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1 量子計算モデル 参考資料 今井浩 東京大学情報理工学系研究科コンピュータ科学専攻 JST ERATO-SORST 量子情報システムアーキテクチャ

2 情報を処理する観点から

3 どうして量子計算 通信を研究するのか? 量子計算によって 古典計算 通信では現実的に解けない問題を 実際に解く 今のコンピュータの高速化に量子力学が障害 その量子力学を肯定的に使って 今解けないものを解けるようにする 古典計算 確率計算の発展としての量子計算 新計算モデルの開拓ー限界の打破 今のコンピュータ 情報処理方式への還元

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

5 計算 ( 情報処理 ; 通信含む ) とは何か 情報を まず何らかのルールで与え ( 入力 ) 何らかの物理状態で表現し ( 符号化 ) 所望の状態になるように操作し ( 計算 ) 所望の結果を得る ( 出力 ) そして通信では その物理状態を伝送する ( 符号化 復号化 )

6 1930 年代 計算モデル Turing Machine Recursive Function Theory λ Calculus 1980 年中ごろ : 計算は対話だ! C. H. Papadimitiriou: Games against Nature. J. Computer & System Sciences, 現代 : Nature is computing R. M. Karp (Turing 賞 (1985), 京都賞 (2008))

7 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 Peggy (Prover) Oracle x = 1, y = 0, z = 1 にして! ( x y z) ( x y z) ( x y z) は Victor (Verifier) NP satisfiable?

8 Computation Theory undecidable decidable intractable= exponential time tractable= polynomial time cn cn 2 2 EXP PSPACE NP-complete exp( O((logn)1/3(loglogn)2/3)) n: input size P n 3.5 Llog n log log n n n log n n Halting problem of Turing Machine Presburger arithmetic traveling salesman graph isomorphism integer factoring linear programming n n matrix multiplication sorting, FFT median

9 量子計算が現代計算を凌駕する点!? Deutch-Jozsa algorithm Exponential gap, but for a partial function Shor s quantum polynomial factoring Classically, only subexponential factoring Polynomial Las Vegas algorithm for primality testing does not provide a rigorous proof Grover s search algorithm Practically important, but theoretical non-exponential そして多数 : 隠れ部分群問題 量子ウォーク / 探索

10 量子力学の観点から

11 1927 Solvay Conference on Quantum Mechanics 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

12 EPR and Bohr

13 Bell-CHSH correlation experiment two measurements apply one two measurements apply one A 1 entangled quantum state B 1 A 2 Alice Bob B 2 Classical correlation: P A1 ) P( B1 ) + P( A1 B1 ) + P( A1 B2 ) + P( A2 B1 ) P( A2 B ) ( 2 Quantum correlation: P( A ) P( B1 ) + P( A1 B1 ) + P( A1 B2 ) + P( A2 B1 ) P( A2 B2 )

14 量子非局所性 量子対話証明 量子計算量理論

15 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 IP=PSPACE Peggy (Prover) Interactive Communication Victor (Verifier)

16 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 Peggy (Prover) Oracle x = 1, y = 0, z = 1 にして! ( x y z) ( x y z) ( x y z) は Victor (Verifier) NP satisfiable?

17 Example: Graph Non-Isomorphism INPUT: Graph Non-Isomorphism Problem (GNI) Two graphs G 1, G 2 of n vertices QUESTION: For all permutation π S n on vertices, π (G 1 ) G 2? Protocol of verifier V: 1. Choose an index i {1,2} of graphs and a permutation π S n at random. Send a graph π (G i ) to prover P to ask which of the two is isomorphic to π (G i ). 2. Receive an index j from P. Accept iff i = j.

18 1 2 同型 非同型

19 Multi-prover Interactive Proof Alice MIP=NEXPTIME X Bob Victor (Verifier)

20 NEXP EXP NEXP=MIP=QMIP EXP QIP 量子計算量クラス IP=IP poly =AM poly PSPACE BQPSPACE=PrQPSPACE=(N)PSPACE=IP PrQP = PP AQMA QMA NQP=co-C = P NP BQP BPP RQMA MA=AM 1 EQMA NP(=EMA) co-np AM=AM 2 =AM 2 =IP 2 RP co-rp ZPP P P

21 EPR paradox and Bell inequalities Einstein, Podolsky, Rosen (1935) quantum entanglement vs. relativity theory Bell inequality (1964) Entanglement/Nonlocalit violation entanglement measure instantly (local) faster than light state change 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 Semidefinite Programming (Avis, Imai, Ito 2006)

22 A Two-Party One-Round Interactive Proof System [Cleve, Høyer, Toner, Watrous CCC 2004] Alice Provers 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 V( a, b; s, t) = 1 0 < 1 Provers win Provers lose 答 a {01, } 答 b {01, } Bob 質問 t {01, } 質問 s {01, } Victor (Verifier)

23 A Two-Party One-Round Interactive Proof System [Cleve, Høyer, Toner, Watrous CCC 2004] Alice Provers 事前に回答戦略を協力して練ってよい 関数 a: S A, b: T Bを確定 質問が始まったら通信できないゲームの値 V( a, b; s, t) = 1 0 < 1 答 a A ={ 01,,, A 1} Provers win Provers lose 答 b B ={ 01,,, B 1} Bob 質問 s S ={ 01,,, S 1} 質問 t T ={ 01,,, T 1} Victor (Verifier)

24 Alice CHSH Game V( a, b; s, t) = 1 0 a b= s t otherwise 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 a( 0) = 0, a(1) = 1; b(0) = 1, b(1) = 0 ゲームの値 3/4 答 a {01, } 答 b {01, } Bob 質問 s {01, } 質問 t {01, } Victor (Verifier)

25 CHSH Game の解析 古典の場合 : ゲームの値 3/4 量子の場合 : φ φ 事前にAliceとBobがエンタングル状態 ( )/ 2 を共有 それぞれ自分のところで部分測定できる X a = φ (0) (0) ( θ) = cosθ 0 + sinθ 1 0 a φa 0 として X a = φ ( /4) ( /4) ( θ) = sinθ 0 + cosθ 1 1 a π φa π 1 Yb = φ ( π /8) φ ( π /8) 0 b b Yb = φ ( π /8) φ ( π /8) 1 b b ゲームの値 cos 2( π /8) 0.85 > で測定 結果を回答 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

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