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## Computing with Quanta for mathematics students

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### Computing with Quantafor mathematics students

Table of ContentsTable of ContentsTable of ContentsTable of Contents

Financial supports from Kinki Univ.,

MEXT and JSPS

Mikio Nakahara

Department of Physics &

Research Centre for Quantum Computing

Kinki University, Japan

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

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I. Introduction: Computing with Physics

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More complicated Example

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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.

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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

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2. Computing with Vectors and Matrices2.1 Qubit

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Qubit |ψ〉

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2.2 Two-Qubit System

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Tensor Product Rule

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Entangled state (vector)

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2.3 Multi-qubit systems

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2.4 Algorithm = Unitary Matrix

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Unitary Matrices acting on n qubits

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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

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3. Brief Introduction to Quantum Theory

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Axioms of Quantum Physics

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Example of a measurement

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Axioms of Quantum Physics (cont’d)

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Qubits & Matrices in Quantum Physics

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- 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

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4. Quantum Gates, Quantum Circuit and Quantum Computer

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4.2 Quantum Gates

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Hadamard transform

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4.3 Universal Quantum Gates

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4.4 Quantum Parallelism

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- 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

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Q Teleportation Circuit

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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

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- 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

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5. Simple Quantum Algorithm- Deutsch’s Algorithm -

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- 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

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Difficulty of Prime Number Facotrization

- Factorization of N=89020836818747907956831989272091600303613264603794247032637647625631554961638351 is difficult.
- It is easy, in principle, to show the product of p=9281013205404131518475902447276973338969 and q =9591715349237194999547 050068718930514279 is N.
- This fact is used in RSA (Rivest-Shamir-Adleman) cryptosystem.

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Shor’s Factorization algorithm

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

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NMR molecule and pulse sequence ( (~300 pulses~ 300 gates)

perfluorobutadienyl iron complex with the two 13C-labelled

inner carbons

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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!

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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!

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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

量子暗号鍵配布 1

三省堂サイエンスカフェ 2009年6月

量子暗号鍵配布 2

三省堂サイエンスカフェ 2009年6月

量子暗号鍵配布 3

三省堂サイエンスカフェ 2009年6月

イブの攻撃

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。

三省堂サイエンスカフェ 2009年6月

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

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7. 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.

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7.1 Computational path in U(2n)

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Map of Kyoto

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7.2 Optimization of 2-qubit gates

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NMR Hamiltonian

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Time-Optimal Path in SU(4)

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Cartan Decomposition of SU(4)

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How to find the Cartan Decomposition

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Example: CNOT gate

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6. Warp-Drive を用いた量子アルゴリズムの加速 (quant-ph/0411153)

奈良女子大学セミナー 28 Jan. 2005

7. 実験結果

- Carbon-13 で置換したクロロフォルム

qubit 1 = 13C, qubit 2 = H

初期状態

出力状態

Qubit 2

Qubit 1

奈良女子大学セミナー 28 Jan. 2005

8. Summary I: Cartan分解

奈良女子大学セミナー 28 Jan. 2005

奈良女子大学セミナー 28 Jan. 2005

Power of Entanglement

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