Feynmanの2つの提唱からグラフと計算量への展開
Exploration from Two Proposals by Feynmanto Graphs and Computational Complexity
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
ナノ量子情報エレクトロニクス研究機構
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Two Proposals by Feynman
• `Nanotechnology’``Why cannot we write the entire 24 volumes of the Encyclopedia Britannica on the head of a pin?''– Annual Meeting of American Physical Society, 1959
• Quantum Computer/SimulationSimulating Physics with Computers:quantum computer may outperform classical one– MIT Physics of Computation Conference, 1981– CLEO/IQEC, 1984
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Arranging Atoms one by one with STM/AFM
Feynman (1959):``What would happen if we could arrange the atoms one by one the way we want them'‘
• STM (Scanning TunnelingMicroscope)– Eigler, Schweizer (1990)
• AFM (Atomic Force Microscope)– Sugimoto, Abe, Hirayama, Oyabu, Custance, Morita (2005)– Sugimoto, Pou, Custance, Jelinek, Abe, Perez, Morita (2008)
Title : The Beginning Media : Xenon on Nickel (110)
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A Boy And His Atom: The World‘s Smallest Movie (IBM Research)
http://www.research.ibm.com/articles/madewithatoms.shtml
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Lattice ー tiling• Tiling by regular polygons– Grünbaum, Shephard (1977), Chavey (1989)
• 3 Platonic tilings: triangular, square, honeycomb+ 8 = 11 Archimedean tilings
Platonic tilings
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Archimedean tilings
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Motion planning for reconfiguring atoms
• Călinescu, Dumitrescu, Path (2008)– Reconfigulation in Graphs and Grids– Minimum‐weight bipartite matching
• Fu, Imai (2008)– Motion planning for square lattice case
• Fu, Imai, Moriyama (2010)– Proximity on lattices ⇒ faster matching algo.
• Graph algorithms for lattices
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From Graphs to Quantum states
• Measurement‐based Quantum Computation(MBQC)– Graph state for square lattice ‐‐‐ universality
• What types of graphs are universal?Van den Nest, Miyake, Dür, Briegel (2006)– Necessary condition:
rank width of graph is unbounded– Besides square lattice
triangular, honeycomb, Kagome: universal11
Graph Minor Theory
• Robertson, Seymour (1983‐)• Origin/example:–Wagner’s theorem (similar to Kuraowski’s theorem)• Graph is planar iff it has no K5, K3,3 as its minor
K5 K3,312
Minor
e
Contractionof e
Deletionof e
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Graph Minor Theory
• Robertson, Seymour (1983‐)
• Main Theorems:– A class of graphs closed under minor operation can be characterized by a finite set of forbidden minors
– A graph with sufficiently large tree width has a large square lattice as its minor
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Minor/Vertex‐minor
v
e
Contractionof e
Deletionof e
Minor
Vertex‐minor
Local complementationof v
Deletionof v
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Measurement‐Based Quantum Computing (MBQC)and Graph Vertex‐Minors
• Vertex‐minor– Vertex deletion σz in MBQC– Local complementation σy in MBQC
• Rank width introduced by Oum (2005)⇒Open Problem: ``A graph with very largerank width has a large grid as its vertex‐minor?’’
• Restricting to some planar tiling (Chavey 1989)Proposition:All 3+8=11 Archimedean lattices are universal.(including 3 Platonic ones and Kagome)
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Returning to Feynman’s 2nd proposal`Quantum computer may execute quantum simulation fast,while classical computer would not’
For a new problem to solve, in computer science,
• Devise an efficient algorithm to solve it
or
• Show its computational intractability, hardness– Computational Complexity: P vs. NP (BPP vs. MA)– Quantum Computational Complexity: BQP vs. QMA,…
So we should do this17
Quantum Complexity Theory
• BQP: quantum simulation (Lloyd 1996)• QMA: ground state energy of a 2‐local Hamiltonian
(Kempe, Kitaev, Regev 2006)• QMA(2): pure‐state N‐representability
(Liu, Christandl, Verstraete 2007)
QMA (Watrous 2000)QMA(2) (Kobayashi,Matumoto, Yamakami 2003,2009)‐‐‐ Quantum (Multi‐prover) Interactive Proof
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QMA(2)(Kobayashi, Matsumoto, Yamakami 2003, 2009)
• Two Merlins (provers) give an advice to Arthur (verifier)
• QMA(2)=QMA(k) (k>2)(Harrow, Montanaro, FOCS 2010)
• Tool to demonstrate the computational intractability much deeper
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Quantum Nonlocality to Graph Problem
• Bell inequality model= 2‐prover 1‐round Quantum Interactive Proof
• Directly connected with – graph cuts, cut polytope– its semidefinite relaxation
• Quantum chromatic number χQ(G)(Avis, Hasegawa, Kikuchi, Sasaki 2006;Cameron, Montanaro, Newman, Severini, Winter 2007)
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Chromatic number and Perfect Graph Theorem
• χ(G): minimum number of colors such that two adjacentvertices have different colors
• ω(G): maximum number of vertices forming a complete graph
• Graph G is perfect if χ(GS)=ω(GS) for induced subgraph GS for any subset S of vertices
• Weak Perfect Graph Theorem (Lovász 1972):Graph is perfect iff its complement is perfect.
⇒• Polyhedral characterization• Semidefinite Relaxation through Lovász theta function (1979)
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Concluding remarks
• Connection with graph theory, especially– Graph minor theory– Perfect graphs
• Quantum complexity theory–We should ask intrinsic complexity of problems,• not only by devising efficient quantum algorithms• but also through quantum complexity classes
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Acknowledgment• Figures on Archimedean tilings were provided by Akihiro Hashikura, who is working jointly for our periodic graph project.
• Problems around arranging atoms were conducted with Norie Fu and Sonoko Moriyama.
• Valuable comments on complexity issues from Hirotada Kobayashi and Keiji Matsumoto are greatly appreciated.
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量子シミュレーションの計算量理論へ
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
東京大学ナノ量子情報エレクトロニクス研究機構
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Quantum Simulation by Quantum Computer
• Universality of Computation• Analog vs. Digital Computation– Classical case / Quantum case
• Computational Complexity Theoryfor any computational problem!
– over the Reals– for Differential Equations– and then for Quantum Simulation!
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注: MIT Physics of Computation Conference 1981
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風洞は「計算装置」Goldstein, von Neumann
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物理シミュレーションと万能計算
力学系シミュレーションをとある物理実験で実行
• 風洞実験装置 ≠ universal computation• 量子コンピュータ= universal computation
汎用性の観点(cf. Universal Turing machine 1936)⇒ universal computationを目指す
(専用計算機の限界)
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Digital vs. Analog
• Digital Computation, now everywhere• Analog Computer– コンピュータ開発黎明期に存在、廃れてきた歴史
–近似精度を思いのままに制御する困難さ
– `Analog’, `アナログ’という言葉の多義さ
• 数値を、長さ・回転角・電流などの連続的に変化する物理量で示すこと。⇔デジタル[大辞泉]
• 離散値に対する連続値、実数
• 類似性
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Church‐Turing Thesis
[P. Shor, SICOMP, 1997]
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(Universal) Quantum Turing Machine
• [Deutsch 1985]• [Bernstein, Vazirani 1997] ,[Adleman, DeMarrais, Huang 1997]
⇒ 3/5, 4/5の振幅でε近似実現
有限離散値・離散時間!
Perfectly, Digital, yet with controllable error
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Computational complexity over the Reals
• Blum, Shub, Smale (1989)モデル- analog?–実数を1 wordに正確に蓄え、演算も正確にできる
–その上でのNP完全性等の理論
• Julia set, undecidability• 4‐feasibility problem, NP‐complete
• Ko, Friedman (1982)モデル- digital?–実数値をm ビット2進数でデジタル近似、
–そのm を入力サイズに入れて計算時間を定義
⇒より高精度の近似解を求める-より時間がかかる
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Computational Compexityfor Solving Differential Equation
• Ko, Friedman (1982):解析的ならP(多項式時間)
• Kawamura (IEEE CCC 2009)Lipschitz連続なものでPSPACE(多項式領域量)完全なもの存在
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Returning to Feynman’s proposal`Quantum computer may execute quantum simulation fast,
while classical computer would not’
For a new problem to solve, in computer science,• Devise an efficient algorithm to solve itor• Show its computational intractability, hardness– Computational Complexity: P vs. NP (BPP vs. MA)– Quantum Computational Complexity: BQP vs. QMA,…
So we should do this 37
Quantum Complexity Theory• BQP: quantum simulation (Lloyd 1996)• QMA: ground state energy of a 2‐local Hamiltonian
(Kempe, Kitaev, Regev 2006)• QMA(2): pure‐state N‐representability
(Liu, Christandl, Verstraete 2007)
QMA (Watrous 2000)QMA(2) (Kobayashi,Matsumoto, Yamakami 2003,2009)‐‐‐ Quantum (Multi‐prover) Interactive Proof QMA(k)=QMA(2) (k≥2) (Harrow, Montanaro, FOCS, 2010)Tool to demonstrate the computational intractability deeper
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課題
• 実数計算量
–多変数関数・多価関数の理論構築
• 量子計算量
– QMA完全性周辺での近似可能性解明
• 物理シミュレーションの計算量解明へ
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Approximability
イジング分配関数の古典・量子計算について
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
東京大学ナノ量子情報エレクトロニクス研究機構
Ising model [Ising 25]
• グラフ 点 枝
• スピン , 相互作用力 , 外部磁場
, ∈ ∈
∈ ,
0: ferromagnetic, 強磁性 0: antiferromagnetic, 反強磁性
Quantum Algorithm for Partition Function
Van den Nest, Dür, Vidal, Briegel (PRA 2007)/
∈ ∈
過去から現在そして未来へ
• グラフ理論と統計物理は、数十年前にPlanar graphでIsingモデル分配関数計算を軸に出会ったことがあった
• グラフ理論・アルゴリズム論からは着実な発展
– MCMCからFPTASまで
– Tutte多項式([Sekine, I, Tani 95]以来の研究も)– Exponential‐time algorithmicsの展開
• 今一度さらなる出会いが量子アルゴリズムでも
本スライド:Ising分配関数計算を軸に古典・量子アルゴリズムの先端へ
Partition function, planar case
平面グラフの場合(正方格子を含む):• 有限平面グラフ:完全マッチングの数の多項式時間よりP[Kasteleyn 61] (Bergeの本参照), [Temperley, Fisher 61]
• 遡るとTutte行列、Pfaffianと完全マッチング[Tutte 47]
• Lovász, Plummer (82)のMatching Theoryの本参照
– Chap. 8 Determinants and matchings, 8.1. Permanents,8.3. The Pfaffian and the number of perfect matchings,8.7. Two applications to physical science, etc.
Partition function, complexity
厳密解法
• NP完全 [Barahona, JPA 82], [Istrail, STOC 00]• Jerrum, Sinclair (SICOMP 93)–#P‐complete even for ferromagnetic case–No FPRAS for general cases unless NP=RP
Partition function, approximability• Ferromagnetic case:– Jerrum, Sinclair (SICOMP 93): FPRASRapidly mixing of a Markov chain ⇒MCMC
– For states of spanning subsets, Not for spin states
• Antiferromagnetic case:– Sinclair, Srivastava, Thurley (SODA 12): FPTASfor graphs of degree inside the region
for a unique Gibbs measure of a ‐regular tree– Based on [Weitz, STOC 06] for FPTAS for #stable sets and more
– Sly, Sun (arXiv:1203.2602; FOCS 12) No FPRAS unless NP=RP for outside the region
FPTAS, FPRASComputation of • FPTAS (Fully Polynomial‐Time Approximation Scheme):
Exponential‐time Algorithms
厳密解法(NP完全なのでexponential algo.で)• 既存研究Tutte多項式に対するexponential algorithmsをIsing partition functionに適用
• ではTutte多項式とは?
– Tutte多項式計算の量子アルゴリズムも重要
Tutte多項式とは• 1912年頃からBirkhoffら
chromatic polynomial• Tutte, Whitney による2変数多項式
枝部分集合Aのランク
縮約削除
)2)(1();( 3 K
Tutte‐Gröthendieck invariant
Tutte多項式の性質・計算量
特殊な場合
• グラフの諸性質– 森・木・spanning sets等個数
– chromatic/flow polynomial– Reliability
• 統計物理– Ising, Potts model– percolation
• 結び目Jones多項式
• マトロイド, 符号理論, etc.
計算量
• 5点を除き#P完全
アルゴリズム(後述)• Exact– mildly/moderately exponential
• Approximate– Randomized– Quantum– Deterministic
Tutte平面[Björklund, Husfeldt, Kaski, Koivisto]より
Ising
Potts
Jones
Partition function & Tutte polynomial
•
•
• の条件は、グラフに 点新たに追加で解決可能
Tutte多項式計算アルゴリズム
厳密解法
•一般 ∗ time
•平面グラフ ∗ [Sekine, HI, Tani 95; 関根, HI, KI 98]
• ∗ , tree‐width of G
• Vertex‐exponential time [Björklund, Husfeldt, Kaski, Koivisto 08]∗ bounded‐degreeなら ∗
17 17正方格子の厳密計算可能
[Andrzejak 98; Noble 98]
Tutte多項式計算
近似アルゴリズム• PRAS [Alon, Frieze, Welsh 95]
量子近似アルゴリズム• 加法近似 [Aharonov, Jones, Landau 06]More recent papers• I. Arad, Z. Landau: Quantum computation and the evaluation of tensor networks. Arxiv:0805.0040v3; SICOMP 2010.
• M. Van den Nest: Simulating quantum computers with probabilistic methods. arXiv:0911.1624v3; QIC 2011.
Quantum Algorithm for Partition Function
Van den Nest, Dür, Vidal, Briegel (PRA 2007)/
∈ ∈
例:
の添字で表記
+ + +
Stabilizer, ∈
)
1
0
Tensor network I: subcubic treeVan den Nest et al. [PRA 07]の図
行列の 上
1 2 3 4 5 6
Tensor network II: Graph Minor Theory
1 2 3 4 5 6
Van den Nest et al. [PRA 07]の図
∈
Oum, Seymour [JCT B 06]
Theorem 5. [Van den Nest et al. 07]TTN description of graph state | , inner product | , and MQC on , can be classically computed/simulatedin poly , 2 time.
Tensor network III: Schmidt decompositions
1 2 3 4 5 6
[Van den Nest, Dür, Vidal, Briegel PRA 07]([Shi, Duan, Vidal PRA 06]の結果利用)
, ∈ 0,1
計算過程の例
Graph state
Tree‐width (83~)[Robertson, Seymour 90]
Branch‐width (88~)[Robertson, Seymour 91]Subcubic tree for widthsGeneralization to matroids
Rank‐width (04~)[Oum, Seymour 06]
AKLT model[Affleck, Lieb, Kennedy, Tasaki 87]
Matrix Product State (MPS)[Fannes, Nactergaele, Werner 92]
Projected Entangles Pair Strategies (PEPS)[Verstraete, Cirac 04]
Subcubic Tensor Tree Network[Shi, Duan, Vidal 06]
MQC and rank‐width[Van den Nest, Miyake, Dür, Briegel 06]
Concluding Remarks1.Originally Classical vs. Classical via Quantum• vs. (N.B. 1)
2.More recent papers towards quantum algorithmics• I. Arad, Z. Landau: Quantum computation and the evaluation of tensor networks. Arxiv:0805.0040v3; SICOMP 2010.•M. Van den Nest: Simulating quantum computers with probabilistic methods. arXiv:0911.1624v3; QIC 2011.
3.There are many quantum algorithms!
4.One direction: Quantum Computing vs. Graph Theory