Computing with Quanta for mathematics students Mikio Nakahara Department of Physics & Research...

Computing with Computing with QuantaQuanta

for mathematics students for mathematics students

Mikio NakaharaMikio NakaharaDepartment of Physics & Department of Physics & Research Centre for Quantum Research Centre for Quantum ComputingComputingKinki University, JapanKinki University, Japan

Financial supports from Kinki Univ.,

MEXT and JSPS

Colloquium @ 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

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I. Introduction: Computing with PhysicsI. 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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2. Computing with Vectors and Matrices2.1 Qubit

Colloquium @ William & Mary 7


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


Entangled state (vector)



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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3. Brief Introduction to Quantum Theory


Axioms of Quantum Physics


Example of a measurement


Axioms of Quantum Physics (cont’d)


Qubits & Matrices in Quantum Physics


Actual Qubits


Trapped Ions

Molecules (NMR)

Neutral Atoms

Superconductors




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4. Quantum Gates,4. Quantum Gates, Quantum Circuit Quantum Circuit and Quantum Computerand 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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5. Quantum Teleportation


Unknown Q State

Initial State

Bob

Alice

Q Teleportation Circuit



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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5. Simple Quantum Algorithm5. Simple Quantum Algorithm- - Deutsch’s Algorithm -Deutsch’s 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-labelledinner carbons 43



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 48

　量子暗号鍵配布 1


量子暗号鍵配布 2






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。



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.


7.1 Computational path in U(2n)


Map of Kyoto


7.2 Optimization of 2-qubit gates


NMR HamiltonianNMR Hamiltonian


Time-Optimal Path in SU(4)Time-Optimal Path in SU(4)


Cartan Decomposition of SU(4)Cartan Decomposition of SU(4)


How to find the Cartan DecompositionHow to find the Cartan Decomposition



Example: CNOT gate


奈良女子大学セミナー 28 Jan. 2005 66

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



奈良女子大学セミナー 28 Jan. 200569

7. 7. 実験結果実験結果

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 1

Qubit 2


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


8. Summary I: Cartan8. Summary I: Cartan 分解分解


Summary II: Warp-DriveSummary II: Warp-Drive


Power of Entanglement

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