計算モデル論ー対話証明計算モデルと量子非局所 ... - imai...

計算モデル論

ー対話証明計算モデルと量子非局所性計算モデルー

今井浩

2014-06-27, 07-04, 07-09

背景・動機1

• 計算量の重要未解決問題– P vs NP

• 計算モデル(1930’s)– Turing Machine– λ-Calculus– Recursive function

• Interactive Proof (1985-)

• Birch and Swinnerton-Dyer Conjecture • Hodge Conjecture • Navier-Stokes Equations • P vs NP • Poincaré Conjecture • Riemann Hypothesis • Yang-Mills Theory

http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/http://www.claymath.org/millennium/Hodge_Conjecture/http://www.claymath.org/millennium/Navier-Stokes_Equations/http://www.claymath.org/millennium/P_vs_NP/http://www.claymath.org/millennium/Poincare_Conjecture/http://www.claymath.org/millennium/Riemann_Hypothesis/http://www.claymath.org/millennium/Yang-Mills_Theory/

背景・動機2

• 量子計算・量子情報– Shorの素因数分解量子多項式時間アルゴリズム– Groverの探索アルゴリズム–量子ウォーク

• 量子暗号– BB84

• 量子entanglement, 量子非局所性–量子情報処理の力の源⇒対話証明との関係

1927 Solvay Conference on Quantum Mechanicshttp://commons.wikimedia.org/wiki/Image:Solvay_conference_1927.jpgより

A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrödinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin,

P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr,

I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. Richardson

http://commons.wikimedia.org/wiki/Image:Solvay_conference_1927.jpg

EPR and Bohr

`teleportation’ between distant 2 parties

Alice Bob

entangled quantum state

Alice measures Bob’s statebecomesinstantaneously

Alice measures Bob’s statebecomesinstantaneously

EPR paradox and Bell inequalities• Einstein, Podolsky, Rosen (1935)

– quantum entanglement vs. relativity theory

• Bell inequality (1964)– Entanglement/Nonlocalit⇒violation

• CHSH inequality (Clauser, Horne, Shimony, Holt 1969)– applicable to a bipartite system

• Aspect et al. (1982)– Experimental verification of violation of

CHSH inequality

• Tsirelson (1980): max. violation value⇒ Experimental realization (Sakai et al. 2006)

entanglement

measure(local)

statechange

instantlyfaster than light

Bell-CHSH correlation experiment

Alice Bob

entangled quantum state1A

2A

1B

2B

0)()()()()()( 2212211111 £-+++-- BAPBAPBAPBAPBPAPClassical correlation:

21)()()()()()( 2212211111 £-+++-- BAPBAPBAPBAPBPAP

Quantum correlation:

two measurementsapply one

two measurementsapply one

対話証明システム

Interactive Proof System[Babai 1985; Goldwasser, Micali, and Rackoff 1985]

• Two players: prover, verifier– Prover tries to convince verifier of her assertion with

unbounded computational power– Verifier must check validity of prover’s assertion

probabilistically and efficiently:• probabilistically ⇒ with bounded error• efficiently ⇒ in time polynomial to input length

Peggy (Prover) Victor (Verifier)Interactive

Communication

IP=PSPACE

Nondeterministic Polynomial (NP)

• Two players: prover, verifier– Prover tries to convince verifier of her assertion by

just given one certificate– Verifier must check validity of prover’s assertion

efficiently:• efficiently ⇒ in time polynomial to input length

e?satisfiabl)()()( はzyxzyxzyx ⁄⁄Ÿ⁄⁄Ÿ⁄⁄Peggy (Prover) Victor (Verifier)

NP

Oracle

にして!1,0,1 === zyx

Example: Graph Non-Isomorphism

◎ Protocol of verifier V:1. Choose an index i Œ {1,2} of graphs

and a permutation p Œ Sn at random.Send a graph p (Gi) to prover P

to ask which of the two is isomorphic to p (Gi).2. Receive an index j from P.

Accept iff i = j.

Graph Non-Isomorphism Problem (GNI)

INPUT: Two graphs G1, G2 of n vertices

QUESTION: For all permutation p Œ Sn on vertices,p (G1) π G2?

1

2 3

4

4

3 1

2

同型

非同型

1 2

34

1

23

4

凸多面体入門

7v6v

凸多面体のV-表現とH-表現

1v

4v 5v

3v2v

}0,1|{5

1

5

1≥== ÂÂ

==ii

iii

ivP lll

I }|{ ii bxaxP £◊=

3点完全グラフK3のcut polytope

x

y

z

(1,1,0)

(0,1,1)

(0,0,0)

(1,0,1)

1

23

K3

0£-- zyx

上限定理・線形計画ー Polyhederal Combinatorics ー

• d 次元 n 点の凸包のファセット数：– (t,t2,t3,…,td) (t=1,2,…,n)の凸包を考えてみて!

• 線形計画問題

• 「凸多面体の簡潔な記述≒多項式時間解法」

• 凸包アルゴリズム：cdd, lrs, etc.–計算幾何(Computational Geometry)

Î ˚)( 2/dnO

}0,|min{ ≥=◊ xbAxxc

Correlation between 2 Events A,B

1)()()(

)()()()(

1)(),(),(0, Events 2

£-+

££

££

ABPBPAP

BPABPAPABP

ABPBPAPBA

)(Cor 2K□

Correlation Polytopealso known as Boolean Quadratic Polytope)(AP

)(BP

)()(

BAPABP

«=

(1,0,0)

(0,1,0)

(1,1,1)B

A

A, B

∅

space-))(),(),(( ABPBPAP

Correlation of 2 events

Cut polytope of K3

x

y

z

(1,1,0)

(0,1,1)

(0,0,0)

(1,0,1)

1

23

K3

)(Cut 2K—=□

zyx +£triangle inequality

Correlation polytope ⇔ Cut polytope

)(Cor 2K□ )(Cut 2K—

□

A B

)(AP

X

)(BP

)(ABP

covariance mapping

23 KK —=

suspension

Correlation of a bipartite system of },{ and },{ 2121 BBAA

))(),( no()(),(),(),()(),( ),(),(

??? amongn Correlatio

2121

22122111

2121

BBPAAPBAPBAPBAPBAPBPBPAPAP

1A 1B)( 11BAP

2A 2B)( 22BAP

)( 21BAP )( 12BAP


)(Cor 2,2K□ )(Cut 2,2K—

□

1A 1B

)( 1AP

X

)( 1BP

)( 11BAP

covariance mapping

2,22,2,1 KK —=

2A 2B)( 22BAP

)( 21BAP )( 2BP

suspension

Correlation of a bipartite system of},,,{ and },,,{ 2121 BA mm BBBAAA LL

))(),( no(),,1;,,1(

)(),(),(??? amongn Correlatio

2121 jjii

BA

jiji

BBPAAPmjmi

BAPBPAPLL ==

1A 1B)( 11BAP

AmA

BmB

)(BA mm

BAP


)(Cor , BA mmK□ )(Cut , BA mmK—

□covariance mapping

BABA mmmmKK ,,,1 —=

)( 1AP

X

)( 1BP

1A 1B)( 11BAP

AmA

BmB

)(BA mm

BAP

Problem:Enumerate all the facets of

)(Cut , BA mmK—□

to know bipartite correlation

量子対話証明へ

1対1量子暗号を超えること計算量の枠組みと物理実験の対応

Interactive Proof System[Babai 1985; Goldwasser, Micali, and Rackoff 1985]

• Two players: prover, verifier– Prover tries to convince verifier of her assertion with

unbounded computational power– Verifier must check validity of prover’s assertion

probabilistically and efficiently:• probabilistically ⇒ with bounded error• efficiently ⇒ in time polynomial to input length

Peggy (Prover) Victor (Verifier)Interactive

Communication

IP=PSPACE

Multi-prover Interactive ProofMIP=NEXPTIME

Alice Bob

Victor (Verifier)

X

量子情報とBell不等式

A Two-Party One-Round Interactive Proof System[Cleve, Høyer, Toner, Watrous CCC 2004]

}1||,,1,0{ -=Œ

TTt

L質問

Alice Bob

Victor (Verifier)

Provers

}1||,,1,0{ -=Œ

SSs

L質問

• 事前に回答戦略を協力して練ってよい• 関数

• 質問が始まったら通信できない

}1||,,1,0{ -=Œ

AAa

L答

}1||,,1,0{ -=Œ

BBb

L答

ÔÔÓ

ÔÔÌ

Ï

最も単純な場合

Alice Bob

Victor (Verifier)

Provers

}1,0{=ŒAa答 }1,0{=ŒBb答

answered] is ),(Pr[)0,0;,( babaC =

相関表t=0

b=0 b=1s=0

a=0

c(0,0) c(0,1)

a=1

c(1,0) c(1,1)

˙˚

˘ÍÎ

È=

==

0001

0,0

c

ba

˙˚

˘ÍÎ

È=

==

0010

1,0

c

ba

˙˚

˘ÍÎ

È=

==

01000,1

c

ba

˙˚

˘ÍÎ

È=

==

10001,1

c

ba

˙˚

˘ÍÎ

È=

-===

-===

qpqpqppqc

qqqb

pppa

1,]0Pr[

1,]0Pr[

0),(

0),(1

0,

≥

=Â=

jic

jicji

Alice Bob

Victor (Verifier)

Provers

}1,0{Œt質問}1,0{Œs質問

•事前に回答戦略を協力して練ってよい•質問が始まったら通信できない

}1,0{Œa答 }1,0{Œb答

|S|=|T|=2, |A|=|B|=2

相関表(|S|=|T|=2, |A|=|B|=2)

t=0 t=1b=0 b=1 b=0 b=1

s=0 a=0 c(0,0;0,0) c(0,1;0,0) c(0,0;0,1) c(0,1;0,1)

a=1 c(1,0;0,0) c(1,1;0,0) c(1,0;0,1) c(1,1;0,1)

s=1 a=0 c(0,0;1,0) c(0,1;1,0) c(0,0;1,1) c(0,1;1,1)

a=1 c(1,0;1,0) c(1,1;1,0) c(1,0;1,1) c(1,1;1,1)

16 deterministic patterns

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0000010100000101

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

1010000010100000

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0101000000000101

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0101000001010000

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

1001000000001001

…

独立試行相関表

t=0 t=1b=0 b=1 b=0 b=1

s=0 a=0 p0q0 p0(1-q0) p0q1 p0(1-q1)

a=1 (1-p0)q0 (1-p0)(1-q0)

(1-p0)q1 (1-p0)(1-q1)

s=1 a=0 p1q0 p1(1-q0) p1q1 p1(1-q1)

a=1 (1-p1)q0 (1-p1)(1-q0)

(1-p1)q1 (1-p1)(1-q1)

一般の古典相関表

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0000010100000101

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

1010000010100000

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0101000000000101

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0101000001010000

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

1001000000001001

確定的相関表の凸一次結合

…

じゃんけんのpay-off matrix

注：この表では行プレイヤーから見た値で記述

ゲームの値＝0

グーチョキパーグー 0 1 -1チョキ -1 0 1パー 1 -1 0

ゲームの値

Alice Bob

Victor (Verifier)

Provers

}1||,,1,0{ -=Œ

TTt

L質問

}1||,,1,0{ -=Œ

SSs

L質問

• 事前に回答戦略を協力⇒C(a,b;s,t)• 質問が始まったら通信できない

}1||,,1,0{ -=Œ

AAa

L答

}1||,,1,0{ -=Œ

BBb

L答

),;,(),;,(,;,

tsbaCtsbatsbaWV Â=

ゲームの値

),;.( tsbaW重み

A Two-Party One-Round Interactive Proof System[Cleve, Høyer, Toner, Watrous CCC 2004]

Alice Bob

Victor (Verifier)

Provers

}1||,,1,0{ -=Œ

TTt

L質問

}1||,,1,0{ -=Œ

SSs

L質問

• 事前に回答戦略を協力して練ってよい• 関数

• 質問が始まったら通信できない

}1||,,1,0{ -=Œ

AAa

L答

}1||,,1,0{ -=Œ

BBb

L答

ÔÔÓ

ÔÔÌ

Ï

CHSH Game

Alice Bob

Victor (Verifier)


}1,0{Œa答 }1,0{Œb答

ÔÔÓ

ÔÔÌ

Ï Ÿ=≈=otherwise0

1),;,( tsbatsbaW

ゲームの値：等確率質問に対して証明者が勝つ確率の最大値

0)1(,1)0(;1)1(,0)0( ==== bbaa4/3　ゲームの値fi

CHSH Game

Alice Bob

Victor (Verifier)


}1,0{Œa答 }1,0{Œb答

ÔÔÓ

ÔÔÌ

Ï Ÿ=≈=otherwise0

1),;,( tsbatsbaV

ゲームの値：等確率質問に対して証明者が勝つ確率の最大値

0)1(,1)0(;1)1(,0)0( ==== bbaa4/3　ゲームの値fi

s t V0 0 0 1 00 1 0 0 11 0 0 0 11 1 1 1 1

tsŸ ba≈

Alice, Bob: prior shared randomness でも3/4は上限

pay-off matrix M

V (0,0¦0,0)(1,1¦1,1)

(0,0¦0,1)(1,1¦1,0)

(0,0¦1,0)(1,1¦0,1)

(0,0¦1,1)(1,1¦0,0)

(0,1¦0,0)(1,0¦1,1)

(0,1¦0,1)(1,0¦1,0)

(0,1¦1,0)(1,0¦0,1)

(0,1¦1,1)(1,0¦0,0)

(0,0) 1 1 0 0 1 1 0 0(0,1) 1 0 1 0 1 0 1 0(1,0) 1 1 0 0 0 0 1 1(1,1) 0 1 0 1 1 0 1 0

),( ts

))1(),0(|)1(),0(( bbaa

43T0,4

1,0,41,0,0,4

1,41

41,4

1,41,4

1: valueGame =˜̃˜˜

¯

ˆ

ÁÁÁÁ

Ë

Ê

˜̃˜˜

¯

ˆ

ÁÁÁÁ

Ë

Ê M

and Optimal Strategies

じゃんけんのpay-off matrix

注：この表では行プレイヤーから見た値で記述

（前の表では列よりの値）

グーチョキパーグー 0 1 -1チョキ -1 0 1パー 1 -1 0

CHSHゲーム

• 重み

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

0110100110100101

CHSH Gameの解析• 古典の場合：ゲームの値 3/4• 量子の場合：

– 事前にAliceとBobがエンタングル状態を共有– それぞれ自分のところで部分測定できる

1cos0sin)(1

1sin0cos)(0qqqf

qqqf

+-=

+=

)8/()8/(1

)8/()8/(0

)4/()4/(1

)0()0(0

pfpf

pfpf

pfpf

ff

--=

=

=

=

bbbY

bbbY

aaaXaaaX

2/)1100( +

としてで測定、結果を回答

⇒ゲームの値 75.085.0)8/(2cos >ªp

まさしくCHSH不等式として知られるBell不等式の話そのもの他にKochen-Specker定理、擬似テレパシー, 量子グラフ彩色など種々の展開

純粋状態での量子情報基礎入門

ノルム1の複素ベクトルユニタリ行列

測定

１ビット

• 古典ビット– “真” か “偽” の２状態のうちどちらかをとる

• 確率ビット– 確率pで“真”をとり、確率qで“偽”をとる

– p＋q＝１ , （0≦ p, q ≦1)

• 量子ビット– 確率|α|2で“真”をとり、確率|β|2で“偽”をとる

– |α|2 +|β|2=1 , (α,β は任意の複素数)

（確率）振幅と呼ぶ10 ba +

n ビット

• 古典 n ビット– 2n 個の状態のうちどれかひとつをとる

• 量子 n ビット– 2n次元の複素内積空間の単位ベクトル– 2n 個の基底状態の重ね合わせをとる

12121100 --+++= nn qqq aaay L

12

12

21

20 =+++= -naaay L

　振幅:,,, 1210 -naaa L

entangled 10

10

11011000

1001

211100

21

separable 102

10

212

1

01

0011

210100

21

1000

11

0100

10

0010

100

101

01

0001

010

011

0000 :)4(qubit 2

122101

010

2 :)2(qubit 1

:??)|(|

:)|(||)|(|

|,|,|,|||

C,,||||,||,|C

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ƒ===ÒÒ+

ÒÒ+ƒÒ=ƒ==ÒÒ+

=Ò=Ò==Ò==ÒÒƒ=Ò=

Œ=+=Ò=Ò=Ò=

gg

bb

gbgbgbgb

bababa

f

　

基底　

N

N

12111,,10100,0000

10

00

111|,,

0

010

0100|

,

0

01

0|0|0|000|2C :qubit

C,,12||2||,|101|,

010| :)2(qubit 1

-=◊◊◊=◊◊◊=◊◊◊

=Ò◊◊◊=Ò◊◊◊

=Ò◊ƒ◊◊ƒÒÒƒ=Ò◊◊◊

Œ=+=Ò=Ò=Ò=

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n

nn

nnnn

N

L

ML

M

M

4847648476

44444 844444 7648476

基底

　 bababa

f

量子並列計算

Ò-+Ò-+

+Ò+Ò+Ò+Ò=Ò

--12|22|

3|2|1 |0||

1222

3210nn

nn aa

aaaay

LL

L

状態の重ねあわせが可能

計算過程はそれぞれの状態の振幅が変わっていく過程

Â-

=

=12

0

2 1||n

iia

→ ユニタリ行列をかける

部分測定

に収斂を測定、全体はで確率

に収斂を測定、全体はで確率

で部分測定最左ビットを

　

yb

jb

yjybjby

ƒÒ

ƒÒ

fi

-ƒÒ+ƒÒ=Ò

1|1||

0|0||

}1,0{

qubit) 1:,(1 |0||

21

20

10 n

最左ビットのみ扱える

˜̃˜˜˜

¯

ˆ

ÁÁÁÁÁ

Ë

Ê

=010

˜̃˜˜˜

¯

ˆ

ÁÁÁÁÁ

Ë

Ê

=101

˜˜˜˜

¯

ˆ

ÁÁÁÁ

Ë

Ê

=ba

j

12||2|| =+ ba

2||a

2||b

に状態はで測定を確率


で測定を直交基底

1,2||10,2||0

1,0

b

a

ba

jǪ̂

Ô˝¸

ÔÓ

ÔÌÏ

˜˜˜˜

¯

ˆ

ÁÁÁÁ

Ë

Ê

=

射影測定1

0

1

˜˜˜˜˜

¯

ˆ

ÁÁÁÁÁ

Ë

Ê

+-+

=ba

baj2

1

12||2|| =+ ba2/2|| ba -

2/2|| ba +



で測定を直交基底

1,2/2||1

0,2/2||0

1,0

¢-¢

¢+¢

¢¢Ǫ̂

Ô˝¸

ÔÓ

ÔÌÏ

ba

ba

j

射影測定20¢1¢

( ) ( )

ÓÌÏ

¢¢==

¢¢==

+=¢+=

BBBB

AAAA

BABA

BBAA

00 ,00,00 ,00

.102

10,11002

1

21

21

r

あまり意味のない測定

0 1 0 10 ½ ¼ ¼1 ½ ¼ ¼0 ¼ ¼ ½1 ¼ ¼ ½

1A

2A

2B1B

( )

( )

( ) ( )

ÔÔÓ

ÔÔÌ

Ï

==

¢¢==

ÔÔÓ

ÔÔÌ

Ï

¢++

=¢++

=

+=¢

+=

BBBB

AAAA

BB

BABA

BB

AA

ffyy

yy

r

21

21

,

,00 ,00

0122

1 ,0022

1

,102

10

,11002

1CHSHに対して有効な測定

1A

2A

2B1B

0 1 0 10 C D D C1 D C C D0 C D C D1 D C D C

)8

(sin21

2481

)8

(cos21

248223

2

2

p

p

=+

=

=++

=

D

C

CHSH Gameの解析• 古典の場合：ゲームの値 3/4• 量子の場合：

– 事前にAliceとBobがエンタングル状態を共有– それぞれ自分のところで部分測定できる

1cos0sin)(1

1sin0cos)(0qqqf

qqqf

+-=

+=

)8/()8/(1

)8/()8/(0

)4/()4/(1

)0()0(0

pfpf

pfpf

pfpf

ff

--=

=

=

=

bbbY

bbbY

aaaXaaaX

2/)1100( +

としてで測定、結果を回答

⇒ゲームの値 75.085.0)8/(2cos >ªp

まさしくCHSH不等式として知られるBell不等式の話そのもの他にKochen-Specker定理、擬似テレパシー, 量子グラフ彩色など種々の展開

CHSH is not the only Bell inequality〈A1B1〉+〈A1B2〉+〈A2B1〉-〈A2B2〉 ≤ 2

-〈A1〉-〈A2〉+〈B1〉+〈B2〉+〈A1B1〉+〈A1B2〉+〈A1B3〉+〈A2B1〉+〈A2B2〉-〈A2B3〉+〈A3B1〉-〈A3B2〉 ≤ 4

CHSH inequality

Correlation inequalities with A1,…,A4, B1,…,B4 [Gisin, priv. comm.]

Tsirelson’s theorem [1980] applicable to inequalities on 〈AiBj〉

13.514

Maximum in quantum case:

(Correlation inequalities; multi-party case consideredby [Werner, Wolf 2001] [Żukowski, Brukner 2002])

???5 (2 qubit system) ≤

I3322 inequality[Pitowsky, Svozil 2001]

[Collins, Gisin 2004]

8.165

2〈A1B1〉+〈A1B2〉+〈A1B3〉+2〈A1B4〉+〈A2B1〉+〈A2B2〉+2〈A2B3〉-2〈A2B4〉+〈A3B1〉+2〈A3B2〉-2〈A3B3〉-〈A3B4〉+2〈A4B1〉-2〈A4B2〉-〈A4B3〉-〈A4B4〉 ≤ 10

Computed by using SDPA http://grid.r.dendai.ac.jp/sdpa/

2〈A1B1〉+〈A1B2〉 +〈A1B4〉+〈A2B1〉-〈A2B2〉+〈A2B3〉-〈A2B4〉

+〈A3B2〉 -〈A3B4〉+〈A4B1〉-〈A4B2〉-〈A4B3〉-〈A4B4〉 ≤ 6

Most Bell inequalitiesuse some of 〈Ai〉or 〈Bj〉

Not a correlation inequality

An implication of our results

We give an upper bound for an arbitrary quantum Bell inequality

Key tools:1. Convex geometry (same as [Tsirelson 1993])

by solving a semidefinite program

〈A1B1〉+〈A1B2〉+〈A2B1〉-〈A2B2〉 ≤ 2CHSH inequalityMaximum in quantum case:

-〈A1〉-〈A2〉+〈B1〉+〈B2〉+〈A1B1〉+〈A1B2〉+〈A1B3〉+〈A2B1〉+〈A2B2〉-〈A2B3〉+〈A3B1〉-〈A3B2〉 ≤ 4

2. Combinatorial optimization(such as cut polytopes [Deza, Laurent 1997])

???5 (2 qubit system) ≤

I3322 inequality[Pitowsky, Svozil 2001]

[Collins, Gisin 2004]

量子計算量クラス確率計算量クラス古典計算量クラス

Multi-prover Interactive ProofMIP=NEXPTIME

Alice Bob

Victor (Verifier)

X

量子計算量クラス

P

NP

PSPACE

EXPNEXP

PZPP

RP co-RPNP(=EMA) co-NPBPP

BQP MA=AM1

QMAAQMA

PrQP = PP

BQPSPACE=PrQPSPACE=(N)PSPACE=IP

EQMARQMA

NQP=co-C=P

AM=AM≧2=AM2=IP2

IP=IPpoly=AMpolyQIP

EXPNEXP=MIP=QMIP

Computation Theoryundecidable

intractable=exponentialtime

tractable=polynomialtime

Halting problem of Turing Machine

nn log n

n1.193

n3.5L log nlog log n

mediansorting, FFT

tionmultiplicamatrix nn¥

linear programming

Presburger arithmeticcn22

cn222

EXPPSPACENP-complete graph isomorphism

integer factoring

traveling salesman

))3/2)log(log3/1)((logexp( nnO

decidable

n: input size

P

古典計算量クラス

P

NP

PSPACE

EXPNEXP

P

PSPACE

NEXPEXP

NP

Polynomial Time

NondeterministicPolynomial Time

Polynomial Space

Exponential Time

Tractable

Intractable

確率計算量クラス

P

NP

PSPACE

EXPNEXP

PZPP

RP co-RPNP co-NPBPP

PP

PSPACE

NEXPEXP

• A language L is in PP

• BPP: Bounded PP• RP: Randomized P• co-RP: complement of RP• ZPP: Zero-error PP

PP: Probabilistic Polynomial

,every for TM, ticprobabilis time-poly: xM$€,2/1] accepts Pr[, if (i) >Œ xMLx.2/1] accepts Pr[, if (ii) £œ xMLx

1/3 (ii) 2/3, i)( £≥0 (ii) 1/2, i)( =≥

1/2 (ii) 1, i)( £=0 (ii) 1, i)( ==

NP in terms of Probabilistic TMs• A language L is in NP

,every for TM, ticprobabilis time-poly: xM$€,0] accepts Pr[, if (i) >Œ xMLx.0] accepts Pr[, if (ii) =œ xMLx

対話計算量クラス

P

NP

PSPACE

EXPNEXP

PZPP

RP co-RPNP co-NPBPP

MA=AM1

PP

PSPACE=IP

AM=AMc≧2=AM2=IP2

IP=IPpoly=AMpoly

EXPNEXP=MIP

量子計算量クラス

P

NP

PSPACE

EXPNEXP

PZPP

RP co-RPNP(=EMA) co-NPBPP

BQP MA=AM1

QMAAQMA

PrQP = PP

BQPSPACE=PrQPSPACE=(N)PSPACE=IP

EQMARQMA

NQP=co-C=P

AM=AM≧2=AM2=IP2

IP=IPpoly=AMpolyQIP

EXPNEXP=MIP=QMIP

FPTAS, FPRAS

Quantum Adiabatic Computation,Quantum Annealing

and Ising Partition Function

Quantum adiabatic computation/quantum annealing

GroundState

GroundState

GroundState

Quantum AnnealingHamiltonian

in the Ising model

Almost Time-Reversible

Hamiltonian witheasily prepared state

Ising Model+1

+1

-1+1

-1

-1

Ising model [Ising 25]• グラフ𝐺𝐺 = 𝑉𝑉,𝐸𝐸 , 𝑣𝑣 ∈ 𝑉𝑉 :点, 𝑒𝑒 = (𝑢𝑢, 𝑣𝑣)(∈ 𝐸𝐸): 枝• スピン 𝜎𝜎 ∈ {−1,1}𝑉𝑉, 相互作用力 𝐽𝐽𝑢𝑢𝑢𝑢, 外部磁場𝑀𝑀𝑢𝑢

Hamiltonian 𝐻𝐻(𝜎𝜎) = − �(𝑢𝑢,𝑢𝑢)∈𝐸𝐸

𝐽𝐽𝑢𝑢𝑢𝑢𝜎𝜎𝑢𝑢𝜎𝜎𝑢𝑢 −�𝑢𝑢∈𝑉𝑉

𝑀𝑀𝑢𝑢𝜎𝜎𝑢𝑢

Partition function 𝑍𝑍 𝐺𝐺,𝛽𝛽 = �𝜎𝜎∈{−1,1}𝑉𝑉

𝑒𝑒−𝛽𝛽𝐻𝐻 𝜎𝜎

ÿ𝐽𝐽𝑢𝑢𝑢𝑢 > 0: ferromagnetic, 強磁性ÿ𝐽𝐽𝑢𝑢𝑢𝑢 < 0: antiferromagnetic, 反強磁性

Simulated Annealing

• Proposed byS. Kirkpatrick, C. D. Gelatt, Jr, and M. P. Vecchi: Optimization by Simulated Annealing. Science, Vol.220, No.4598 (1983), pp.671–680.

• Considered applying the Metropolis’ algorithm to the Ising model in the beginning,

Quantitative Church-Turing Thesis

• Thesis 1.1 (quantitative Church’s thesis). Any physical computing device can be simulated by a Turing machine in a number of steps polynomial in the resources used by the computing device.[Short, SICOMP 1997]

Scott Aaronson’s Research Statement

… Either

1. the Extended Church-Turing Thesis is false,

2. quantum mechanics as conventionally understood is false, or

3. the factoring problem is solvable in polynomial time on a classical computer.

http://www.scottaaronson.com/research.pdf

80

arXiv:1212.1739

arXiv:1305.4904

arXiv:1401.7087

arXiv:1404.6499

arXiv:1305.5837

arXiv:1304.4595Nature Phys. 10, 218 (2014)

arXiv:1403.4228

Series of papers

arXiv:1401.2910Science 1252319, Published online 19 June 2014.

レポート問題

1. Bell不等式について、以下の問に答えよ。1. 凸包ソフトウェア(cdd, lrsなど)を用いて、一般の

Bell不等式を求め、考察を加えよ。2. Correlation polytopeとCut polytopeそれぞれが、Bell不等式の形にどう影響するか。具体的にcovariance mapを与えて議論せよ。

2. 次の論文を読んで議論せよ。Scott Aaronson: NP-complete problems and physical reality. ACM SIGACT News, Vol.36, No.1 (2005), pp.30-52.

提出

• レポートは1問のみ回答でよい• 理7号館1階情報科学科レポートボックス• 締切：8月末（できる限り7月中旬に）

計算モデル論��ー対話証明計算モデルと�量子非局所性計算モデルー背景・動機1スライド番号 3背景・動機21927 Solvay Conference on Quantum Mechanics �http://commons.wikimedia.org/wiki/Image:Solvay_conference_1927.jpgよりEPR and Bohr`teleportation’ between distant 2 partiesEPR paradox and Bell inequalitiesBell-CHSH correlation experiment対話証明システムInteractive Proof System� [Babai 1985; Goldwasser, Micali, and Rackoff 1985]Nondeterministic Polynomial (NP)Example: Graph Non-Isomorphismスライド番号 14凸多面体入門凸多面体のV-表現とH-表現3点完全グラフK3のcut polytope上限定理・線形計画�ー Polyhederal Combinatorics ーCorrelation between 2 Events A,BCorrelation of 2 eventsCut polytope of K3Correlation polytope ⇔ Cut polytopeCorrelation of a bipartite system ofCorrelation polytope ⇔ Cut polytopeCorrelation of a bipartite system ofCorrelation polytope ⇔ Cut polytope量子対話証明へInteractive Proof System� [Babai 1985; Goldwasser, Micali, and Rackoff 1985]Multi-prover Interactive Proof量子情報とBell不等式A Two-Party One-Round Interactive Proof System�[Cleve, Høyer, Toner, Watrous CCC 2004]最も単純な場合相関表|S|=|T|=2, |A|=|B|=2相関表(|S|=|T|=2, |A|=|B|=2)16 deterministic patterns独立試行相関表一般の古典相関表じゃんけんのpay-off matrixゲームの値A Two-Party One-Round Interactive Proof System�[Cleve, Høyer, Toner, Watrous CCC 2004]CHSH GameCHSH Gamepay-off matrix Mじゃんけんのpay-off matrixCHSHゲームCHSH Gameの解析純粋状態での量子情報基礎入門１ビット n ビットスライド番号 51スライド番号 52量子並列計算部分測定射影測定1射影測定2あまり意味のない測定CHSHに対して有効な測定CHSH Gameの解析CHSH is not the only Bell inequalityAn implication of our results�量子計算量クラス�確率計算量クラス�古典計算量クラスMulti-prover Interactive Proof量子計算量クラスComputation Theory古典計算量クラス確率計算量クラスPP: Probabilistic PolynomialNP in terms of Probabilistic TMs対話計算量クラス量子計算量クラスFPTAS, FPRASQuantum Adiabatic Computation,�Quantum Annealing�and �Ising Partition FunctionQuantum adiabatic computation/quantum annealingIsing ModelIsing model [Ising 25]Simulated Annealingスライド番号 78Quantitative Church-Turing ThesisScott Aaronson’s Research StatementSeries of papersレポート問題提出

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