Computing with Quanta for mathematics students Mikio Nakahara Department of Physics & Research Centre for Quantum Computing Kinki University, Japan Financial supports from Kinki Univ., MEXT and JSPS

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 2

William & Mary I. Introduction: Computing with Physics 3

William & Mary More complicated Example 4

William & Mary Quantum Computing/Information Processing Quantum computation & information processing make use of quantum systems to store and process information. Exponentially fast computation, totally safe cryptosystem, teleporting a quantum state are possible by making use of states & operations which do not exist in the classical world. 5

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 6

2. Computing with Vectors and Matrices 2.1 Qubit William & Mary 7

Qubit |ψ 〉 8

Bloch Sphere: S 3 → S 2 William & Mary π 9

2.2 Two-Qubit System 10

Tensor Product Rule William & Mary 11

Entangled state (vector) William & Mary 12

William & Mary 2.3 Multi-qubit systems 13

William & Mary 2.4 Algorithm = Unitary Matrix 14

Unitary Matrices acting on n qubits William & Mary 15

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 16

3. Brief Introduction to Quantum Theory William & Mary 17

Axioms of Quantum Physics William & Mary 18

Example of a measurement William & Mary 19

Axioms of Quantum Physics (cont’d) William & Mary 20

Qubits & Matrices in Quantum Physics William & Mary 21

Actual Qubits William & Mary 22 Trapped Ions Molecules (NMR) Neutral Atoms Superconductors

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 23

William & Mary 4. Quantum Gates, Quantum Circuit and Quantum Computer 24

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William & Mary 4.2 Quantum Gates 26

William & Mary Hadamard transform 27

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William & Mary 4.3 Universal Quantum Gates 29

William & Mary 4.4 Quantum Parallelism 30

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 31

5. Quantum Teleportation William & Mary 32 Unknown Q State Initial State Bob Alice

Q Teleportation Circuit William & Mary 33

William & Mary 34 As a result of encoding, qubits 1 and 2 are entangled. When Alice measures her qubits 1 and 2, she will obtain one of 00, 01, 10, 11. At the same time, Bob’s qubit is fixed to be one of the four states. Alice tells Bob what readout she has got. Upon receiving Alice’s readout, Bob will know how his qubit is different from the original state (error type). Then he applies correcting transformation to his qubit to reproduce the original state. Note that neither Alice nor Bob knows the initial state Example: 11

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 35

William & Mary 5. Simple Quantum Algorithm - Deutsch’s Algorithm - 36

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William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Quantum Teleportation 6. Simple Quantum Algorithm 7. Shor’s Factorization Algorithm 39

William & Mary Difficulty of Prime Number Facotrization Factorization of N= is difficult. It is easy, in principle, to show the product of p= and q = is N. This fact is used in RSA (Rivest-Shamir- Adleman) cryptosystem. 40

William & Mary Shor’s Factorization algorithm 41

William & Mary Realization using NMR (15=3×5) L. M. K. Vandersypen et al (Nature 2001) 42

William & Mary NMR molecule and pulse sequence ( (~300 pulses~ 300 gates) perfluorobutadienyl iron complex with the two 13C-labelled inner carbons 43

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William & Mary Foolproof realization is discouraging … ? Vartiainen, Niskanen, Nakahara, Salomaa (2004) Foolproof implementation of factorization 21=3 X 7 with Shor’s algorithm requires at least 22 qubits and approx. 82,000 steps! 45

William & Mary Summary Quantum information is an emerging discipline in which information is stored and processed in a quantum-mechanical system. Quantum information and computation are interesting field to study. (Job opportunities at industry/academia/military). It is a new branch of science and technology covering physics, mathematics, information science, chemistry and more. Thank you very much for your attention! 46

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4. 量子暗号鍵配布 三省堂サイエンスカフェ 2009 年 6 月 48

量子暗号鍵配布 1 三省堂サイエンスカフェ 2009 年 6 月 49

量子暗号鍵配布 2 三省堂サイエンスカフェ 2009 年 6 月 50

量子暗号鍵配布 3 三省堂サイエンスカフェ 2009 年 6 月 51

量子暗号鍵配布 4 三省堂サイエンスカフェ 2009 年 6 月 52 イブがいなければ、 4N の量子ビットのうち、平均し て 2N 個は正しく伝わる。

イブの攻撃 三省堂サイエンスカフェ 2009 年 6 月 53 2N 個の正しく送受された量子ビットのうち、その半 分の N 個を比べる。もしイブが盗聴すると、その中 のいくつか (25 %) は間違って送受され、イブの存在 が明らかになる。

William & Mary Table of Contents 1. Introduction: Computing with Physics 2. Computing with Vectors and Matrices 3. Brief Introduction to Quantum Theory 4. Quantum Gates, Quantum Circuits and Quantum Computer 5. Simple Quantum Algorithm 6. Shor’s Factorization Algorithm 7. Time-Optimal Implementation of SU(4) Gate 54

William & Mary Time-Optimal Implementation of SU(4) Gate Barenco et al’s theorem does not claim any optimality of gate implementation. Quantum computing must be done as quick as possible to avoid decoherence (decay of a quantum state due to interaction with the environment). Shortest execution time is required.

William & Mary Computational path in U(2 n )

William & Mary 57 Map of Kyoto

William & Mary Optimization of 2-qubit gates

William & Mary 59 NMR Hamiltonian

William & Mary 60 Time-Optimal Path in SU(4)

William & Mary 61 Cartan Decomposition of SU(4) Cartan Decomposition of SU(4)

William & Mary 62 How to find the Cartan Decomposition

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William & Mary 64 Example: CNOT gate

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奈良女子大学セミナー 28 Jan Warp-Drive を用いた量子アルゴリズ ムの加速 (quant-ph/ )

奈良女子大学セミナー 28 Jan

奈良女子大学セミナー 28 Jan

奈良女子大学セミナー 28 Jan 実験結果 Carbon-13 で置換したクロロフォルム qubit 1 = 13 C, qubit 2 = H 初期状態 出力状態 Qubit 1Qubit 2

奈良女子大学セミナー 28 Jan Field Gradient 法による NMR スペクトル 10 パルス  4 パルス， 1/J  1/2J によるスペクトルの 改善

奈良女子大学セミナー 28 Jan Summary I: Cartan 分解

奈良女子大学セミナー 28 Jan Summary II: Warp-Drive

William & Mary Power of Entanglement 73