量子酔歩netsci/wp-content/uploads/2017/10/...contents part 1：量子ウォーク概説 1....

なぜ量子ウォークは着目され続けているのか

（抜粋）

第10回情報ネットワーク科学研究会首都大学東京秋葉原サテライトキャンパス 20171017

量子酔歩

今野紀雄(横浜国立大学)

Contents

Part 1 量子ウォーク概説

sect1 Introduction

sect2 Important properties of QWs

sect3 Measures of QWsStationary measure

Limit measure

sect4 Summary

Part 2 量子ウォークの解析方法

Part 3 今後の展望いくつか

A quantum analogueof the random walk

Random walk Quantum walk

time

positionQuantum walk

Random walk

1 ballistic spreading

3localization

2inverted bell shape

(Hadamard walk)

(Grover walk)

Hadamard walkFourier walk

QW on 2D

RW

Grover walk

QW on tree

Definition of DTQW on Z

left chirality right chirality

Chirality

Time evolution

Quantum coin

Two dimensional unitary operator

Total state space

(RW)

～Probability that a particle is found at position x and at time n

is determined by

Example

Review and BooksN Konno (2008) Quantum Walks Lecture Notes in Mathematics 1954 309ndash452 Springer

S E Venegas-Andreca (2008)Quantum Walks for Computer Scientists Morgan and Claypool

S E Venegas-Andreca (2012)Quantum Information Processing 11 1015-1106 arXiv12014780

R Portugal (2013) Quantum Walks and Search Algorithms Springer

K Manouchehri J Wang (2014) Physical Implementation of Quantum Walks Springer

今野紀雄 (2008) 量子ウォークの数理産業図書

今野紀雄 (2014) 量子ウォーク森北出版書評数学セミナー日本物理学会誌

町田拓也 (2015) 図で解る量子ウォーク入門森北出版

[NEW]今野紀雄 (2016) 四元数森北出版

四元数量子ウォークのテーマ佐藤（小山高専）三橋（法政大）

Japanese Books on Quantum Walks

(RW p+q=1)

組合せ論的手法

＝

p q

RW case

【an explicit expression (K (2002) QIP K (2005) JMathSocJpn)】

量子ウォークの確率測度

QW at time nwith the initial qubit

定理

分布の対称条件

弱収束の極限定理

(K (2002) QIP K (2005) JMathSocJpn)

19

RW QW

spreading

Limit density

time

positionQW

RW

U

(diffusive ) (ballistic )

Classical Mark

Quantum Mark

Lazy random walk

Localization （局在化）Lazy

Lazy

3-state QWOn the other handhellip

localization

Inui K Segawa (2005) PRE

Lazy RW

3-state QW

Grover matrix

Co-existence of ldquoBallistic spreadingrdquo and ldquoLocalizationrdquo

Where is the value 23

where Z has the following limit measure


Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times

( diffusive ) ( ballistic )

There are two stages of limit theorems in the three-state QW

Stage 1

Stage 2

Definition

(quantum walk) belongs to universality class

QW on homogeneous trees

Multi-state QW on 1D lattice

（QW on 2D lattice

Time dependent QW on 1D lattice

QW with memory on 1D lattice

A universality class of quantum walk

Miyazaki Katori K (2007) PRASegawa K (2008) IJQI

K Machida (2010) QIC

Liu (2012) QIP

Machida K (2010) Proc Info C Tech

Examples

Chisaki Hamada K Segawa (2009) IIS

Watabe Kobayashi Katori K (2008) PRA）

polynomial of x (mainly)

QW on joined half lines Chisaki K Segawa (2012) QIC

QW with one defect K Luczak Segawa (2013) QIP

introduced by K Luczak and Segawa (2013) QIP

Liu Petulante (2013) IJQIQW on the half line

Liu Petulante (2009) PRA

Inui K Segawa (2005) PRA K (2008) LNM

1 Path counting method

2 Fourier analysis

3 Spectral analysis (CGMV method)

K (2002) QIP K (2005) JMathSocJpn

Grimmett Janson Scudo (2004) PRE

Cantero Gruumlnbaum Moral Velaacutezquez (2010)

Comm Pure Appl Math

CMV matrix

4 Generating functional approach

Other approaches Stationary phase method helliphellip

Chisaki Hamada K Segawa (2010) IISK Luczak Segawa (2013) QIP

K Segawa (2011) QICCantero Gruumlnbaum Moral Velaacutezquez (2012) QIP ltreviewgt

Analytical methods of QWs

5 Graph zeta functionK Sato (2012) QIP

今後の展望マルコフ過程の一般理論

量子過程の一般理論第一歩定常測度極限測度の特徴づけ「定常測度=極限測度 1次元格子3状態 Grover walk」

Endo Kawai K (2016) RIMS Kokyuroku （むしろ稀）

正再帰性に対応するもの有限系でも定常測度の集合を決定するのは難しい

「定常測度=極限測度」の例が多い

Segawa (2013) J Comput Theor Nanosci (半直線) Konno Segawa Obata (2013) CMP （スパイダーネット）

定常測度 = 極限測度1次元3状態 Grover walk

定常測度

極限測度

初期状態

Recent Results on Stationary Measures

[1] K (2014) Quantum Information Processing 13 1103-1125

[2] K Takei (2015)

Quantum Information and Computation 15 1060-1075

2-state multi-defect model

3-state modelone-defect model[3] Wang Lu Wang (2015)

Quantum Information Processing 14 867-880

[4] Endo Kawai K (2017) IIS 23 57-64

[5] Endo Kawai K (2016) RIMS Kokyuroku 2010 pp45-55

[6] Kawai Komatsu K (2017) arXiv170201523

Multi-state model (Grover walk on Z^d)

[7] Komatsu K (2017) arXiv170307059

2-state model one-defect model （eg）

[1]

量子ウォークの定常測度の集合の決定及び量子ウォークの定常測度時間平均極限測度との関係

Wojcik et al (2012 PRA) K Luczak Segawa (2013 QIP) K (2014 QIP) Endo K (2014 YMJ) Endo et al (2014 YMJ) Wang Lu Wang (2015 QIP) K Takei (2015 QIC) Endo et al (2015 QIC) Endo K Obuse (2015 arXiv) Endo Kawai Konno (2016 RIMS 2017 IIS) Kamai Komatsu K (2017 arXiv) Komatsu K (2017 arXiv)

[2]

量子ウォークの弱収束極限定理の代数的構造Tate (2013 IDAQPRT)

[3]

初期条件を一般化したときの弱収束極限定理Machida (2013 IJQI) Machida (2013 QIC)

[4]

量子ウォークとハミルトニアンとの関係Tate (2013 IIS) Ko Yoo (2013 KMJ)

[5]

量子ウォークの大偏差原理とエントロピーとの関係Sunada Tate (2012 JFA) Ide K Machida (2012 QIC) K Segawa (2012 YMJ)

[6]

極限の密度関数とホインの微分方程式（確定特異点が4個）との関係（ガウス分布の場合には超幾何微分方程式（確定特異点が3個）が対応）

K Machida Wakasa (2012 YMJ)

[7]

極限密度関数の背後にある「独立性」の構造Hamada K Mlotkowski (2009 IIS)

[8]

パス空間を通しての量子ウォークの理解K (2012 QIP) K Segawa (2012 YMJ) K (2013 JCTN) K Segawa (2013 YMJ)

[9]

再帰性の定義と計算手法例えばStefanak Kiss Jex (2008 PRL) Stefanak Kiss Jex(2008 PRA) Bourgain et al (2014 CMP) Grunbaum et al (2013 CMP)

[10]

特異連続スペクトルを持つ量子ウォークの解析Grunbaum Velazquez (2012 Proc of FoCAM2011)

[11]

ゼータ関数と量子ウォークとの関係

K Sato (2012 QIP) Higuchi K Sato Segawa (2014 PJMI)

[12]

d次元格子上のグローヴァーウォークの挙動Inui Konishi K (2004 PRA) Watabe Kobayashi Katori K (2008 PRA) Higuchi K Sato Segawa (2014 JFA) Komatsu (2015 YMJ) Komatsu K (2017 arXiv)

[13]

グラフ上のグローヴァーウォークの挙動Higuchi K Sato Segawa (2013 JMI) Higuchi K Sato Segawa (2014 JFA)

[14]

複雑ネットワーク上の量子ウォークに向けての研究例えばIde K Segawa (2012 QIP) Xu Ide K (2012 PRA) K Obata Segawa (2013 CMP) Ide K Segawa Xu (2014 Entropy)

[15]

時空間に依存した量子ウォークの研究Shikano Wada Horikawa (2014 SciRep)

[16]

観測頻度による古典から量子へのクロスオーヴァーChisaki K Segawa Shikano (2011 QIC)

[17]量子グラフと量子ウォークとの関係

Higuchi K Sato Segawa (2013a YMJ) Higuchi K Sato Segawa (2013b YMJ)

[18]

Open quantum random walk との関係例えばAttal Petruccione Sabat Sinayskiy (2012 JSP) K Yoo (2013 JSP)

[19]四元数量子ウォークへの拡張

K (2015 QSMF) K Mitsuhashi Sato (2016 QIP)

[20]

サイクル上の量子ウォークの周期性円分多項式K Shimizu Takei (2017 IIS)

[21]

staggered量子ウォークとグラフゼータ量子探索K Sato Segawa (2017 arXiv) K Sato Portugal Segawa (2017 arXiv)

Contents

Part 1 量子ウォーク概説

sect1 Introduction

sect2 Important properties of QWs

sect3 Measures of QWsStationary measure

Limit measure

sect4 Summary

Part 2 量子ウォークの解析方法

Part 3 今後の展望いくつか



time


Random walk


3localization


(Hadamard walk)

(Grover walk)


QW on 2D

RW

Grover walk

QW on tree



Chirality

Time evolution

Quantum coin


Total state space

(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]




time


Random walk


3localization


(Hadamard walk)

(Grover walk)


QW on 2D

RW

Grover walk

QW on tree



Chirality

Time evolution

Quantum coin


Total state space

(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]



QW on 2D

RW

Grover walk

QW on tree



Chirality

Time evolution

Quantum coin


Total state space

(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


QW on tree



Chirality

Time evolution

Quantum coin


Total state space

(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]




Chirality

Time evolution

Quantum coin


Total state space

(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


Time evolution

Quantum coin


Total state space

(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


(RW)


is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]



is determined by

Example












(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]













(RW p+q=1)


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


＝

p q

RW case




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]




定理




19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


19

RW QW

spreading

Limit density

time

positionQW

RW

U


Classical Mark

Quantum Mark

Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


Lazy random walk


Lazy


localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]



localization


Lazy RW

3-state QW

Grover matrix





Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]






Press Press

weird feature

ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


ＲＷ 3-QW

Spreading

Limit measure

Localization times



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]



Stage 1

Stage 2

Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


Definition




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]




（QW on 2D lattice






Liu (2012) QIP


Examples











2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]



2 Fourier analysis





Comm Pure Appl Math

CMV matrix














定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]









定常測度

極限測度

初期状態



[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]




[2] K Takei (2015)





[4] Endo Kawai K (2017) IIS 23 57-64




[7] Komatsu K (2017) arXiv170307059


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


[1]



[2]


[3]


[4]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


[5]


[6]



[7]


[8]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


[9]


[10]


[11]



[12]


[13]


[14]


[15]


[16]




[18]




[20]


[21]


[13]


[14]


[15]


[16]




[18]




[20]


[21]




[18]




[20]


[21]


量子酔歩netsci/wp-content/uploads/2017/10/...contents part 1：量子ウォーク概説 1....

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