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Computing with Quanta for mathematics students

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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  1. 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

  2. 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 Colloquium @ William & Mary

  3. I. Introduction: Computing with Physics Colloquium @ William & Mary

  4. More complicated Example Colloquium @ William & Mary

  5. 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. Colloquium @ William & Mary

  6. 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 Colloquium @ William & Mary

  7. 2. Computing with Vectors and Matrices2.1 Qubit Colloquium @ William & Mary

  8. Qubit |ψ〉 Colloquium @ William & Mary

  9. Bloch Sphere: S3→　S2 π Colloquium @ William & Mary

  10. 2.2 Two-Qubit System Colloquium @ William & Mary

  11. Tensor Product Rule Colloquium @ William & Mary

  12. Entangled state (vector) Colloquium @ William & Mary

  13. 2.3 Multi-qubit systems Colloquium @ William & Mary

  14. 2.4 Algorithm = Unitary Matrix Colloquium @ William & Mary

  15. Unitary Matrices acting on n qubits Colloquium @ William & Mary

  16. 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 Colloquium @ William & Mary

  17. 3. Brief Introduction to Quantum Theory Colloquium @ William & Mary

  18. Axioms of Quantum Physics Colloquium @ William & Mary

  19. Example of a measurement Colloquium @ William & Mary

  20. Axioms of Quantum Physics (cont’d) Colloquium @ William & Mary

  21. Qubits & Matrices in Quantum Physics Colloquium @ William & Mary

  22. Actual Qubits Trapped Ions Neutral Atoms Molecules (NMR) Superconductors Colloquium @ William & Mary

  23. 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 Colloquium @ William & Mary

  24. 4. Quantum Gates, Quantum Circuit and Quantum Computer Colloquium @ William & Mary

  25. Colloquium @ William & Mary

  26. 4.2 Quantum Gates Colloquium @ William & Mary

  27. Hadamard transform Colloquium @ William & Mary

  28. Colloquium @ William & Mary

  29. 4.3 Universal Quantum Gates Colloquium @ William & Mary

  30. 4.4 Quantum Parallelism Colloquium @ William & Mary

  31. 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 Colloquium @ William & Mary

  32. 5. Quantum Teleportation Unknown Q State Bob Initial State Alice Colloquium @ William & Mary

  33. Q Teleportation Circuit 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

  35. 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 Colloquium @ William & Mary

  36. 5. Simple Quantum Algorithm- Deutsch’s Algorithm - Colloquium @ William & Mary

  37. Colloquium @ William & Mary

  38. Colloquium @ William & Mary

  39. 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 Colloquium @ William & Mary

  40. 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. Colloquium @ William & Mary

  41. Shor’s Factorization algorithm Colloquium @ William & Mary

  42. Realization using NMR (15=3×5)L. M. K. Vandersypen et al (Nature 2001) Colloquium @ William & Mary

  43. NMR molecule and pulse sequence ( (~300 pulses~ 300 gates) perfluorobutadienyl iron complex with the two 13C-labelled inner carbons Colloquium @ William & Mary

  44. Colloquium @ William & Mary

  45. 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! Colloquium @ William & Mary

  46. 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! Colloquium @ William & Mary

  47. Colloquium @ William & Mary

  48. 4. 量子暗号鍵配布 三省堂サイエンスカフェ　2009年6月

  49. 　量子暗号鍵配布 1 三省堂サイエンスカフェ　2009年6月

  50. 量子暗号鍵配布 2 三省堂サイエンスカフェ　2009年6月

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