computing with quanta for mathematics students mikio nakahara department of physics & research...
TRANSCRIPT
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
2
Colloquium @ William & Mary
I. Introduction: Computing with PhysicsI. Introduction: Computing with Physics
3
Colloquium @ William & Mary
More complicated Example
4
Colloquium @ 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
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
6
2. Computing with Vectors and Matrices2.1 Qubit
Colloquium @ William & Mary 7
Colloquium @ William & Mary
Qubit |ψ 〉
8
Bloch Sphere: S3 → S2
Colloquium @ William & Mary
π
9
Colloquium @ William & Mary
2.2 Two-Qubit System
10
Tensor Product Rule
Colloquium @ William & Mary 11
Entangled state (vector)
Colloquium @ William & Mary 12
Colloquium @ William & Mary
2.3 Multi-qubit systems
13
Colloquium @ William & Mary
2.4 Algorithm = Unitary Matrix
14
Unitary Matrices acting on n qubits
Colloquium @ William & Mary 15
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
16
3. Brief Introduction to Quantum Theory
Colloquium @ William & Mary 17
Axioms of Quantum Physics
Colloquium @ William & Mary 18
Example of a measurement
Colloquium @ William & Mary 19
Axioms of Quantum Physics (cont’d)
Colloquium @ William & Mary 20
Qubits & Matrices in Quantum Physics
Colloquium @ William & Mary 21
Actual Qubits
Colloquium @ William & Mary 22
Trapped Ions
Molecules (NMR)
Neutral Atoms
Superconductors
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
23
Colloquium @ William & Mary
4. Quantum Gates,4. Quantum Gates, Quantum Circuit Quantum Circuit and Quantum Computerand Quantum Computer
24
Colloquium @ William & Mary 25
Colloquium @ William & Mary
4.2 Quantum Gates
26
Colloquium @ William & Mary
Hadamard transform
27
Colloquium @ William & Mary 28
Colloquium @ William & Mary
4.3 Universal Quantum Gates
29
Colloquium @ William & Mary
4.4 Quantum Parallelism
30
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
31
5. Quantum Teleportation
Colloquium @ William & Mary 32
Unknown Q State
Initial State
Bob
Alice
Q Teleportation Circuit
Colloquium @ William & Mary 33
Colloquium @ 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
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
35
Colloquium @ William & Mary
5. Simple Quantum Algorithm5. Simple Quantum Algorithm- - Deutsch’s Algorithm -Deutsch’s Algorithm -
36
Colloquium @ William & Mary 37
Colloquium @ William & Mary 38
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
39
Colloquium @ William & Mary
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.
40
Colloquium @ William & Mary
Shor’s Factorization algorithm
41
Colloquium @ William & Mary
Realization using NMR (15=3×5)L. M. K. Vandersypen et al (Nature 2001)
42
Colloquium @ William & Mary
NMR molecule and pulse sequence ( (~300 pulses~ 300 gates)
perfluorobutadienyl iron complex with the two 13C-labelledinner carbons 43
Colloquium @ William & Mary 44
Colloquium @ 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
Colloquium @ 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
Colloquium @ William & Mary 47
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。
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. Simple Quantum Algorithm 6. Shor’s Factorization Algorithm 7. Time-Optimal Implementation of SU(4) Gate
54
Colloquium @ William & Mary 55
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.
Colloquium @ William & Mary 56
7.1 Computational path in U(2n)
Colloquium @ William & Mary 57
Map of Kyoto
Colloquium @ William & Mary 58
7.2 Optimization of 2-qubit gates
Colloquium @ William & Mary 59
NMR HamiltonianNMR Hamiltonian
Colloquium @ William & Mary 60
Time-Optimal Path in SU(4)Time-Optimal Path in SU(4)
Colloquium @ William & Mary 61
Cartan Decomposition of SU(4)Cartan Decomposition of SU(4)
Colloquium @ William & Mary 62
How to find the Cartan DecompositionHow to find the Cartan Decomposition
Colloquium @ William & Mary 63
Colloquium @ William & Mary 64
Example: CNOT gate
Colloquium @ William & Mary 65
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. 2005 67
奈良女子大学セミナー 28 Jan. 2005 68
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
奈良女子大学セミナー 28 Jan. 2005 70
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. 200571
8. Summary I: Cartan8. Summary I: Cartan 分解分解
奈良女子大学セミナー 28 Jan. 200572
Summary II: Warp-DriveSummary II: Warp-Drive
Colloquium @ William & Mary
Power of Entanglement
73