計算モデル論 ー対話証明計算モデルと 量子非局所 ... - imai...
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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
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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-)
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• 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/
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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
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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
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EPR and Bohr
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`teleportation’ between distant 2 parties
Alice Bob
entangled quantum state
Alice measures Bob’s statebecomesinstantaneously
Alice measures Bob’s statebecomesinstantaneously
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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
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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
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対話証明システム
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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
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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
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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?
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1
2 3
4
4
3 1
2
同型
非同型
1 2
34
1
23
4
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凸多面体入門
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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 £◊=
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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
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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
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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
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Correlation of 2 events
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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
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Correlation polytope ⇔ Cut polytope
)(Cor 2K□ )(Cut 2K—
□
A B
)(AP
X
)(BP
)(ABP
covariance mapping
23 KK —=
suspension
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Correlation of a bipartite system of },{ and },{ 2121 BBAA
))(),( no()(),(),(),()(),( ),(),(
??? amongn Correlatio
2121
22122111
2121
BBPAAPBAPBAPBAPBAPBPBPAPAP
1A 1B)( 11BAP
2A 2B)( 22BAP
)( 21BAP )( 12BAP
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Correlation polytope ⇔ Cut polytope
)(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
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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
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Correlation polytope ⇔ Cut polytope
)(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
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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
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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
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Multi-prover Interactive ProofMIP=NEXPTIME
Alice Bob
Victor (Verifier)
X
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量子情報とBell不等式
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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
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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 =
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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
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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)
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16 deterministic patterns
˙˙˙˙
˚
˘
ÍÍÍÍ
Î
È
0000010100000101
˙˙˙˙
˚
˘
ÍÍÍÍ
Î
È
1010000010100000
˙˙˙˙
˚
˘
ÍÍÍÍ
Î
È
0101000000000101
˙˙˙˙
˚
˘
ÍÍÍÍ
Î
È
0101000001010000
˙˙˙˙
˚
˘
ÍÍÍÍ
Î
È
1001000000001001
…
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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)
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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
…
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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
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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
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CHSH Game
Alice Bob
Victor (Verifier)
}1,0{Œt質問}1,0{Œs質問
}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
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CHSH Game
Alice Bob
Victor (Verifier)
}1,0{Œt質問}1,0{Œs質問
}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
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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
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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
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 +
に状態はで測定を確率
に状態はで測定を確率
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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