Computing with Computing with Quanta Quanta for mathematics students for mathematics students Mikio Nakahara Mikio Nakahara Department of Physics & Department of Physics & Research Centre for Quantum Research Centre for Quantum Computing Computing Kinki University, Japan Kinki University, Japan Financial supports from Kinki Univ., MEXT and JSPS
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Computing with Quanta for mathematics students Mikio Nakahara Department of Physics & Research Centre for Quantum Computing Kinki University, Japan Financial.
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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
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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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
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
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
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
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
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
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
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
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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
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量子暗号鍵配布 1
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量子暗号鍵配布 2
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量子暗号鍵配布 3
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量子暗号鍵配布 4
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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
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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。
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
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 HamiltonianNMR Hamiltonian
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Time-Optimal Path in SU(4)Time-Optimal Path in SU(4)
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Cartan Decomposition of SU(4)Cartan Decomposition of SU(4)
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How to find the Cartan DecompositionHow to find the Cartan Decomposition