scales remain.

Problem in TRG: increase of truncation error
Singular value distribution(2D Ising model）

G. Evenbly and G. Vidal
Phys. Rev. Lett. 115,
T 180405= 1.1 Tc . Now TNR(2015)

ial fixed-point tensor A
zero singular value, w
ature, disordered pha
scheme, for any T >fixed-point tensor At
TRG generates a flow
pends on the initial teure to remove some ofthat (RG irrelevant)retained during coarse
the RG flow. For T <picture as for T > Tc
the spectrum has dege
Renormalization steps
fixed-point tensor AZ
the Z2 symmetry brea
In TRG, the width of the singular value distribution flip
increases
symmetry acts on
Atriv . Finally, Fig. 4(
along renormalization.
a function of scale
Increase of truncation error (decrease ofasaccuracy)
one of the three fixed-Improvement of TRG：Entanglement Filtering
Try to remove CDL structure at renormalization steps.
Z.-C. Gu and X.-G Wen, Phys. Rev. B 80, 155131 (2009)

Idea:

=

S:
S’ =

1/4

S’:
S

Insert this "approximation" into the TRG algorithm.

Tensor Entanglement Filtering Renormalization
Z.-C. Gu and X.-G Wen, Phys. Rev. B 80, 155131 (2009)(after approximation)

Change of SVD:

CDL structure isperfectly removed!
In the case of CDL tensor

Remaining problem in TEFR•TEFR works well far from the critical point.
••

In the vicinity of the critical point, the accuracy is still poor.
••

Because it can remove CDL structure.

Because the actual entanglement is not necessarily
perfect CDL structure.

In order to improve further, we need to consider theentanglement structure beyond CDL tensor.

will be particularly important at criticality, where correlations are present at all length scales.
Example: Partition function of the 2D classical Isingmodel.— We consider the partition function!
!
−H({σ})/T
σi σj (4)e
, H ({σ}) = −Z=

FIG. 3. Benchmark results for the square lattice Ising model
on a lattice with 239 spins. (a) Relative error in the free energy
per site δf at the critical temperature Tc , comparing TRG andTNR over a range of bond dimensions χ. The TRG errors fit
2
δf ∝ χ−3.02 (the inset displays them using log-logaxes), while
2
G. Evenbly
and G. Vidal,Extrapolation
Phys. Rev. suggests
Lett. 115, 180405(2015).
TNR errors
fit δf ∝ exp(−0.305χ).
that TRG would require bond dimension χ ≈ 750 to match
arXiv:1412.0732.
the accuracy of the χ = 42 TNR result. (b) Spontaneous
magnetization M (T ) obtained with TNR with χ = 6 [23].
Even very close to the critical temperature, T = 0.9994 Tc , themagnetization M ≈ 0.48 is reproduced to within 1% accuracy.

Recent progress: Tensor Network Renormalization
{σ}

⟨i,j⟩

on the square lattice, where σk ∈ {+1, −1} is an Ising
spin on site k and T denotes the temperature. Recall

eeEq.Tensor
5 and Fig.
an network
explicit
example),imately
expressed
as a1(a,b)
courser for
tensorof⇣Renormalization⌘ madeNetwork
N/4
just N/4!copies of A(1) , Z ⇡ tTr ⌦x=1 A(1) . By "itera- #N
Ztion,= a sequence
Aijkl A
Akrst
· ·!
· ≡AtTr(2)
opqr
x=1 A . (1)ofmnojtensors
A(0)A!
A(1)
···⊗
will

Point of TNR

ijk··· such that, for any length scale s,
be produced⇣⌘Nssere eachZ index
hosts
aof freedom(s)⇡ tTr ⌦x=1 A χ-level,Ns local⌘ N/4degree.
(2).g. i = 1, 2, · · · , χ), the tensor components Aijkl are Use of a disentangler(Unitary tensor)
We seethat after
justthes̃ ⌘ tensor
log4 (N ) iterations[assumingcalweights,
andtrace tTr
denotes a sum that this model has a global Z2 symmetry: it is invariants̃N =4 ], the partitionoffunction
Zindices
becomesijk
the ·traceunder the simultaneous flip σk → −σk of all the spins.
verconfigurations
all· · . of
P the
(s̃)
(s̃)
aOur
singletensorA produce, Z ⇡ ijan
Aijij
, which we
can finally
A in remove short
It
can
goal
is to
eﬀective
tensorA(1) , roughly We obtained an exact representation for the tensor
evaluate [10]. Alternatively, in the thermodynamic limit (0)Eq. 1 in terms of four Boltzmann weights eσi σj /T ,counting
forfour
copiesoftheoriginal
tensorA≡A,(s)range entanglement
N ! 1, we can study the RG flow of tensors A as
ch
that
Z
can
be
approximately
expressed
asacoarser
(σ
σ
+σ
σ
+σ
σ
+σ
σ
)/T
we progress to larger length scales s. In particular, theAijkl = e i j j k k l l i ,(5)efficiently.
(1)
possible
fixed-point
tensor
of
this
flow
will
containthensor
network
made
of
just
N/4
copies
of
A
,
Z
≈
$
%

universal
N/4properties of the phases and phase transition

which corresponds to having one tensor A for every
two only CDL)
(Not
spins, and a tensor network with a 45 degree tilt with
Tensor Network Renormalization.— Our coarseApproximation
by seeusing
respect to the spin lattice,Fig. 1(a,b). We actu(1)tensor network
graining transformation
A(0) →forAthe→ A(2) → ·of· ·the par(2) ally built our starting tensor A(0) , with bond dimension
tition function Z in Eq. 1 will result from applying one
cluster:χ =two-tensor
4, by joining a square
block of four tensors A tosimple
rule and following
a guidingillbe produced
such that,
forprinciple.
any length scale s,gether. We then applied up to 18 TNR transformationsFirstly, the rule: “We$ are allowed% to insert resolutionsof the identity in the tensor”, since this will not
Ns network
to a system made of N = 218 × 218 tensors A(0) , or equivs
(s)
Z
≈
tTr
⊗
A
,
N
≡
N/4
.
(3)
s
x=1 Z that it represents.
a↵ect the partition function
For this
FIG. 1. (a) alently
We consider
2 ×a4square× N lattice
Ising (slanted
spins. 45 ) of classical spins, where k 2 {+1, 1} is an Ising spin on site k.
purpose, we regard each index of the network as hostingFig.
3(a) shows the relative error δf in the free enrepresentation of a part of the tensor network,
hus,
after s̃ ≡complex
log4 (Nvector) iterations[assuming
N = (b)4s̃ Graphicalfora -dimensional
space V ⌘C . We conergy denotes
per sitef ≡A,log(Z)/(8N), function
at the critical temperwhere each circle
a tensorfor√the partitionsider integer
two cases,s̃ see(i) a joint changeof basisme> Fig.
0], 1(c):the partition
function
Z becomes
Z of the classicalmodel,
see Eq.
tensor Afor
ijkl both TRG andature spinTc ≡2/ ln(1+ 1.2)Here≈ 2.269,&
on two indices, which mixes the corresponding
degreesof(s̃) encodes the Boltzmann(s̃)
weights
of
the
spins
{
,
,
,
j
k
l}
e tracebyofmeansa singletensortransformation
A , Z ≈ u, ijwhere
Aijij , which TNR as a function of the bondi dimensions
χ [24]. The
freedomof a unitaryaccording to the Hamiltonian function H,Better
see Eq.4. (c)accuracy
Inthanthe†††an example, while
FIG.
1.
(a)
As
we
considerasquarelatticeeu can[16].
hand, sertionin theof aTRG
error
decayspolynomially,
the
TNR
error
is
: V ⌦finally
V ! V evaluate⌦ V, with uu= uOn
u =theI ⌦2other, analogous
pair of disentanglers u u between four tensors,(slanted45◦ )fromof tensors
classical
spins,
where{+1, −1}tensor
is antoa disentangler limit
in the N
context
of entanglement
renorwhereintensorsÃ are obtained
A throughgauge σk ∈single
reducedexponentially,
showinga aSVD
qualitatively
diﬀerent
ermodynamic→ ∞,we can study
the flow
simpleoftransformationon theirhorizontal[9], Graphicalfollowed by anmalization [6]; (ii) a combination of two indices into a
Ising
spin
onimplying
siteindicesk. (b)
representation of a part
behaviourande space of tensors given by Eq. 2. The fixed-point†that significantly more accurate
insertion of fourof theform vv wherein termseachof isomesingle one, by means of an isometry v (v or w in Fig.
of projectors
thecantensor
network,
circle3(b)denotestensor A,
results
be obtainedwith TNR.
Figureshowsathe

Trof the⊗x=1
A(1) . By iteration, a sequence of tensors
system.

Power of TNR

G. Evenbly and G. Vidal,Phys. Rev. Lett. 115, 180405 (2015),arXiv: 1412.0732v2 (free energy).

Singular value distribution(technically, this corresponds to a so-called
tensor,(2D IsingCDL
model）

Error of the free energy

T = 1.1 Tc . Now TNR generates
triv
see Appendix B). If we set the disentanglers to be trivial
fixed-point
tensorA, charac⊗2
ial, u = I , then the coarse-graining transformation
zero singular value, which represe
reduces to TRG [18] and fails to remove the short-range
ature, disordered phase. As exp
correlations. However, a judicious choice of disentangler
scheme, for any T > Tc the flowu removes these correlations and produces an uncorretriv
fixed-point
tensorA. In contriv
lated, trivial tensor A . Therefore the disentangler u
TRG generates a flows to a fixe
removes short-range correlations. Importantly, the expends on the initial temperatureample below demonstrates that this is also true in theure to remove some of the short-ra
far more challenging scenario of a system near or at critthat (RG irrelevant) microscopi
icality.
retained during coarse-graining,
Example: Partition function of the 2D classical Ising
the RG flow. For T < Tc (not sho
model.— We consider the partition function
picture as for T > Tc . However,the spectrum has degeneracy 2, a
!
!
Z2
triv
Renormalization steps−H({σ})/T
fixed-point
tensorA≡A⊕
σi σj (4)e
, H ({σ}) = −Z=
the Z3.2 Benchmark
symmetry breaking,
⟨i,j⟩
{σ}
FIG.
results for order
the s
Z
2
•
The singular value distribution is narrower
thanof TRG.
In TNR:
flipa symmetryon134
A million
by exchon
lattice thatwithacts
oversp
Atriv . per
Finally,Fig.
4(c) uses Tthe
• It is almost unchanged at Tc.
energysite attemperature
, coe
with
dimensions{4, 8}
as a bondfunctionof scaleχto= visualiz• Indicating scale invariance of the criticalsystem.
isone
roughlyordersof magnitudeof thetwo
three
fixed-point
tensoInteresting topics in tensor network renormalization•

Try to find efficient algorithm to remove "short range" entanglement•

TNR, Loop-TNR, GILT, Gauge fixing
TNR: G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)
Loop-TNR: S. Yang, Z.-C. Gu and , X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)
GILT: M. Hauru, C. Delcamp. S. Mizera Phys. Rev. B 97, 045111 (2018)
Gauge fixing: G. Evenbly, Phys. Rev. B 98, 085155 (2018)•

Application to lattice QCD••

TRG with Grassmann algebra Z.-C. Gu, F. Verstraete, and X.-G. Wen, arXiv:1004.2563
S. Takeda, and Y. Yoshimura PTEP 2015, 043B1 (2015).

Property at the criticality•

Relation between TNR and MERA•

Relation to Conformal invariance
G. Evenbly and G. Vidal,Phys. Rev. Lett. 115, 200401 (2015)
G. Evenbly, Phys. Rev. B 95, 045117 (2017)

Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

情報のエンタングルメントと行列積表現

Outline•

Outline of tensor network decomposition•

Entanglement••

Schmidt decomposition•

Entanglement entropy and its area law

Matrix product states•

Matrix product states (MPS)•

Canonical form•

infinite MPS

Outline of tensor network decomposition

Classification of Information Compression by
Memory Costs
Linear algebra for huge data:(1) A matrix can be stored
Required memory~(2) Although a matrix cannot be stored, vectors can be stored
Required memory~(3) A vector cannot be stored
Required memory
We try to approximate a vector in a compact form.
Memory ~Exponential

Polynomial:problem size (e.g. system size)When we efficiently compress a vector?

If we can find a basis where the coefficients have a structure (correlation).
(1) Almost all Ci are zero (or very small).
We store only a few finite elements
E.g.

Fourier transformation
If we can neglect larger wave numbers, we can efficiently
approximate the vector with smaller number of coefficients.

Classical state
In this case, we know that only a specific Ci is non-zero.
We need only an integer corresponding to the non-zero element.

When we efficiently compress a vector?

(2) All of Ci are not necessarily independent.
We store "structure" and "independent elements".
E.g. Product state ("generalized" classical state)
A vector is decomposed into product of small vectors.
e.g.
structure: "product state"
independent elements: small vectors

Tensor network decomposition of a vector
Target:Exponentially large Hilbert spacewith+Total Hilbert space is decomposed asa product of "local" Hilbert space.

Tensor network decomposition: index of local Hilbert space: local tensor for "state" iGraphical representations for tensor network•

Vector：•Matrix：Tensor：•* n-rank tensor = n-leg object

When indices are not presented in a graph, it represent a tensor itself.

Graphical representations for tensor network
Matrix product
C

＝

A

B

C

＝

A

B

Generalization to tensors
C
B

A

Contraction of a network = Calculation of a lot of multiplications（縮約）Graph for a tensor network decomposition•

Vectorv*Vector looks like a tensor•

TensorTensor network decomposition*We treat i as an index
of the tensor.

*We can consider tensors
independent on i.

Entanglement（エンタングルメント）

N-qubit system (S=1/2 quantum spin system)
Example vector: Wave function of N-qubit systems…
takes two states

Coefficients = vector:* Inner product:

Schmidt decomposition
General vector:
Schmidt decomposition

Orthonormal vectors
Schmidt coefficient
Schmidt decomposition is unique.

Schmidt decomposition for wave function
Wave function:
Schmidt decomposition

A

B

Divide system into two parts, A and B:General wave function can be represented bya superposition of orthonormal basis set.

B<latexit sha1_base64="ZU1UFVOI4sQ5r55FgImhtnwW0DE=">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</latexit>

A
Orthonormal basis:
Schmidt coefficient:B<latexit sha1_base64="JzhxCAIkIqTxgCdAElfwhSUhGpw=">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</latexit>

APartial trace and reduced density matrix
For
Density matrix:（密度行列）*Note:<latexit sha1_base64="NuLat7dCh/8ymHp3FBjOmHidMOk=">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</latexit>

Orthonormal basis:
Basis for:Index:
Reduced Density matrix:（縮約密度行列）: a positive-semidefinite square matrix inRelation between SVD and Schmidt decomposition
Singular value decomposition (SVD):
For a K × L matrix M,

Singular values:

Singular vectors:

Relation to the Schmidt decomposition:

By using SVD, we can perform Schmidt decomposition.
(and can calculate entanglement entropy.)

Entanglement entropy
Entanglement entropy:
Reduced density matrix of a sub system (sub space):A

B

Entanglement entropy = von Neumann entropy of ρA

B

Schmidt decomposition(*Exercise)

Entanglement entropy is calculated throughthe spectrum of Schmidt coefficientsA

Intuition for EE: two s=1/2 spins
1.
A product state

A B,

Product state : S=02.
Another product state,3.
Spin singlet

Maximally entangled State,

4.
Complicated state,

Large entanglement entropy ~ Large correlation between two parts

Area law of the entanglement entropy in physics
General wave functions:

B

EE is proportional to its volume (# of qubits).

Ground state wave functions:L(c.f. random vector)

A

For a lot of ground states, EE is proportional to its area.
J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys, 277, 82 (2010)

In the case of one-dimensional system:
Gapped ground state for local Hamiltonian
M.B. Hastings, J. Stat. Mech.: Theory Exp. P08024 (2007)

A

Ground state are in a small partof the huge Hilbert spaceBExercise: examples of Schmidt decomposition

1-1: Random wave function (Sample code: Ex1-1.py or Ex1-1.ipynb)••

Make a random vector
SVD it and see singular value spectrum and EE

1-2: Ground state of S=1 Heisenberg chain (Sample code: Ex1-2.py
H=
•
•

X
i

or Ex1-2.ipynb)~i · S~i+1
S

Calculate GS by diagonalizing Hamiltonian
SVD it and see singular value spectrum and EE*Note: the ground state of this model is gappedshow help: python Ex1-2.py -h* Try to simulate different system size "N"* You can simulate other S by changing "m"Result: N=10 spectrum
A

RandomB

Heisenberg

Ground state wave function has lower entanglement!

Matrix product states（行列積状態）

Data compression of wave functions (vectors)
General wave function:

Coefficient vector can represent any points in the Hilbert space.

Ground states satisfy the area law.

Hilbert space

In order to represent the ground state accurately,
we might not need all of aN elements.

Data compression by tensor decomposition:

Tensor network states

Area lawTensor network state
G.S. wave function:=

Vector (or N-rank tensor):``Tensor network”
decomposition
Matrix Product State(MPS)# of Elements＝aN
i1

i2 i3

i1

i2

i4 i5

=
：Matrix for state m

i3
i5

i4

i5
i3

General network
X,Y : Tensors
Tr : Tensor network contraction

i1

i4

i2

By choosing a ``good” network, we can express G.S. wave function eﬃciently.
ex. MPS: # of elements ＝2ND2

D: dimension of the matrix A

Exponential→ Linear*If D does not depend on N…

Examples of TNS
MPS:Good for 1-d gapped systems

MERA:

RG

PEPS, TPS:

For higher dimensional systems
Extension of MPS

Scale invariant systems

Good reviews:

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

MPS：Matrix for state i

Note:
•

MPS is called as "tensor train decomposition" in applied mathematics(I. V. Oseledets, SIAM J. Sci. Comput. 33, 2295 (2011))•

A product state is represented by MPS with 1×1 "Matrix" (scalar)Matrix product state without approximation
General wave function (or vector) can be represented by MPS exactly
through successive Schmidt decompositions
SVD

SVD of<latexit sha1_base64="6m/gbzqlDKBTUkwq2Gwhy8OFn50=">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</latexit><latexit 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(row)

…

(column)(row)(column)

In this construction, the sizes of matrices
depend on the position.
Maximum bond dimension = aN/2
At this stage, no data compression.

Matrix product state: Low rank approximation=

Maximum bond dimension = aN/2

Constant bond dimension = χ

If the entanglement entropy of the system is O(1) (independent of N),
matrix size "χ" can be small for accurate approximation.
MPS is good for gapped 1d systems.
On the other hand, if the EE increases as increase N,"χ" must be increased to keep the same accuracy.

Upper bound of Entanglement entropy:MPS with χ

Reduced density
matrix of region A:

Structure of ρA:A

BRequired bond dimension
in MPS representation

The upper bound is independent of the "length".
length of MPS

EE of theoriginal vector

size of the problem

Required bond dimension
in MPS representation

Gauge redundancy of MPS
MPS is not unique: gauge degree of freedom
We can insert a pair of matrices GG-1 to MPS

=

=

=

=

Gauge fix: Canonical form of MPS

Ref. U. Schollwöck, Annals. of Physics 326, 96 (2011)

Canonical form of MPS: (Vidal canonical form)(G. Vidal, Phys. Rev. Lett. 91, 147902 (2003):Diagonal matrix corresponding
to Schmidt coefficient:Virtual indices corresponding
to Schmidt basis

Left canonical condition:=

Right canonical condition:=(Boundary)=Canonical forms: Left and Right canonical forms

Ref. U. Schollwöck, Annals. of Physics 326, 96 (2011)

Other "canonical" forms of MPS
Left canonical form:

Right canonical form:

Satisfies (at least) left or canonical condition:==

Canonical forms: Mixed canonical forms

Ref. U. Schollwöck, Annals. of Physics 326, 96 (2011)

Mixed canonical form:
is identical with
the Schmidt coefficient.

Left canonical condition:=

Right canonical condition:=

Canonical forms: Note•

Vidal canonical form is unique, up to trivial unitary transformation to
virtual indices which keep the same diagonal matrix structure(Schmidt coefficients).

•

Left, right and mixed canonical form is "not unique". Under generalunitary transformation to virtual indices, it remains to satisfy the
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MPS for infinite chains
By using MPS, we can write the wave function ofa translationally invariant infinite chain…

…

Infinite MPS (iMPS) is made by repeating T and λ infinitely.
Translationally invariant system

T and λ are independent of positions!

* Infinite MPS can be accurate when the EE satisfies the 1d area low (S~O(1)).
If the EE increases as increase the system size,
we may need infinitely large χ for infinite system.

Calculation of expectation value
2
i =1

i

T⇤

h |Ô| i ===h | i=X

Canonical form

Ô

=

Ô

For iMPS, if it is in the (Vidal) canonical form,
the final graph is identical to the above finite system.

Example of iMPS: AKLT state
S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian:…
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The ground state of AKLT model:

S=1 spin:
S=1/2 spin

χ=2 iMPS: (U. Schollwock, Annals. of Physics 326, 96 (2011))

Spin singlet

Exercise 2: Make MPS and approximate it
2-1: Make exact MPS from GS wave function obtained by ED(We can easily check that the MPS obtained by successive SVD
satisfy the canonical condition.)

Sample code: Ex2-1.py or Ex2-1.ipynb
show help: python Ex2-1.py -h

2-2: Approximate the MPS by truncating singular values•

Calculate approximate GS energy and compare it with ED•

Change chi_max and see energies
Sample code: Ex2-2.py or Ex2-1.ipynb
show help: python Ex2-2.py -hRequirement for running sample scripts
File: Exercise_No10.zip
Python environment: python2.7 or python3
Modules: numpy, scipy and matplotlib
Usage:For jupyter notebook, typejupyter notebook
and select Ex?-?.ipynb .
For python (command line), typepython Ex?-?.py -h, then you can know how to change the parameters.

Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

行列積表現の固有値問題への応用

Outline•

Application to Eigenvalue problem(Ground state of quantum many-body systems)••

Application to time evolution of quantum system••

Variational algorithm
TEBD algorithm

Application to Machine learning

Application to eigenvalue problem

Calculation of minimum (or maximum) eigenvalue
Target vector space:

Exponentially large Hilbert spacewith+Total Hilbert space is decomposed asa product of "local" Hilbert space.

Target matrix::Hermitian, square, and sparse(Typically, only O(M) ( =O(aN) ) elements are finite.)
Notice:We consider the situation where
we cannot store O(M) variables in the memory.

Problem:Find the smallest eigenvalue and its eigenvector

Variational calculation using MPS:
Cost function:
Find the MPS which minimizes F
by optimizing matrices in MPS.

Problem in graphical representation
Cost function:

Find

which minimizes F.

Iterative optimization(F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004))

Local optimization problem when we focus on a "site" i :iiRemove Ai

Minimize

Solve generalized eigenvalue problem(Find the lowest eigenstate)
Notice: matrix size =Iterative optimization(F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004))

Update Ais by "sweeping" sites i =1 to N…

Backward "sweeping" sites i =N to 1…

Repeat sweeping until convergence.

Compact representation of an operator
Notice!
We can conduct this algorithm when we can represent
the matrix efficiently.
We consider the situation where we cannot store the matrix in the memory.

In practical applications, we usually represent the matrix
in so called Matrix Product Operator (MPO) form.
=
E.g.

The Hamiltonian of the Heisenberg model isrepresented by MPO with bond dimension.

Relation to Density Matrix Renormalization GroupThe variational MPS method is essentially same with
Density Matrix Renormalization Group (DMRG) algorithm.
(S. R. White, Phys. Rev. Lett. 69, 2863 (1992))（密度行列繰り込み群）(U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005))
U. Schollwöck: Density-matrix renormalization group

DMRG selects compact basis
based on entanglement between"System" and "Environment" blocks.

(U. Schollwöck, Annals. of Physics 326, 96 (2011))

DMRG is a powerful tool in physics and chemistry• One-dimensional spin systems
266
U. Schollwöck: Densi• One-dimensional
electron
FIG. 1. Lowest-lying eigenstatesof !dashed systems
line" blocks A and
FIG. 2. System meets environment: Fundamental den!solid line" AA for the problem of a single particle in a box in• Smallmatrix renormalization-group !DMRG" construction of athe molecules
continuum limit.
perblock from two blocks and two single sites.
• Small two-dimensional systems
cretization can be neglected, the lowest-lying states of A
all have nodes at the lattice ends, such that all productstates of AA have nodes at the compound block center.
The true ground state of AA has its maximum amplitude right there, such that it cannot be properly approximated by a restricted number of block states.
Merely considering isolated blocks imposes wrong

The original DMRG did not use MPS explicitly.
But, MPS gives us a theoretical backgroundfor why DMRG works well.

Because the final goal is the ground state in the tmodynamic limit, one studies the best approximatio
it, the ground state of the superblock, obtained bymerical diagonalization:
MS Nsite Nsite

$"% =

ME

) ) ) ) "m ! ! m $mS!S%$mE!E%
S S E

E

n!1
with # minimum, which in turn gives the energy at thisstep. Then one chooses another site and proceeds in theand (ii) for each k > M other four similar operators whichsame way until the energy converges. At the end we have"k# to E"N# . With them, one cancontainproducts
fromE
all the Pk and can evaluate all expectation values.
build HM and N0 by few matrix multiplications and thus
Let us now explain how one can make the method
determine PM by solving (12). From it, QM is determined.
efficient. Let assume that we have a set of spins in a
Then, we construct rM(1 , sM(1 , tM(1 , and hM(1 starting
ring. The idea is to determine operators Pk in a clockwisefrom rM , sM , tM , and hM . We continue in the same vein,order (first P1 , then P2 , until PN*1 ), then improve themfinding four matrices at each step, and storing them, until
following a counterclockwise ordering (from PN to P2 ),we reach N. Then we start moving counterclockwise andthen again clockwise, until the fixed point is reached. At
start constructing the corresponding four matrices at
each step, a normalization condition similar to (3) iseach step. Notice that in order to construct the matrices
imposed, depending on whether we are in a clockwise
HMD.andN0 weto usePhys.
the Rev.
storedmatrices(15)(2004))
which
(F. N
Verstraete,
Porras,and have
J. I. Cirac,Lett.
93, 227205
or counterclockwise cycle, which makes the matrix
M
were determined when we were moving clockwise. Thus,
well behaved. On the other hand, at each step only thewith this procedure
we have to store of the order of 4N
open:
operators Accuracy
which are strictly
needed
in
later
steps
are
2
becomesworse
if
wematricesofdimensionD. Using sparse matrix multicalculated in an efficient way and stored.
plications to solve the eigenvalue problem [17] and tosystems
The normalizationconsider
condition is basedon the following
update the matrices rM ; sM ; tM ; hM , the number of operafact. Given the state !, characterized by matrices B, if we
5
with
periodic
condition.
tions
per
step
scalesasD"M#;s
"M#;s boundary"M#;s"M(1#;speriodic: and is independent of N. At thesubstitute B!B
X :! Uand B!
end, when we have reached the fixed point, we can deterX*1 B"M(1#;s , where X is a nonsingular matrix, we obtainmine the expectation value of any operator by using (8)"M#;s!
the same state. Analogously, we can substitute B
S=1/2 Heisenberg chain, (N=40)
YB"M#;s :! V "M# and B"M*1#;s ! B"M*1#;s Y *1 . We choose
X in the clockwise cycles to impose (3) and Y in the-4
DMRG (PBC)
10counterclockwise
ones
to
imposeIf we use periodic MPS instead

Relation to Density Matrix Renormalization Group

10

-10

C)

B
(P

FIG. 3.

0

20<mSi S i+1 > /E0 - 1

-8

)
BC
(O

10

ew

~ M(1 ))) + Q~ N j$i +N ; (14)j!i ! Q1 + )))QM*1 + PM + Q~ k are defined as in (7) but with U and V
where Qk and Q
instead of B, respectively. Thus, the operators X and Y are-6

N

Thus, at the point of determining the operator PM ,

10

RG
M
D

d
of openX
MPS,
we can represent&
V "M#;s 'y &V "M#;s ' ! 1:
(13)
s!1
the groundstate more efficiently.

∆E 0 /|E0|

Conventional DMRG algorithm corresponds tovariational calculation using open boundary MPS.

-7

10
0

-7

10

0 10 20 i
40

60

Left: comparison between DMRG (squares) [2] andApplication to time evolutions of quantum system

Time evolution of a quantum system
Schrödinger equation:
Formal solution:
Time evolution operator（時間発展演算子）

Time evolution using MPS:
1. Multiply the time evolution operator to a MPS.
2. Find an approximate MPS representation for it.
When the time step (t) is small,
we can perform the above step efficiently.

Time evolution of a quantum system using MPS
Target: (Basically) one-dimensional quantum systemwith short range interaction
Typical example: Chain of qbits or quantum spins
Transverse field Ising model

Heisenberg model

Typical situation: Quantum quench
Initial state: Ground state of a Hamiltonian which well approximated by MPS
t > 0: Hamiltonian suddenly changes from the initial one.
For a "short" time interval, evolving state isapproximated by MPS efficiently.

Suzuki-Trotter decomposition
Suzuki-Trotter decomposition: (M. Suzuki, Commun. Math. Phys. 51, 183 (1976))
Systematic approximation of exponential operator(1st order)(2nd order)(2nd order)
If our Hamiltonian is represented as a sum of "local" operators,
E.g. transverse field Ising modelTime evolution operator can be approximated as(1st order)Graphical representation of Suzuki-Trotter decomposition
Suppose the Hamiltonian can be decomposed
into the sum of two-body local terms

Suzuki-Trotter decomposition of time evolution operator(1st order)

Multiplication of time evolution operator
If we have MPS representation ofMultiplying the time evolution operator is represented as

If we can perform the transformation(Generally, all matrices change
for better approximation)

We continue the time evolution repeatedly.
Notice: we want to keep the bond dimension χ constant along time evolution.

TEBD algorithm:(G. Vidal, Phys. Rev. Lett. 91, 147902 (2003))

Time Evolving Block Decimation (TEBD)
We can perform the accurate transformation locally by using canonical MPS.

ApproximationT̃1 ˜ 2 T̃2Only the two matrices which are
directly applied TE operator
changes in MPS.

TEBD algorithm:(G. Vidal, Phys. Rev. Lett. 91, 147902 (2003))

Apply TE operator⇥
Combineandmake matrix

SVD
˜2
V†

U

ApproximationT̃1=T̃1 ˜ 2 T̃2

1

1

Truncation

U

˜2
U

T̃2

=

3

U

1

Make tensor

V†Why TEBD is accurate?
1. For accurate calculation, the canonical form is important.
If λ is equal to the Schmidt coefficient, it contains
all information of the remaining part of the system.
Truncation based on local SVD can be globally optimal,
even if we look at a part of the MPS.

2. If the operator is unitary, MPS keeps canonical form within truncation error
If we chose the initial MPS as the canonical form,
TEBD algorithm almost keep it.
(So, TEBD is almost "globally optimal")Extension to infinite sytem iTEBD:
Finite system: TEBD
Sequentially apply ITE operators(G. Vidal, Phys. Rev. Lett. 98, 070201 (2007))(R. Orús and G. Vidal, Phys. Rev. B 78, 155117 (2008))

SVD for each step

Infinite system: iTEBD
Due to the translational invariance,
all SVD are equivalent.

SVD for each step

*Note
Because of SVD in iTEBD algorithm, we need at least two independent
tensors even in translationally invariant system…

…

(i)TEBD can be used for eigenvalue problem
Method to optimize MPS for GS of a specific Hamiltonian
1. Variational optimization
Change matrix elements to reduce the energy:
2. Imaginary time evolution
Simulate imaginary time evolution:（虚時間発展）
For a initial state
By replacing the time evolution operator to
the imaginary time evolution operator,
We can use (TEBD) algorithm for eigenvalue problem.

Difference between TE and ITE:Time evolution operator is unitary:Imaginary time evolution operator is not unitary
In general, by multiplying imaginary time evolution operator to MPS,
the canonical form is destroyed and TEBD becomes less accurate.
However, when τ is small the operator is almost unitary.
(Because it is almost identity matrix)
If we chose the initial MPS as the canonical form,
TEBD algorithm almost keep it.
(So, TEBD is almost "globally optimal" even in thecase of the imaginary time evolution.)*Instead, we can transform the MPS into the canonical form
after multiplying ITE operator in every steps.

3-1: Energy dynamics in TEBDExercise 3: (TEBD and) iTEBD simulation (ITE)
3-1: TEBD simulation
Simulate small finite size system and compare energy with ED
Sample code: Ex3-1.py or Ex3-1.ipynb
show help: python Ex3-1.py -h

3-2: iTEBD simulation
Simulate infinite system and calculate energy
Sample code: Ex3-2.py or Ex3-2.ipynb
show help: python Ex3-2.py -h* Try simulation with different "chi_max", "T_step"Requirement for running sample scripts
Python environment: python2.7 or python3
Modules: numpy, scipy and matplotlib
Usage:For jupyter notebook, typejupyter notebook
and select Ex?-?.ipynb .
For python (command line), typepython Ex?-?.py -h, then you can know how to change the parameters.

Application to machine learning
E. Miles Stoudenmire and D. J. Schwab, NIPS 2016

Machine learning for classification problem
Problem: we want to classify an input vector by several labels
E.g Classification of handwriting images
Standard procedure:First, input vector
is mapped onto higher dimensional space(non-linear feature map)
Then it is transformed to labels through a linear map

In the case of supervised machine learning, we optimize W based on thecorrect labels of a lot of input vectors.

MPS representation of the classification problem
E. Miles Stoudenmire and D. J. Schwab, NIPS 2016

If we select a "product state" as a feature map

The dimension of vector space isThen we can apply MPS approximation for W

W

discussed previously [38, p. 193] and have been argued
The tensor s1 s2 ···sN is the tensor product of the same
to be useful for data where no relationship is assumed
local feature map sj (xj ) applied to each input xj , where
between di↵erent components of the input vector prior
the indices sj run from 1 to d; the value d is known as
to learning [39].
the local dimension. Thus each xj is mapped to a dThough we will use only the local feature map Eq. (3)dimensional vector, which we require to have unit norm;in our MNIST experiment below, it would be interesting
this implies each (x) also has unit norm.
to try other local maps and to understand better the role
The full feature map (x) can be viewed as a vector
2 the costthey play in the performance of the model andin a dN -dimensional space or as an order-N tensor. The
Miles Stoudenmire
and D. J. Schwab, NIPS 2016of E.
optimizing
the model.
s1 s2 s3 s4 s5 s6tensor diagram for (x)Representing
is shown thein weights
Fig. 2.W Thisof Eq.type(1) as ofan MPS allows since
us to efficiently
optimize thesetensor is said be rank-1
it is manifestly
theweights
prod-and adap= s
tively change their number by varying W locally a few
s2 s3 s4 s5 s6
uct of N order-1 tensors.
Inatphysics
terms,(x) tohas
tensors
a time, inclose analogy
thethedensity ma- III. MULTIPLE1 LABELCLASSIFICATION
same structure as a productstate or unentangled
trix renormalizationgroup algorithm waveused in physics[26, 35]. Similar alternating least squares methods for
FIG. 2. Input data is mapped to a normalized order N tensor
function.
tensor trains have also been explored in applied mathe-In whatwith a trivial(rankwe1) productstructure.
follows
are interested
in multi-class learnFor a concrete example
of
this
type
of
featuremap,
matics [36].
ing, for which we choose a “one-versus-all” strategy,Thispaper is organizedfollows:Nwepixels,
propose our genconsider inputs which aregrayscale
imagesas withwhich we take to mean generalizing the decision funcapproachfrom
then describe
algorithm
where each pixel valueeralranges
0.0 foranwhite
to for
1.0optimizing
the weight vector W in MPS form. Wetion Eq. (4) to a set of functions indexed by a label `thtest our approach,for black. If the grayscale
pixel
value
of
the
j
pixel
both on the MNIST handwritten digit set
and on twotoy data
better feature
understandmap
the role of theis xj 2 [0, 1], a simpledimensional
choice forthetolocalf ` (x) = W ` · (x)(4)
local feature-space dimension d. Finally, we discuss thesj(xTheirj ) isfeature classmap:of functions realized by our proposed models as wellasapossibleh⇣ ⇡ ⌘generative⇣ ⇡interpretation.
⌘iand classifying an input x by choosing the label ` forThose wishing to reproduce our results can findsj(xj ) = sample
cos codes
xj ,basedsin on xthe(3) [37] whichj ITensor library|f ` (x)| is largest.
at:22
https://github.com/emstoudenmire/TNML
Since we apply the same feature map
to all input
FIG. 3. For the case of a grayscale image and d = 2, eachtheis mappedcasegrayscale
image
and is illustrated in Fig. 3. The full image is represented
data, For
thevalue
only
quantitythat
depends
on vector.
the label ` is
pixel
to of
a normalized
two-component`
The full image
is mapped
the tensor productall thelocal W ` as a
as a tensor product of these local
vectors. Froma physics
the weightvector
W . to Though
one ofcan
view
II. ENCODING
INPUT
DATA
pixel vectors as shown in Fig. 2.
Application
to MNIST
database
handwritten
sjperspective,is the normalized
wavefunctionof aofsincollection digitsof vectors labeled by `, we will prefer to view
The most
use of tensor
gle qubit where the “up”
statesuccessfulcorresponds
to anetworks
whitein physics
W ` as an order N +1 tensor where ` is a tensor index and

MPS representation of the classification problem
Feature map(handwritten numbers
fromstate
0 to
pixel, the “down”
to a 9)black pixel, and a superpo-

so far has been in quantum mechanics, where combining
N independent systems corresponds to taking the tensor
product of their individual state vectors. With the goalof applying similar tensor networks to machine learning,
we choose a feature map of the form

sition corresponds to a gray pixel.
While our choice of feature map (x) was originallymotivated from a physics perspective, in machine learning terms, the feature map Eq. (2) defines a kernel whichis the product of N local kernels, one for each component xj of the input data. Kernels of this type have been
discussed previously [38, p. 193] and have been argued
to be useful for data where no relationship is assumed
between di↵erent components of the input vector prior
to learning [39].
Though we will use only the local feature map Eq. (3)in our MNIST experiment below, it would be interestingχTest set error
10~5%500/10000s s ···s
s
s
s
(x) =
(x1 ) ⌦
(x2 ) ⌦ · · ·(xN ) . (2)20~2%200/10000s s ···s
The tensor
is the tensor product of the sameslocal feature map(xj ) applied to each input xj , where120~0.97%
97/10000
1 2

1

N

1 2

2

N

N
j

the indices sj run from 1 to d; the value d is known as
the local dimension. Thus each xj is mapped to a ddimensional vector, which we require to have unit norm;
this implies each (x) also has unit norm.

It is comparable withthe state of the art!
<1%

We can enjoy demo: http://itensor.org/miles/digit/index.html

References for application to machine learning
Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale
Optimization Problems: Perspectives and Challenges PART 1
A. Cichocki, N. Lee, I.V. Oseledets, A.-H. Phan, Q. Zhao, D. Mandic
Foundations and Trends in Machine Learning, vol. 9, no. 4-5, pp. 249-429, 2016

(arXiv.1609.00893)

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations.
Part 2 Applications and Future Perspectives
A. Cichocki, A-H. Phan, Q. Zhao, N. Lee, I.V. Oseledets, M. Sugiyama, D. Mandic
Foundations and Trends in Machine Learning: Vol. 9: No. 6, pp 431-673, 2017 (arXiv.1708.09165)

Topics:• Supervised Learning with Tensors• Tensor Train Networks for Selected Huge-Scale Optimization Problems• Tensor Networks for Deep Learning• ...

Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

テンソルネットワーク表現への発展Breakdown of MPS representation

Required bond dimension
in MPS representation

The upper bound is independent of the "length".
length of MPS

EE of theoriginal vector

size of the problem

Required bond dimension
in MPS representation

Phase transition
Transverse field Ising chain:
1

2

3

…

Ground state:Ferromagnetic state:Disordered state(Field induced ferro)

In between these two limits,
there is a phase transition.
At the phase transition,order parameter becomes zero.
（秩序変数）(Spontaneous)
Magnetization（自発磁化）

Ferro

DisorderN-1 N

Critical point and correlation length:Critical point （臨界点）
Behavior of a correlation function:

Ferro

Disorder:Ferromagnetic state:Disordered state:Critical point

Correlation length diverges at critical point!

:Correlation length (相関長)

FerroScale invariance at the critical point:Critical point （臨界点）
Power low decay!
After a scale transformation

Change in the correlation function is only a constant factor.
If we scale spins as
the correlation function becomesThis property is called as "scale invariance".（スケール不変性）
Physics (properties) in different scale is essentially same.

DMRG (variational MPS) calculation of TFI model
FIG. 1. Relative errors of the ~a! ground-stateenergy, and ~b! first excited-state energy as afunction of the chain length N for different valuesof M . Data obtained by the infinite lattice algorithm (I51) in the disordered regime at g 52.

Ö. Legeze, and G. Fáth, Phys. Rev. B 53, 14349 (1996)

Errors of the ground and the 1st excitedstates energies varying system size N.
For a fixed dimension m,

Until the algorithm keeps all states, i.e., for<2(lnM11), the DMRG is exact and only the machine’s
merical inaccuracy is seen, limiting the precision on theder of 10214. For longer chains, however, the errors assoted with the reduction of degrees of freedom also come in.
For the off-critical values, g 52 and 0.5, the ground-stateor d E GS shows practically no size dependence. It depends,
wever, crucially on the value of M . The behavior is theost clear for g 52, where the infinite system ground state isique with a gap above it. Even a small value of M ,;10, is enough to reach the machine’s precision limit
g. 1~a!#. There is, on the other hand, an interesting size
pendence in the error of the first excited state d E 1XS , escially for greater M values @Fig. 1~b!#. A maximum
olves, above which d E 1XS begins to dwindle again. Thiscan be understood, however, since the approximate excitedstate, yielded by the DMRG process, is not necessarily exactly orthogonal to the real ground state, and can have afinite overlap with it. This fact leads to an overcompensationof the otherwise increasing positive absolute error for long
chains.
For g 50.5, where the ground state is asymptotically doubly degenerate, the behavior is more subtle @Figs. 2~a,b!#.
The splitting of the two lowest levels decreases exponentially as N increases. When this difference goes below themachine’s precision limit ;10214, the algorithm is incapable to further resolve the two levels, and the target statethat it yields is a linear combination of the two exact eigenvectors with random coefficients. As a consequence, in eachstep of the DMRG algorithm the target state changes unpre-

Ferro

DisorderFerro and disordered states:
The errors are almost independent of N.
Critical point:
The errors gradually increases as increase N.
14 354

53
ÖRS LEGEZA AND GÁBOR FÁTH

FIG. 2. Same as Fig. 1, but in the orderedregime at g 50.5. Curves labeled by TS52 show
the case, when the excited state was targeted together with the ground state.

ACCURACY OF THE
53 DENSITY-MATRIX RENO

FIG. 3. Same as Fig. 1, but at the critical pointg 51. Curves with M 54,8,16 are dominated by
the truncation error, while those with M 548 by
the environment error. The M 532 curves show acrossover between the two types.

Entanglement entropy of TFI model
Entanglement entropy:A

EE of theoriginal vector

State

B

Required bonddimension

Ferro or
Disordered

CriticalWe need polynomially large bond dimension for critical system!
More efficient tensor network for critical systems?
Key point: Scale invariance of the system

Higer dimensional system
Transverse field Ising model on square lattice:Two-dimensional array

Ly:Summation over thenearest neighbor pair

Area law
Even in ferro and disordered phases,
the entanglement entropy depends on size N.

Lx
Phase diagram

Ferro

DisorderMPS for two-dimensional system
When we apply MPS representation for a square lattice system:
Setting (1):Satisfying area law?

Setting (2):Break down of the area law!
MPS cannot cover the area law of the entanglement
entropy in higher (d =2,3, ...) dimensions.

Possible MPS(Snake form)

Two settings of system and environment
(2)
(1)
A'
B'
A

Ly
B
Lx

MPS for two-dimensional system: comment
MPS can treat "rectangular" or "quasi one dimensional" lattice.
In setting (1), MPS can satisfy the area low partially.
We can increase Lx easily with keeping Ly constant.

(1)
A

B
Quasi one dimensional system ("strip" or "cylinder")

(2)
A'

Ly

Lx

B'

Tensor network for critical systems:
Multi-scale Entanglement Renormalization Ansatz

Hierarchical structure: tree tensor network
Critical system

Scale invariance

A simple scale invariant tensor network: tree tensor network
Tree tensor network state"scale"(renormalization)

MPS

Notice:Unitary tensors
Unitary tensor

k

l

i

j

k

l

k

l

k'

l'

k'

l'

i

j

i

j

i'

j'

i'

j'

Isometric tensors
Isometric tensor (half unitary tensor) = Isometry
k

k

k'

k'

k

i

j

Unitarity condition only for "bottom" legs.
Isometry works as a "projector" from the bottom space to the top space.

dim(bottom)

dim(top)

It is also related to the "renormalization" of degree of freedoms.
We pick up "important" degree of freedoms by isometries.

Isometric tree tensor network and its scale invariance
Consider an (infinite) tree tensor network consistsof identical isometries as a wave function.
Properties:
1. It is normalized as(Trivial from the definition of the isometry)

2. It can be scale invariant.

spin"renormalized" spin

If

, thenScale invariant!

Entanglement entropy of TTN
Entanglement entropy of tree tensor networks (TTN):
Due to the tree structure, two regions are connected by only "one bond".
(or a few)

Cut here!
Cut here!

A A A A A A A A B B B B B B B B

A A A A B B B B B B B B B B B B

MERA(G. Vidal, Phys. Rev. Lett. 99, 220405 (2007))(G. Vidal, Phys. Rev. Lett. 101, 110501 (2008))

Multi-scale Entanglement Renormalization Ansatz (MERA)
Before applying isometry, insert a unitary tensor.
Isometry:

TTN

MERA

Unitary tensor:

Normalization
Scale invariance (if we set the identical tensors)

Entanglement entropy of MERA
Due to the unitary matrices, # of bonds connecting two regions
logarithmically increase.
Nc (N )

log N

rank ⇢A ⇠
SA = Tr ⇢A log ⇢A  (log ) log N

A A A B BB B B B B B B B B B B
Minimum # of cuts = 2

Nc (N )

# of minimum cut
for a N-site region

A A A A A A A B B B B B B B B B
Minimum # of cuts = 3

,-'r( 'r+1(
M x,3−a ,HPotts = & &M z'r(+ & M x,a$76%

HXX = & $''xr(''xr+1(+ ''yr(''yr+1(%,

$77%

r

a=1,2

Application of MERAr
Transverse field
Ising chain:

HHeisenberg = & $''xr(''xr+1(+ ''yr(''yr+1(+ 'z'r('z'r+1(%, $78%
r

MERA can representvery large (Infinite)critical system!

Energy errors:

where 'x, 'y, and 'z are the spin-1/2 Pauli matrices andare the matrices
M x,1, M x,2, and M zMERA(Infinite chain)(G. Evenbly and G. Vidal, Phys. Rev. B. 79, 144108 (2009))

14 354DMRG (finite chain)

ÖRS LEGEZA AND GÁBOR FÁTH

Ö. Legeze, and G. Fáth (1996)

dictably, leading to the loss of optimality of the truncation
process, and hence, to considerably higher, rather scatterederror rates. As Fig. 2~b! shows, the DMRG can even ‘‘lose’’

curves, where
since the trunc
tude in the sm

Interesting topics related to MERA•

By using scale invariance of MERA, we can calculate properties of critical system(G. Evenbly and G. Vidal, Phys. Rev. B. 79, 144108 (2009))
accurately.
(R.N.C. Pfeifer, (G. Evenbly and G. Vidal, Phys. Rev. A. 79, 040301(R) (2009))••

Critical exponents and Operator product expansion coefficients in the Conformal
Filed Theory (CFT)

We can consider MERA in higher dimensions•

It is scale invariant but satisfies the area low(G. Evenbly and G. Vidal, Phys. Rev. Lett. 102, 180406 (2009))•

For the system with logarithmic correction in the EE, such as metal, "branching
MERA" has been proposed.
(G. Evenbly and G. Vidal, Phys. Rev. Lett. 112, 220502 (2014))(G. Evenbly and G. Vidal, Phys. Rev. B. 89, 235113 (2014))•

Relation between MERA and other fields•

Wavelet transform•

AdS/CFT (quantum gravity, black hole)(G. Evenbly and S. R. White, Phys. Rev. Lett. 112, 140403 (2016))(M. Nozaki, S. Ryu, and T. Takayanagi, J. High Energy Phys. 10, 193 (2012))

Tensor network for higher dimensional systems:
Tensor Product State(Projected Entangled Pair State)

Entanglement entropy in higher dimensions
Ground state wave functions:

B

For a lot of ground states, EE is proportional to its area.
J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys, 277, 82 (2010)

Area low:LA

In d=1, MPS satisfies the area low.
Q. What is a simple generalization of MPS to d > 1?
A. It is Tensor Product State (TPS)!

Tensor Product State (TPS)

TPS on square lattice

TPS (Tensor Product State) (AKLT, T. Nishino, K. Okunishi, …)
PEPS (Projected Entangled-Pair State)(F. Verstraete and J. Cirac, arXiv:cond-mat/0407066)

d-dimensional tensor network representation
for the wave function of a d-dimensional quantum system
Tensor = Projector
Tr: tensor network “contraction”: Rank 4+1 tensor

x,y,x’,y’ = 1,2, ... D
mi= 1,2, ... m

D = “bond dimension”
m = dimension of the local Hilbert space*D can be larger than m. “Virtual state “Maximally entangled state
between D-state spinsEntanglement entropy of TPS (PEPS)
Bond dimension = DAB# of bonds connecting regions A and B(square lattice)
Nc (L) = 4L
(d-dimensional
Nc (L) = 2dLd 1
hyper cubic lattice)rank ⇢A  D

SA =Nc (L)⇠D

2dLd

Tr ⇢A log ⇢A  2dL

d 1

1

log D

TPS can satisfies the area law even for d >1.

We can efficiently approximate vectors
in higher dimensional space by TPS.
* Similar to the MPS in 1d, TPS can approximate infinite system!

may be found in Kitaev (2003).
We consider a square lattice, possibly embedded into a nontrivial surface su
a torus, and place spins on the edges, as in Fig. 3.1. The Hamiltonian is given b
HT = −Je!
s

As − Jm

Example: Ground state represented by TPS

!

Bp

p

where s runs over the vertices (stars) of the lattice and p runs over the plaqu
The star operator acts on the four spins surrounding a vertex s,

The two-dimensional toric code

Toric code model(A. Kitaev,!Ann. Phys. 303, 2 (2003).

σjx

As =

(3.2)

j∈star(s)nal toric code
while the plaquette operator acts on the four spins surrounding a plaquette,
!
!
Bp =
σjz .
σjx
(3.2)
As =

Bp

(3.3)Asj∈∂pj∈star(s)operator acts ons thespins surrounding
plaquette,
Itsfourgroundstate is soa called
Zp2 spin liquid state.
Fig. 3.1 A piece of the toric code. The spins live on the edges of the square lattice!
As Bp = Bp As(3.4)
spins adjacent to a star operator
A and a plaquette operator B are shown.
z"Spinliquid"
is
a
novel
phase
different
fromconventional
magnetic orders.
Bp =
σj .
(3.3)
Clearly, the A all commute with one another, as do the B . Slightly less trivially,spbecause any given star and plaquette share an even number of edges (either none orj∈∂pxztwo) andIttherefore
minus signs arisingcommutation
of
σ
and
σ
on Rev. Lett. 96, 220601 (2006).
can betherepresented
by from
D=2theTPS.
(F. Verstraete, et al, Phys.
those withedges onecancel.
Since allof HT commute,we expect to be able toommuteanother,
as of
dothe
theterms
Bp . Slightly
less trivially,solve it term by term.
0,1: eigen
state(Non-zero elementsof tensor)
As BweBof
Aσs x HT working(3.4)
In particular,
will
solve
in the σ z basis. Define
classical variablesp =psj = ±1 to label
the σ z basis
For1each1classical0spin0configuration {s}, we can00
1 states.
1
tar define
and plaquetteshare
an even0 number of 1edges (either1 none or
the plaquette
0 flux!
00
1 σ x and σ z on
the minus signs
arising 1from the w
commutationofsj . , 1(3.5)
p (s) =001

Since all of the terms of HT commute, we
expect to be able toj∈∂p0
0 that there
1
1 a vortex1 on plaquette0m. If wp = −1, 0we say
is
p. 1
0
0z
1
1variables
will solve H0T working
in
1 the σ basis.
1 Define classical0101{s}, we canstatesσ z 3.1
basis Groundstates.0For
each classicalspin configuration"Difference between MPS and TPS
Cost of tensor network contraction:
d-dimensional cubic lattice

MPS:
TPS:

N =L

d

O(N )
O(e

L

d

1

)

It is impossible to perform exact
contraction even if we know
local tensors in the case of TPS.

In the case of TPS,
usually we approximately
calculate the contraction.

MPS

TPS
(PEPS)Example of approximate contraction: CTM method
For (infinite) open boundary system
Infinite PEPS

(T. Nishino and K. Okunishi, JPSJ 65, 891 (1996))(R. Orus et al, Phys. Rev. B 80, 094403 (2009))

Environment

Corner transfer matrix
Representation(with a translational invariance)

Environment

C

e

e

e
C

C

e

Corner transfer matrix

C

Edge tensor

Double tensor=
Mapping to a "classical" systemeC=bond dimension

Cost of (approximate) contraction
MPS:

TPS:
When we use CTM environment in 2D,Ce

We can treat very small bonddimensions in TPS!

=

Application of TPS to eigenvalue problem
For calculation of minimum eigenvalues and its eigenvector,
we can use similar techniques to those in MPS
Variational method:(P. Corboz, Phys. Rev. B 94, 035133 (2016))(L. Vanderstraeten, Phys. Rev. B 94, 155123 (2016))

minimize cost function:
Imaginary time evolution:
Simulate imaginary time evolution:（虚時間発展）
For a initial state(H. G. Jiang et al, Phys. Rev. Lett. 101, 090603 (2008))(J. Jordan et al, Phys. Rev. Lett. 101, 250602 (2008))

Example of application: Honeycomb lattice Kitaev
Model
A. Kitaev, Annals of Physics 321, 2 (2006)

Kitaev modelx-bond:bond direction

y-bondDepending on the bond direction, only
specific spin components interact.

Exactly solvable by introducing Majorana fermion

Phase diagram

Isotropic region (B) : gapless spin liquid
Anisotropic region (A) : gapped spin liquid
gappedCf. The anisotropic limit corresponds to the Toric code.
*Recently, researchers have realized that thistype of models might appear in real materials.
Hot topic!

gapless
gappedgappedz-bondx-bond

Application : Kitaev spin liquidz-bondy-bondHoneycomb lattice Kitaev modelAt, the ground state isa gapless spin liquid.

In the present (super)computers,
we can access around D=10 (maybe 16)
by using massively parallel code.

Energy error obtained by iTPS
(T. okubo et al, unpublished)

0.009
0.008

full update

0.007
0.006

The error of the ground stateenergy is less than 10-3
for infinite system!

0.005
0.004
0.003
0.002
0.001

iTPS can represent Kitaev spin liquidin the thermodynamic limit accurately.

0

0

0.05

0.1

0.15

0.2

0.25

Interesting topics related to TPS•

Application to itinerant electron system, which may break the area law(P. Corboz et al, Phys. Rev. B. 81, 165104 (2010))(P. Corboz, Phys. Rev. B. 93, 045116 (2016))•

Characterization of topologies in wave function•

Symmetric tensor network and modular matrix(J.-W. Mei et al, Phys. Rev. B. 95, 235107 (2017))•

Application to three dimensions•

So far, there is no practical calculations for non-trivial models.

•Mainly, due to the scaling:Tentative lecture schedule

1日目

2日目

3日目
Optional

1.現代物理学における巨大なデータと情報圧縮
2.格子スピン模型の統計力学
3. 線形代数の復習
4. 特異値分解と低ランク近似
5. テンソルネットワーク繰り込みによる情報圧縮
6. 情報のエンタングルメントと行列積表現
7. 行列積表現の固有値問題への応用
8. テンソルネットワーク表現への発展
9. フラストレート磁性体への応用

フラストレート磁性体への応用（optional: 時間があれば。）

フラストレートレーション
複数の最適化条件を同時に満たせない状態
あちらを立てればこちらが立たず磁性体におけるフラストレーション
最適化条件：エネルギーを最小にしたい
反強磁性

局所的なエネルギー最小：各ペアでスピンを反平行
フラストレートスピン系
イジングスピンベクトルスピン・古典スピン模型：•••?
全ての辺で同時にスピンを反平行に出来ない

大規模な基底状態の縮重
弱い摂動による新規秩序
傾いた磁気秩序・量子スピン模型：
傾いた磁気秩序（120 構造）

•
•

スピン液体
隠れた秩序、トポロジー

フラストレート量子スピン系の数値計算
フラストレート量子スピン系における数値計算の課題
熱力学極限に迫れる手法が少ない•••

量子モンテカルロ法： 負符号問題のため、精度が出ない
厳密対角化法： 厳密だが、少数クラスターしか取り扱えない
変分モンテカルロ法： 大きな系を扱えるが、バイアスが存在

テンソルネットワーク法
変分波動関数として、（基底状態の）波動関数を“効率的”に表現したテンソルネットワーク状態を用いる
・フラストレーションがあっても問題なし！
・無限系の波動関数を直接取り扱える！

変分波動関数としてのTPS•

Area lawを満たせるので、ボンド次元Dを十分に大きい有限の値にとれば、多くの基底状態を表現できる•

並進対称性を仮定すれば、無限系でさえ有限のDで計算可能
並進対称性の例：•

テンソルの最適化、期待値の計算の手法がかなり発達してきた•

スピン液体相等の非自明な基底状態でもOK
e.g. ハニカム格子上のKitaev スピン液体
J. O. Iregui, P. Corboz, and M. Troyer, et al., PRB 90, 195102 (2014).

Application of TPS to eigenvalue problem
For calculation of minimum eigenvalues and its eigenvector,
we can use similar techniques to those in MPS
Variational method:(P. Corboz, Phys. Rev. B 94, 035133 (2016))(L. Vanderstraeten, Phys. Rev. B 94, 155123 (2016))

minimize cost function:
Imaginary time evolution:
Simulate imaginary time evolution:（虚時間発展）
For a initial state(H. G. Jiang et al, Phys. Rev. Lett. 101, 090603 (2008))(J. Jordan et al, Phys. Rev. Lett. 101, 250602 (2008))

Application1: Honeycomb lattice Kitaev Model
A. Kitaev, Annals of Physics 321, 2 (2006)

Kitaev modelx-bond:bond direction

y-bondDepending on the bond direction, only
specific spin components interact.

Exactly solvable by introducing Majorana fermion

Phase diagram

Isotropic region (B) : gapless spin liquid
Anisotropic region (A) : gapped spin liquid
gappedCf. The anisotropic limit corresponds to the Toric code.
*Recently, researchers have realized that thistype of models might appear in real materials.
Hot topic!

gapless
gappedgappedz-bondx-bond

Application1 : Kitaev spin liquidz-bondy-bondHoneycomb lattice Kitaev modelAt, the ground state isa gapless spin liquid.

In the present (super)computers,
we can access around D=10 (maybe 16)
by using massively parallel code.

Energy error obtained by iTPS
(T. okubo et al, unpublished)

0.009
0.008

full update

0.007
0.006

The error of the ground stateenergy is less than 10-3
for infinite system!

0.005
0.004
0.003
0.002
0.001

iTPS can represent Kitaev spin liquidin the thermodynamic limit accurately.

0

0

0.05

0.1

0.15

0.2

0.25

Eq. (3), the spin coupling, e.g., on a (x)-bond, is of S i S j% þ s2sin theverytype,
formpreciselyof the exchange
Hamiltonian
dependsonbondand excit
third
as in the Kitaev model. The honeycomb
geometry
profileof Kramersstates,as we pies, withlattice throughprovidesa adensity
particularly
striking
exampleof newHunddemonstratephysics below.
introduced by strong SO coupling: the[For ! ¼
andHeisenbergcouplings
model is ofconverted
into theKitaev states.—model
the 180!
Exchangeneighboring
KramersHamwith a spin-liquid
We consider
the limitgroundof the state.
strong spin-orbit coupling, i.e., canting aWhila physicalex-the is indepe
Theiscompound
Li2 RuO
3 [28] represents
when !
larger thanexchange
interaction
betweenfrom
distortion
ample of the A2 BO3 structure. By replacement of spin-one
isospins.
for isospins are then4þ The exchange Hamiltonians
4þ! intoSranwith spin-one-half Ir ions, one may realize aRu
2I
obtainedbyprojecting
thecorresponding
superexchangeAsstrongly spin-orbit-coupled Mott insulator with low-energy
theinmspin-orbital
modelsonto
theisospinstatesEq.
(1).
First,
we
Hamiltonphysics described by the Kitaev model.
mode
present‘‘Weak’’
thepresults
for
the
case
of
cubic
symmetry
(!
¼
0,
IrO
.—As
an
exampleofferromagnetism
of
Sr
2
4
S~~i
model
ﬃﬃﬃ
enou
Mott insulator,
discuss
layeredsin"a¼spin-orbit-coupled
1= 3), and discusslater the weeffectsofthe
a tetragonal
gered
rot
pling
of theundopedhigh-Tc
compoundSr
2 IrO4 , a ttwo
2g analog
distortion.
We
considercommoncases
of
TM-O-TMtanð2#ÞStrong spin-orbit couplingcally¼
Effectivebond``spin”
moment:" square lattice of Ir4þ ions is
cuprate
La
CuO
.
In
Sr
IrO
,
a
2
4
2
4
geometries: (A) a 180 -bond formed by corner- anisotrop! whformed by corner-shared IrO6 octahedra, elongated along
shared octahedra as in Fig. 2(a), and! (B) a 90" -bond Hamilton(B
the c-axis and rotated about it by ! ’ 11 [19] (see Fig. 4).
formedbyedge-shared
ones,
Fig.
2(b).
on a
a magnetic transition at # 240 K disSr2 IrO4 undergoes
G.Jackeli, et al., PRL 102, 017205 (2009) (A)A 180" bond: For this geometry, the nearest3
2
3
2 x
the
z z
S 1S 3 is diagonal in thezzorbital space
J. Chaloupka, et al., PRL 105, 027204 (2010)neighbor t2g
nond(a) hopping matrix(b)3and, on a given bond,only two orbitals are active, e.g., jxyi
a cha
z
xx
1.5
1
and jxzi orbitals along a bond in x-direction [Fig. 2(a)].
ThisThe spin-orbital exchange
Hamiltonian
for such a system
y
Ham
y y
Ir ions form an honeycomb lattice.
S 2S 3x x
x
S1S2
has alreadybeen reported: see Eq. (3.11) in Ref. [12]. After
omet
1
yy
projecting it onto the ground state doublet, we find an
gen i
2exchange Hamiltonian for isospins in a form ofof th
Ir - Ir direct exchange:
Heisenberg interaction
0.5
inter
FIG. 3 (color online). Examples of the structural units formed
ture(a)90! TM-O-TM bonds and corresponding(b)
spin-coupling patbyoterns. Gray circles
stand for magnetic ions, and small open
180
of0 tho
Ir - O - Ir exchange:
Anisotropic Kitaev interaction
circles denote oxygen sites. (a) Triangular 90unit
cell oftion
ABO2 -type layered compounds, periodic sequence of this unit
plane
forms a triangular lattice of magnetic ions. The model (3) on thisstructure is a realization of a quantum compassp model on a
FIG.With
4. T

A2IrO3

Iridium Oxides

φ/α

Na2IrO , Li IrOztriangular lattice: e.g., on a bond 1-2, laying perpendicular to
x-axis, the interaction is S x1 S x2 . (b) Hexagonal unit cell ofyz
A2 BO3 -type layered compound, in whichmagneticxzions(B-sites) form a honeycomb lattice. (Black dot: nonmagnetic
xy On an pxx-bond,xy
A-site).
the interaction is S xi S xj , etc. For thisystructure, the model (3) is identical to the Kitaev model.

Depending on the bond direction, only
specific spin component interact.

of the tetr
an IrO2 -pl
about z-ax
% ’ with'=5,
the in-planquan
z-axisocc

017205-3
xz

pz

xz

xz

yz
pz

intro
Jahn
studi
dege
How

ies experdiagonalization.
IrO3 , which includes
coux, y, and isotropicz axes in superexchange
defining the t2g -orbitals
areillustrated
rder[7, 8]. z axes in defining the t2g -orbitals,gs inandare illustrated
out of thehoneycomb
plane. The honeycomb
addition as
to directions
the Kitaev-type
anisotropic
nearestoneycomb
planeis thenperpendicular
toaxes(x, y,dependz) = plane.
(1, 1, 1). TheThe
dashed honeycomb
hbor Ising interactions
whose
anisotropy
irections
out
of
the
honeycomb
additional
anisotropic couplings asboundary
represents
a 24-siteout
clusterhedescribebond directions.
However,
it turned
that used
this later for the exact
eel isthenbe perpendicular
to (x, y, z) = (1, 1, 1). The!
dasheddiagonalization.
ange
cou! ⃗T ⃗cannotstraightforwardly consistent with theĤ = the exact Ŝ ℓ JΓ Ŝ m ,(1)
c nearestndary
a 24-siteclusteragorder represents
either. This discrepancy
inspiredfurtherused later for
Γ=X,Y,Z ⟨ℓ,m⟩∈Γesiesdepend
on suitableadditional
low-energy
eﬀective hamiltonians
anisotropic
couplings as foronalization.
thatwith
this A =Na or Li. First, models with further(Y. Yamaji et al. Phys. Rev. Lett. 113, 107201(2014))
rO
T
3
⃗
T
!
!
y
x
z
with couplingsthe⃗Addi-⃗= (Ŝℓ , Ŝℓ , Ŝℓ ) is a vector of SU(2) spin opwhere Ŝ ℓ (1)hbor[7, 8, 12,Ĥ13]
were studied. Ŝ
Ŝ
,
=
J
Γ
m
ℓ
d Isingfurthererators. The exchange couplings are given in matricesalanisotropy[14] due to Γ=X,Y,Z
a strong⟨ℓ,m⟩∈Γ
trigonal disonians forJΓ . The summations are over the nearest-neighbor pairs
on, which actually contradicts the distortions in the(1, 1, 1)
h furtherT⟨ℓ, m⟩. The group of bond Γ with Γ=X, Y and Z is de⃗ ab initio
wasalso
eriments [8] andintheyx treatments,zwhere Ŝ ℓ = (Ŝℓ , Ŝℓ , Ŝℓ ) is a vector of SU(2) spin opx-bondd. Addifined in Fig. 1. The exchange matrices are parametrized
z-bondmined. Quasimolecularorbitals[15],insteadofthezerators. The exchange couplings are given in matrices
gonal disasTmic orbitals assumed
intheKitaev-Hubbard
model
JΓ . The summations are over the nearest-neighbor pairs ⎡ons in the⎤⎡ ′ ′′ ′ ⎤xe claimed as a ⟨ℓ,properchoiceofthestarting
point.
So
J
I
I
K
I
I
y-bondm⟩. The group of bond Γ with Γ=X,
Y
and
Z
is
de1
2
Γ
m
, was also
22ℓyhe origin of the
zigzagtype
antiferromagnetic
order
′′
′′
⎣
⎦
⎣
JZ = I1 J I2 , JX = I2 J I1′ ⎦ ,fined in Fig. 1. The exchange matrices are parametrized
ad of therved for Na2 IrO
the possible⟨ℓ,m⟩∈Γ
route to realize
as 3 andΓ=X,Y,Z
I2 I2 KI2′ I1′ J ′rd model⎡ ′′ ′′ ′ ⎤quantum spin liquid are controversial.
⎡⎤⎡ ′ ′′ ′ ⎤J2point. So
J I2 I1
J I1 I2
K I2 I2
T
thisorder
Letter,we derive an ⎣ab initio spin
etic
′′
′′
′ ⎦ J = ⎣ I ′′ K ′ I ′ ⎦ ,⎦ , Jmodel
⎣for
(2)
J
=
I
J
I
=
I
J
I
, Y
Z
1
2
X
2
2
2
1
y
x
z
IrO
and
show
that
trigonal
distortions
presentin′′′to realize
3
I1 I2 J
I2 ℓI2 K
I2′ I1′ J ′ℓℓℓ
IrO3 in addition to the spin-orbit
holds the⎡ ′′ couplings⎤
J modelI2′′ forI1′ A2 IrO3 will
J3
where we choose a real and symmetric parameterizationThe
simplest
andrealistic
spin
model for
′′
′
′
⎦and
KitaevHeisenberg
likeJ(2) coupling
JYcoupling= ⎣ I2 K hamiltonian
IK
,
2
by
using
U(1)and
SU(2)-symmetryof electron wave
out to modify
the Kitaev-Heisenberg
bypresent
inI1′ I2′ J ′+holds theOﬀ-diagonal
couplings
I1 and I2
where we choosea real and symmetricparameterization
2 IrO3 willby using U(1)- and SU(2)-symmetry of electron wave
tonian by
It also contains J2 and J3 interaction term.

ab initio Hamiltonian of Na2IrO3itional anisotropic couplings as

ab initio Hamiltonian
Ĥ =

!

! ⃗
⃗
Ŝ J Ŝ ,(1)⃗
re Ŝ = (Ŝ , Ŝ , Ŝ ) is a vector of SU(2) spin opors. The exchange couplings are given in matrices
The summations are over the nearest-neighbor pairsm⟩. The group of bond Γ with Γ=X, Y and Z is ded in Fig. 1. The exchange matrices are parametrized⎡

J2 : only “z-bond” which is perpendicular to NN z-bond.

⎤

⎡

⎤

′
′′
′
three
third
neighbors
JJ3 : Iall1 ofIthe
K
I
I
2
2
2
′′
′′
JZ = ⎣ I1 J I2 ⎦ , JX = ⎣ I2 J I1′ ⎦ ,I2 I2 KI2′ I1′ J ′to the trigonal
distortion, the ab initio Hamiltonian contains strong
⎡ Due
⎤
′′
J ′′ I2oﬀ-diagonal
I1′
couplings, together with J2 and J3 interaction(2)
JY = ⎣ I2′′ K ′ I2′ ⎦ ,-6.19
-6.2
-6.21
-6.22
-6.23
iPEPS
-6.24
DMRG: Lx × 6 T. Okubo et al, PRB 96, 054434 (2017).
-6.25
ED
-6.26
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

Results: comparison with other methods
Energy (per sites)-6.17 (a)
-6.18
-6.19
-6.2
-6.21
-6.22
-6.23
iPEPS
-6.24
DMRG: Lx × 6
-6.25
ED
-6.26
0
0.05
0.1

0.15

0.2

0.25

Order parameter

0.3

0.4
(b)
0.35
0.3
0.25
0.2
0.15
iPEPS
0.1 DMRG: Lx × 6
0.05
ED
0
0
0.05
0.1

0.35

0.3
iPEPS:(1,1,0)(1,-1,0)(0,0,1)

0.2
0.1
0

0.15

0.2

0.1

0.25

Stability of the Zigzag(Z) state in the thermodynamic
limit is confirmed by iPEPS calculation.

0.4
(b)
0.35
0.3 of iPEPS, DMRG and ED are consistent.
Energies
0.3
0.25
iPEPS:
0.2
0.2• For 4 × 6 lattice, DMRG and(1,1,0)ED
0.15(1,-1,0)
iPEPS(0,0,1)
0.1
0.1givealmost
DMRG:
Lx × 6same energy.
0.05
0• Finite ED
D
of
iPEPS
and
finite0.1 Lx0.2of DMRG
0.3
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

are overlapped.

0.2

0.3

0.3

0.35

Zigzag(Z) order parameters are consistent.
•

Extrapolations of them are <M> ~ 0.3•

Spins are almost along (1,1,0) direction,
which is consistent with the experimental
observations.

Phase diagram varying the trigonal distortion
T. Okubo et al, PRB 96, 054434 (2017).
(a)Phase diagram-5

Zigzag-5.4 (X,Y) (Z)Zigzag (Z)16-sites

-5.2

-6

IC

iPEPS
DMRG

iPEPS
DMRG

-6.4
-20

0

20

40

(d)
120°/6-site(e-1) 16-site (XY)

0.01
0
60

DMRG: 6x8 cluster(e-1)6x10
16-site(XY)cells
iPEPS: 4x4, 2x6, 6x8, 8x12,
unit
•
•

120°/3-site
120°/6-site

0.03

120°/3-site

-6.2

-40

(c) (d)

0.02
(c)

-6.6
-60

Zigzag (X)120°/3-site

0.04

120°

-5.6
-5.8

0.05

(c) (b)(e-2) 16-site (Z)Energies obtained by iPEPS and DMRG are consistent
New phases are stabilized compared with the previous ED reports

(e-1)
(XY)
(e-2)16-site
16-site(Z)応用例2：Shastry-Sutherland 格子模型
P. Corboz and F. Mila, PRL 87, 115144 (2013)

S=1/2 Shastry-Sutherland 格子ハイゼンベルグ模型

Shastri-Sutherland 格子

PHYSICAL REVIEW B 87, 115144 (2013)

PHILIPPE CORBOZ AND FRÉDÉRIC MILA

基底状態相図Dimer phase

Plaquette phase(a) MPS

Néel phase

Cij32 j3
j1

ci1 i2 i3 i4 i5 i6

中間
相？

j2

j3

≈ A
B
C
D
PHILIPPE
CORBOZ AND FRÉDÉRIC MILA
波動関数のiTPS表現iii1 i2 i3 i4 i5 i6
(a)

1

i3

2

Setup O

Setup P

(b)

j4

i5

i4

(b) iPEPS
00 .675(2) 0 .765(15)DJ

FIG. 1. (Color online) The phase diagram of the ShastrySutherland model as a function of nearest-neighbor coupling J(J ′ = 1), obtained with iPEPS. The width of a bond is proportional
to the magnitude of the bond energy, where full (dashed) lines
correspond to negative (positive) energies. The arrows in the rightpanel illustrate the Néel order. In between the well-established dimer
and Néel phase we find a phase with plaquette long-range order.

無限系のTPSを用いて解明！

A

H

|Ψ

E

D
(c)

A

D
H

E

(d)

B
F

H

G

H

A

Setup C

E

D

C

E

D

G

A

H

C

F

D

G

B

E

C

F

A
Setup D

H

B

A
E

E

j5

F
i6

then acquire
matrix. Det
symmetries
For themethod expl
isometries b
a column (orand 32 for deand 36.

Aldru
p

Ourumain
Fig. 1: iPEPS
Adimerrphl theJ < 0.765(1p indivi
thedof
the phase tra

FIG. 2. (Color online) Graphical representation of an infinite
The dime
projected entangled-pair state (iPEPS) made of a 4 × 2 unit
cell

m

Es

For
finite Jm′ thesystemsuppressedis frustratedwith
becausethe spins
on it issetup P
However,is clearly
increasing
D and−0.3840.
setup D
diagonal
bond
would
like
to
be
antiparallel
to
each
other
very likely to vanish in all simulation setups in the large-D
setup C0insteadparallelas inthat
the Néelphase).
This
competitionof is −0.386
limit,ofwhichshows
the SU(2)spinrotationsymmetrynteraction
leadsintothea suppressionof the local
magnetic moment
not broken
plaquette phase,as expected.
−0.3880.
m. For Finally,
example,inforFig.
J =6(d)1 (Fig.
the local moment
is reduced as athe5)plaquetteorder parameter−0.39 and F. Mila, PRB 87, 115144 (2013)o mfunction= 0.21(1),further suppressedwith decreasing
J . Its P. Corboz
0
0.05
0.1
0.15
0.2
of Jand
forit isdifferentbond dimensions
D is shown.
1/D
The difference in bond energies !E 中間相での物理量
also vanishes in this case
エネルギー
Plaquette VBS 秩序パラメタ
自発磁化
n the large-D limit,
indicating the absence of translational
0.25
0
−0.38
0.25
(c)
(b)
(a)
ymmetry breaking.

応用例2：Shastry-Sutherland 格子模型

0.2

0.15
Plaquette phase

0.15

0.
∆E

0.2

∆E

Es

−0.384

setup O
setup P
setup DC.
setup Cm

−0.38200.1−0.3866 we present simulation results
0.1 for J = 0.7, where
In Fig.
0.
ll setups
consistently predict a plaquettephase for large bond
0.05
−0.3880.05
imension. The lowest variational energy obtained is EsD=10 =
0
−0.39
0
0.3881
with
setup
D.
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
1/D
1/D
1/D
The differenceinbondenergies
!E
in
Fig.
6(c)
clearly
PHILIPPE CORBOZ AND FRÉDÉRIC MILA
PHYSICAL REVIEW B 87, 115144 (2013)
emains finite in alliPEPSによる基底状態相図
simulation setups, and we find that the
FIG. 6. (Color online) Results for th
0.25
0.2
j j
(a) MPSC
trong bonds(c)form
plaquettes
as illustratedin(d)Plaquette
phase
NéelFig.
phase1 for large
Dimer phaseiin the plaquette phase. (a)–(c) Energy
0.2
中間相はD. The dependence
of !E on D varies
from one setup to ci i iand
j
j
j
j
j
differenceas
0.15≈ A in Bbond
C energies
D
E !EF
1 2 3 i4 i5 i6
nother;0.15
nevertheless, a rather large value between 0.1 anddimension, obtained for J = 0.7 with
i1
i2
i3
i5
i6
i4
i3 i4 i5 i6i1 i2Plaquette型のVBS状態.15 seems to be compatible with all simulation
setups.
Thelinearextrapolations
are
a
guide
to
0.1
D=6
0.1 magnetic moment in Fig. 6(b) is finite
D=7
The local
for small D,
parameter as a function of J in the pl
D=80.05(b) iPEPSsetup D.
which would
suggestcoexistenceofplaquetteandNéel order.
0.05
D=9
D
A
B
C
D
A
D=10 J
2 3

3

2

3

4

5

∆E

∆E

1

0 0
0
0.05

0.1

0.15
1/D

0 .675(2) 0 .765(15)0
0.2

0.68

0.7

0.72
J

0.74115144-5
0.76

|Ψ

H

D

E

A

F

B

G

C

H

D

E

A

Aldru
p

2

a single-phase product with no impuri
except for a quasi-two-dimensional (quasi-2D) system
susceptibility was measured using a supe
CaV4 O9 [5], for which a model based on the plaquettetum interference device magnetometer
singlets is considered to explain the origin of the spin
T range from 1.7 to 400 K in an appl
gap [6]. For this reason, it is important to discover other
H of 1.0 T. The Cu NQR spectra wer
examples of 2D spin systems having a finite spin gap tohomemade phase-incoherent-type spect
an excited state.
quency was scanned from 16.0 to 27
In this Letter, we investigated a 2D quantum spin sysproximate intervals of about 0.1 MHz.
tem SrCu2 sBO3 d2 by means of magnetic susceptibility,
at 63 Cuy65 Cu nucl
relaxation
rate
1yT
1
nuclear quadrupole resonance (NQR), and high-field magSrCu(BO3)2
netization, and have found that SrCu2 sBO3 d2 is a new
H. Kageyama et al, PRL 82, 3168 (1999)
spin-gap system with an exact ground state. Moreover,in the magnetization curves直交dimerat 1.7 and 0.5 K, we have
1
1
Shastri-Sutherland 格子
succeeded in observing 4 and 8 plateaus of the saturation
magnetization, which is the first observation of the quantized magnetizations in 2D spin systems.
SrCu2 sBO3 d2 has a tetragonal unit cell with the cell constants of a ≠ 8.995 Å and c ≠ 6.649 Å at room tempera1
tures [7]. All Cu21 ions with a localized spin S ≠ 2 arelocated at crystallographically equivalent sites. The structure is characterized by the layers of interconnected rect- 2+
angular planar CuO4 and triangular planar BO3 groups
as shown in Fig. 1(a). These layers extend parallel to
FIG. 1. (a)Schematic viewthe c
the c axis, and are structurally separated from each otherweekofending
磁化過程
21Y S I C A L R E V I E W L E T T SrCuPH
E R S2 sBO3 d2 along f001g.
The closed
8 FEBRUARY2013
PRL 110,by
067210(2013)
ions. A unique 2D magplanes
composed of Sr
M. Takigawa
et al,large
PRLopen110, circles
067210(2013)
circles, anddenote, resp
netic linkage of the Cu21 spins is formed as illustrated in
3 2
31
22
B , and O
ions. The unit cellis rep1/4
Fig. 1(b): The first-nearest-neighbor (1 NN) Cu pairs 0.12
(the21
lines. ( b) 2Dof the Cu sp
z = 10coordinatesmm
dimeric units,
0.10
31 distance of 2.905 Å) share an edge to form
1/4
0 the 2NN Cu pairs are denoted by the soli
1/6
respectively.
which are connected 34orthogonally with each other through

応用例2：Shastry-Sutherland 格子模型の磁化過程
実際の物質に実現例が存在！

Cu・磁化過程に複数の磁化プラトー
30 31680.06

0031-9007y99y82(15)y3168(4)$15.00

H (T)

33

M/MSat = 1/8, 2/15, 1/6, 1/4, 1/3, 1/2
29

0.08

2/15

0.04

2/15
z = -10 mm

5

1/8© 1999 The American Physical Societ
10

15

20

25

30

35

H (T)

0.02
0.00

32

τ/H (a.u.)

・基底状態はDimer 状態

Magnetization <Sc>

SrCu(BO )

15

20

H (T)

25

30

35

応用例2：Shastry-Sutherland 格子模型の磁化過程
P. Corboz and F. Mila, PRL 112, 147203 (2014)

week endingcf.
MERAの計算：J.
Lou
et
al,
arXiv:1212.1999
無限系のTPSによる計算
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
E
T
T
E
R
S
11 APRIL 2014
L 112, 147203 (2014)
x 10

0.03

ポイント

0.02

波動関数に仮定する周期性を変えることで、0
様々な並進対称性の破れを表現できる

0.04
0.1

0.2

0.3

0.4
J’/J

0.5

E/J

0.6

0

複数の磁化プラトーが実現

3 (color online). Estimate of the binding energy between
M/M as a=function
1/8,of2/15,
1/6, 1/5, 1/4,…
Sz ¼ 1 tripletsSat
J 0 =J, for different values of D.

0.4

0.25
0.2
s

0.03

(a)

2/15 : (3, 3),(8,2)
1/7 : (4, 2),(2,6)
1/6 : (3,1),(0,4)
2/11 : (1, 7),(3,1)
1/5 : (1, 3),(3,1)
1/4 : (2, 2),(4,0)
4x4 supersolid
DW phases

1

M/M

0.02

2

D=2
D=4
D=6
D=8
D=10
extrap

iTPSによる磁化過程

zero field
1/32 : (8,0),(0,8)
1/12 : (1, 5),(5, 1)
1/11 : (2, 4),(5,1)
1/10 : (2, 4),(10,0)
2/19 : (8,2),(1,5)
1/9 : (2, 4),(3,3)
1/8 : (4,2),(0,4)

0.01

0.01

4

0.5

(b)

0.15

実験とよく一致！
dilutetions for electronic bands, which
are well localized if aregion0.05
d is separated from the others by a＊DM相互作用によりlarge gap, it must be
ible to reconstruct well-localized wave functions from00.4
実験に無い1/5プラトーは消失
owest pair of bands; the energy of these localized states
be of the order of the average energy of the bands. It(c)1/8, 2/15,
1/4,
1/3,
s out実験：that the averageenergy is1/6,
alwaysbelow
that 1/2
of(8,2)
triplets [44], leading to a binding energy of the same

dense
plateau
1/8
region

0.6

h/J

1/5

1/4

1/6
2/15

0.1

intermediate
DW phases
0.45

0.5

h/J

0.55

h/J

(d)

0.6

0.65

応用例3：カゴメ格子ハイゼンベルグ模型
T. Okubo and N. Kawashima, in preparation

カゴメ格子

ハミルトニアン•ゼロ磁場での基底状態
古典模型：

各三角形で“120 度構造”を満たす任意の状態
立体構造を含んで大規模に縮退

量子ゆらぎ：
S=1/2量子スピン系

籠目
スピン液体の候補•
Z2 spin liquid• U(1) Dirac spin liquid•…

＊最近のiTPSによる計算：
H. J. Liao, et al, arXiv:1610.04727

S. Jiang, et al, arXiv:1610.02024

U(1) Dirac spin liquidを示唆

Taken from http://koharu2009.blogspot.jp/

To realize
inste
3
DMRG (S.Nishimoto,(2013)
selectetaal,particul
1.5
we pave this tri
1
gain energy.
6
In each config
c
7/9
0.5
4
bonds of the tria
0
As every
trian
0
0.5
1
d
M/M
2
(S.Nishimoto, et al, (2013)
triangles, the str
・“グランドカノニカル”DMRG
1/3
5/9
7/9
0
until it meets its
00.5
1５つの磁化プラトーが存在
M/M
In addition to目的
representative t2
01
M/MSat = 0, 1/9, 1/3, 5/9, 7/9
stringforming
J. Phys. Soc. Jpn. 83, 104710 (2014)
H. Nakanoand T. Sakai
厳密対角化(H. hexagon.
Nakano, et al, (2014)
1/3プラトー状態を明確にしたい
One t
1/2
1/2
1/2
1/3 プラトー状態はresonate状態
1longer string o(a)“ramp”
mechanical res
is the latter (seNakano,plot
and ofT. Sakai(2012, 2014)
3 | (H.
Density
the magnetizationin the long-range-ordered・厳密対角化（NFigure
≤ 42）
0.5
tiled with hexa
plateau states. The diameter of the blue circle on each lattice site scales
30.
formedM/MSat =1/3は“磁化ランプ”
the magnetization density. Hexagram consisting of nine sites indicate
Once all the v
the magnetic unit cell, extended from the original one by three0 times
0
1
2
3
filledwitha(Qmag ¼ 9). Hexagons
inlightyellow
colour
are
the
guide
to
the
eye.
h/J
(b)
M/M
sat ¼ 1/3, a(d) Schematic alignment of spins on a hexagram, whereFig.thenumbers(T.
Picot,
et undistortedal, (2016)
2. (Color) Magnetization
process
for thekagome-latticeTN(T. Picot, et al, (2016), T. Okubo et al, (2015年秋物理学会)) antiferromagnet. The results of ﬁnite-size clusters・テンソルネットワーク¼ 42, 39, 36,
and 27
onfor Neachhexag
indicate the magnetic density (Sz) on each vertices andareon
hexagons.
illustrated by red circles, blue triangles, black squares, and green reversed
respectively.
neglecting the q
Numerically, the exact magnetic density of the vertices triangles,of hexagramof a
DMRGと同じ位置に４つの磁化プラトー
vertices of the hand b are shrunk because of quantum fluctuation by about 2–5%
from
the
2
cf. M/MSat = 0にはギャップがないように見える
fractional values given in d, which is not the case for the 7/9 plateau in c. is valid as far as
The interactions
1.5
M/MSat =1/3は古典的な、up-up-down状態(c)work as an in
1plaquette. Figur
0

Fig. 1. Cluster shapes of the (A) one- and (B) two-dimensional systems
of interacting S ¼ 1 dimers. Our calculations havepbeen
ﬃﬃﬃﬃﬃ carriedpﬃﬃﬃﬃﬃ out for
N ¼ 20 in both cases. In (B), the tilted square of 20 # 20 is shownwith dotted lines. The thick green and thin black bonds denote the intraand interdimer interactions, respectively.

1

h/J

2

(b)

χ

S=1/2模型の磁化過程

0M/M sat

1

(b)

0.5

0

χ

(a)

0

1

h/J1

2

0

1

h/J1

2

sat

sat

Fig. 2. Field-derivative of the magnetization ! of the one- and twodimensional systems of interacting S ¼ 1 dimers, denoted by crosses andcircles, respectively. Insets (a) and (b) show the magnetization process ofthe one-dimensional case (A) in Fig. 1 for J2 =J1 ¼ 0:15 and that of thetwo-dimensional case (B) for J2 =J1 ¼ 0:05, respectively. The main panel
shows ! as a function of the magnetization divided by the saturation withcorresponding colors and symbols.

(D)

1

(E)

M/M sat(C)

Fig. 4. Results for the kagome-lattice antiferromagnets. (a) Magnetization
processes shown for the N ¼ 36 cluster (D) and the N ¼ 33 cluster(C) with circles and squares, respectively. (b) Field-derivative of themagnetization ! as a function of the magnetization divided by thesaturation. Circles and squares in (b) denote the results of ! corresponding to the cases in (a). Crosses correspond to the N ¼ 36 cluster (E). Notethat part of ! for the N ¼ 36 cluster (E) is missing because ! cannot be
defined when the lowest-energy state in the subspace does not become theground state of the system in the magnetic field. Diamonds represent thecase of the N ¼ 30 cluster investigated in ref. 12.

M/Ms

Fig. 3. Shapes of the finite-size clusters in the kagome lattice. Cluster (C)is the same as that for N ¼ 33 in ref. 12. Cluster (D) is the new cluster for
N ¼ 36, while cluster (E) is that for N ¼ 36 in ref. 15.

M=Msat ! 0:6 in Fig. 4(b), ! is enhanced irrespective of thesystem size except for the case of the green cluster (E) for
N ¼ 36, in which ! cannot be obtained near M=Msat ! 0:6;
the enhancement suggests the existence of some anomaly
around M=Msat ! 0:6. It should be clarified in future studies
whether this anomaly is a jump in the magnetization process
or another.

0hFig. 5. Schematic shape of the magnetization ramp.

Finally, let us discuss the relationship between ourobservation of the magnetization process of the ideal S ¼
1=2 kagome-lattice Heisenberg antiferromagnet and themagnetization measurement of the actual compounds vol-053707-3

h c– , h c+s

手法：PEPS テンソルネットワーク法
波動関数のExtended PEPS (PESS) 表現
スピンに依存しないテンソル

D. Poilblanc et al, PRB 87, 140407(R) (2012)
Z. Y. Xie et al, PRX 4, 011025(2014))

18-sites ユニットセルによる(√3 × √3 構造に整合)

D

D

D

D
D

D

スピンに依存するテンソルm

D
D

D:ボンド次元

m: Sz = ±1/2

二段階の計算：１．テンソルの最適化: 虚時間発展
打ち切り法: Simple update
２．物理量の計算:
近似的な縮約：Corner Transfer Matrix 法（＋ Suzuki-Trotter 分解）

結果：磁化過程
４つの明確な磁化プラトー
・D=7までで、ほぼ収束
M/MSat=1/9, 1/3, 5/9, 7/9
DMRGとコンシステント

・ゼロ磁場近傍にギャップは見えない
DMRGとは異なる

PEPSによる磁化過程

1

D=4
D=5
D=6
D=7

0.8

7/9

0.6

5/9

0.4

1/3

0.2
0

1/9
0

0.5 1

1.5 2

2.5 3

q=0

:up
:down

1/3プラトーは円の半径＝

結果：1/3プラトー状態-0.521テンソル最適化

DMRGと同様のResonated状態
エネルギー-0.522

従来：•

2サイト（T. Okubo et al)-0.523•

3サイト（T. Picot et al)-0.524の虚時間発展演算子

期待値の大きさHz = 1.1√3 × √3-0.525六角形ループの相関を十分に取り込めない？

今回：-0.526-0.527-0.528q=0-0.529 √3 × √3
Resonated
-0.53
3
4

6サイトの虚時間発展
3サイト•

Resonated
5

6

6サイトの最適化を行うことで、DMRGと同様の Resonated 状態が準安定状態として出現

2サイト •

7

Resoneted 状態のエネルギーはDの増大で、
up-up-down状態よりも低くなるmagnitude of approximately 10% of J. Other possible perturbations are the next-nearest-neighbor interaction J2 and diagonalinteraction Jd bridged via the Cd ion in the center of the hexagon
as shown in Fig. 2b, which was observed as a low value (Supplementary Fig. 2).

blunt pla
temperatu
conventio
or a spinweak anistransition
Faraday
rotationmeasurements
up ettoal,160
T.Commun.
We measuredacR. Okuma,
D. Nakamura,
T.OkuboNat.
10, 1229deﬁne(2019).
small magnetization from a tiny hexagonal single crystal of CdK at which
Real compounds often contain Dzyaloshinskii-Moriya
Interaction:with a diameter of approximately
1 mm and a thickness of the Bk sh
150 µm (Fig. 3a) by a Faraday rotation technique optimized to a the platea
single-turn coil in magnetic ﬁelds of up to 160 T. The (mk = Mk
COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09063-7

Comparison with experiments

J + Dz model

1.2

1

1.1
1.0

5/9

3/9
1

0

2

0.8

M7
M6

BS
B7

M3

3

B5

0.7

B4

0.6

2

B3

M2
B2

0.5
M1

M
1
m7
m6
m5
m4
m3
m2
m1

1

B1

0.3
0.2

Magnon

1/9

M5

CdK: B // c, T ~ 5 K

B6

0.4

Spin

0

M4

0.9
M (!B Cu –1)

M /MS

7/9

b

4

1.3

dM /dB (10–2 !B Cu –1 T–1)

a<latexit sha1_base64="iQ1jxaN4Vy+KjGtbqpV/X/jYGx4=">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</latexit>

0.1

0

1

2
B /J

3

0.0

0

50

100
B (T)

150

0

0

最後に
•

テンソルネットワーク表現を用いると、効率的に情報を圧縮できる場合がある•

分配関数のテンソルネットワーク表示は、テンソルネットワーク繰り込みを用いた実空間
繰り込み群により精度よく計算できる。
•

基本となる情報圧縮手段は、特異値分解（SVD）による低ランク近似•

適切なテンソルネットワークにより（基底状態の）波動関数を非常に効率的に表現できる•

適切なネットワークを探すには、エンタングルメントエントロピーの面積則が大事•

１次元量子系では、行列積状態（MPS）が非常に成功している•

２次元以上の量子系では、MPSは大きな系を効率的に近似できないが、テンソル積状態（TPS）ではうまくいく•

テンソルネットワーク表現のアイデアは、物理に限らず、様々な分野へ応用可能•

MPS、MERAなどの（量子）機械学習への適用•

量子回路デザインへの応用

今回紹介しなかったけど重要な話題•

基底状態に限らず、有限温度の計算手法の開発も進んできている•

密度行列のテンソルネットワーク表現
P. Czarnik et al., Phys. Rev. B 99, 035115 (2019).
A. Kshetrimayum et al, Phys. Rev. Lett. 122 070502 (2019).

•

Fermion系にも適用可能で、Hubbard模型の計算では最先端に匹敵する精度を出しつつある••(P. Corboz et al, Phys. Rev. B. 81, 165104 (2010))(P. Corboz, Phys. Rev. B. 93, 045116 (2016))（注）金属状態ではエンタングルメントエントロピーの面積則が破れる対称性を持つテンソルを用いた、波動関数トポロジーの特徴付け(S. Jiang and Y. Ran, Phys. Rev. B. 92, 104414 (2015))(J.-W. Mei et al, Phys. Rev. B. 95, 235107 (2017))

Report problem
以下の二点を含めて、A4で1枚くらい（多くても可）のレポートを提出してください
提出先：大久保へメールで提出（11/21〆切）t-okubo@phys.s.u-tokyo.ac.jp（pdf、wordなど。手書きの場合は写真でも可）•講義の感想と講義を通して理解できたことの概要（まとめ）•

自身の研究分野（もしくは興味のある分野）でテンソルネッ
トワーク（または情報圧縮）が役に立ちそうかどうかの検討

モンテカルロ法

乱拓アルゴリズム（擬似）乱数を実行中に参照し乱拓アルゴリズム：
その値によって振る舞いを変える
例：モンテカルロ積分
円の面積<latexit sha1_base64="BMLUQznYw3ggsXzWbHMplGpCwIU=">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</latexit>

アルゴリズム初期化<latexit sha1_base64="h2Dw3uDZhFjiJh4vRCkP2bSyOBY=">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</latexit>

loop i
一様乱数からとってくる<latexit sha1_base64="GuxcTj8AWvrg/TGIdygN+sOVmu4=">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</latexit><latexit sha1_base64="fuAVTOAuzdAnwdT9v2rsCn08IhM=">AAAClHichVHLSsNAFL2Nr1ofrQoiuAmWiiCUiQgWcVEogqvSh31AW0oSp3VomoRkWqjFH3AvLgRFwYX4B27d+AMu+gnisoIbF96kAdFivWEyZ87cc+fMXMXUmM0J6fmEsfGJySn/dGBmdm4+GFpYzNtGy1JpTjU0wyoqsk01ptMcZ1yjRdOiclPRaEFpJJz9QptaNjP0Q94xaaUp13VWY6rMkSolxXKdcltMbkrVUJhEiRviMJA8EAYvUkboEcpwBAao0IImUNCBI9ZABhu/EkhAwESuAl3kLETM3adwCgHUtjCLYoaMbAP/dVyVPFbHtVPTdtUqnqLhsFApQoS8kHvSJ8/kgbySzz9rdd0ajpcOzspAS81q8Gwl+/Gvqokzh+Nv1UjPHGoQc70y9G66jHMLdaBvn1z0s7uZSHed3JI39H9DeuQJb6C339W7NM1cjvCjoBe3ioOwTdLvpgyD/FZUQpzeDsdjXsP8sAprsIFd2YE4HEAKcu67nsMVXAvLwp6QEPYHqYLP0yzBjxCSX/dPlEM=</latexit>

if

end loop

then<latexit sha1_base64="ovmyaQJWslvCv+/iIDXZ28P1UvY=">AAACmnichVHLSsNAFL2Nr1ofjboRdFEsFUEoExEsropuFEH6bqEtIYnTGpoXybRQiz/gD7joSrEL8Q/cuvEHXPQTxGUFNy68SQOixXpDMmfO3HNzhiNbmuowQvoBbmJyanomOBuam19YDPNLywXHbNoKzSumZtolWXKopho0z1Sm0ZJlU0mXNVqUG4fuebFFbUc1jRxrW7SqS3VDramKxJAS+fCpKEUqdcqcCKJtQeSjJE68iowCwQdR8Ctl8o9QgTMwQYEm6EDBAIZYAwkcfMogAAELuSp0kLMRqd45hUsIobaJXRQ7JGQb+K3jruyzBu7dmY6nVvAvGr42KiMQIy/kngzIM3kgr+Tzz1kdb4brpY2rPNRSSwxfrWY//lXpuDI4/1aN9cygBgnPq4reLY9xb6EM9a2L60F2PxPrbJJb8ob+b0ifPOENjNa70kvTTHeMHxm9eFNchDEJv0MZBYWduIA4vRtNJvzAgrAGG7CFqexBEo4gBXkvuS7cQY9b5w64Y+5k2MoFfM0K/Cgu9wVkEZYc</latexit>

統計誤差は

に比例

次元の呪いこのやり方（棄却法）は高次元積分に無力
n次元立方体（1辺の長さ2）に対する、d次元球の体積の割合
Γ関数の漸近形<latexit sha1_base64="/SIYBXxKdXMnOfwVjoPm1jFp7pg=">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</latexit>

dが大きくなると指数的に球の体積の割合が減る

球に“ヒット”する確率が減り、誤差が指数的に増大してしまう（誤差を一定に保つには、サンプリング数Nを指数的に大きくする必要がある）

Importance Sampling•

積分に寄与する部分を重点的にサンプリングする
スピン模型の例：
ボルツマン重みが大きいところを重点的にサンプリングする
ランダムに発生した部分空間での和
規格化定数（分配関数）が未知なので、「マルコフ連鎖モンテカルロ法」で重点的サンプリングを実現する。

マルコフ連鎖モンテカルロ法
ボルツマン分布を確率過程の定常分布として生成したい。
状態Γ を"ランダムに" Γ’ に変えていく<latexit sha1_base64="eMb5jI4SFiLd5/I6KcsA2DqhYmY=">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</latexit><latexit sha1_base64="P/6Pwfgg3Mr1znIMRR9DIXPJj54=">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</latexit>：状態Γ が Γ’ に変わる確率
このとき、時刻tの確率分布

は、マスター方程式を満たす（確率の保存則）。

＊マルコフ＝右辺がひとつ前の時刻tの情報だけ

この確率過程が時間無限大で、ある確率分布に収束<latexit sha1_base64="N8TRel/SNPUg+u3ypY8YP7jse/I=">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</latexit>：今の場合、ボルツマン分布<latexit sha1_base64="N8TRel/SNPUg+u3ypY8YP7jse/I=">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</latexit>

この確率過程で現れる点の集合はボルツマン分布になっている

マルコフ連鎖モンテカルロ法の収束条件
P(Γ)に収束するための十分条件
1. “エルゴード性”•

任意の二つの状態 Γ とΓ’ が有限の回数でつながる。
•

W を行列としてみると、この条件は<latexit sha1_base64="V6EAz098ok9DaMPWp1dwC348Clg=">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</latexit>

2. “釣り合い条件”•

確率の流れが、P(Γ)に対して釣り合う

特別な場合: 詳細釣り合い条件イジング模型への適用：メトロポリス法
Local update:

スピンの一部（典型的には１つのスピン）の状態を変えるマルコフ連鎖モンテカルロ法Γ から Γ’への変更で S1, S2, .. S12, S14, S15, … ,S25,

21

22

23

24

25

16

17

18

19

20

11

12

13

14

15

6

7

8

9

10

1

2

3

4

5

を固定して、 S13だけを変更する。
このような local update では,
条件を満たす遷移確率 Wを
比較的簡単に見つけられる。

メトロポリス法：イジングスピン
ひとつスピンを選んで（13番）スピンを反転する

•

21

22

23

24

25

16

17

18

19

20

11

12

13

14

15

6

7

8

9

10

1

2

3

4

5

<latexit sha1_base64="Jqm5Ny7B+h5KkWH6LTbhMFuPOE4=">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</latexit>

この時、詳細釣り合い条件を満たすWは<latexit sha1_base64="3gvtvI87Z1fzTi46Ii+j+dEVhlk=">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</latexit>を満たす。したがって、例えば<latexit sha1_base64="mLisxoTr4DHt+cNNYw+AbVuPxAc=">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</latexit><latexit sha1_base64="ppBtkctTB7cg3E2wLF74oXJN5/g=">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</latexit>とすれば良い。

メトロポリス法の流れ
Step 0: 初期状態を準備する
loop ti 番目のスピンを選ぶ

典型的には
random state (T→∞)• ordered state (T→0)•

1. Siから候補、Γ’を作る•

イジング模型 : Si’ = -Si

2. エネルギー差を計算

エネルギーが減る場合ΔE < 0),

3. 乱数を生成

新しい状態は確率1で採択.

4. r の値に応じて、次の状態 Γt+1を通常、物理量はNスピンの試行が終わる毎に計算することが多い。

で決める
5. 物理量 O(Γt)を計算する<latexit sha1_base64="ES1Vzov84ZRnD0sLmkpYgZ5RlU8=">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</latexit>"1 モンテカルロステップ"= N スピンの反転の試行

角転送行列繰り込み群（CTMRG）•

奥西・西野ら (1995)による逐次的な“繰り込み”によるテンソ
ルネットワークの計算方法••

Corner Transfer Matrix Renormalization Group（CTMRG）

分配関数のテンソルネットワーク表現をL→L+2のように数サイトずつ大きくしていくことで、徐々に計算する•

近年、２次元量子多体系の基底状態計算アルゴリズム（PEPS法、TPS法）の一部にも使われる

CTMRGでやりたいことL×Lの分配関数が(L+2)×(L+2)の分配関数を計算（近似的に）計算できた

系を少し大きくする

角転送行列表現

C

C

C

C

: D×D

C

系を少し大きくする

e e

e
e

e
e
C

C

e e

（近似）

C

Cの大きさを変えずに系を大きくする: D×D

角転送行列の意味（近似なし）

4×4

6×6
C

=

=

=
e

角転送行列表現

C

C

C

C

C

系を少し大きくする

e e

C

C

e
e

e
e
e e

=

C

（近似）

C
e

e

CTMRGのレシピ
1. SVDによる分解
C行列と思ってSVDeM=
Mは実対称（エルミート）行列e
M : 2D×2D
C

e e

e
e

e
e
C

C

e e

=

=

C
＊対称性を仮定

特異値が大きいものD個を残せば良い近似！

CTMRGのレシピ: 2D×D

2. SVDを使って近似
C

e

=

e

: D×D
対角成分が
の対角行列

C

e e

e
e

e
e
C

C

e e

C

=

≃

近似

CTMRGのレシピ
くりこみ変換まとめ
1. LxL の系の角転送行列をSVD

C

e

e

3. (L+2)×(L+2)の角転送行列を作成

C

=
=

e

e

2. Projectorを作る
特異値が大きい方からD個だけ残す=大きさLの系の分配関数が逐次求まるe