Computing with Quanta for mathematics students

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# Computing with Quanta for mathematics students - PowerPoint PPT Presentation

Financial supports from Kinki Univ., MEXT and JSPS. Computing with Quanta for mathematics students . Mikio Nakahara Department of Physics & Research Centre for Quantum Computing Kinki University, Japan. Table of Contents. 1. Introduction: Computing with Physics

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## Computing with Quanta for mathematics students

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Financial supports from Kinki Univ.,

MEXT and JSPS

### Computing with Quantafor mathematics students

Mikio Nakahara

Department of Physics &

Research Centre for Quantum Computing

Kinki University, Japan

• 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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• 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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Qubit |ψ〉

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Bloch Sphere: S3→　S2

π

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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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• 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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Actual Qubits

Trapped Ions

Neutral Atoms

Molecules (NMR)

Superconductors

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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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4.2 Quantum Gates

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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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5. Quantum Teleportation

Unknown Q State

Bob

Initial State

Alice

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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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• 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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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

量子暗号鍵配布 1

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

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。

• 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)

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

Qubit 2

Qubit 1

10パルス  4パルス，1/J  1/2Jによるスペクトルの改善

8. Summary I: Cartan分解

Summary II: Warp-Drive

Power of Entanglement

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