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

2. 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

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

4. Colloquium @ William & Mary More complicated Example 4

5. 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

6. 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

7. 2. Computing with Vectors and Matrices 2.1 Qubit Colloquium @ William & Mary 7

8. Qubit |ψ 〉 8

9. Bloch Sphere: S 3 → S 2 Colloquium @ William & Mary π 9

10. 2.2 Two-Qubit System 10

11. Tensor Product Rule Colloquium @ William & Mary 11

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

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

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

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

16. 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

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

18. Axioms of Quantum Physics Colloquium @ William & Mary 18

19. Example of a measurement Colloquium @ William & Mary 19

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

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

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

23. 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

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

25. Colloquium @ William & Mary 25

26. Colloquium @ William & Mary 4.2 Quantum Gates 26

27. Colloquium @ William & Mary Hadamard transform 27

28. Colloquium @ William & Mary 28

29. Colloquium @ William & Mary 4.3 Universal Quantum Gates 29

30. Colloquium @ William & Mary 4.4 Quantum Parallelism 30

31. 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

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

33. Q Teleportation Circuit Colloquium @ William & Mary 33

34. 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

35. 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

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

37. Colloquium @ William & Mary 37

38. Colloquium @ William & Mary 38

39. 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

40. Colloquium @ William & Mary Difficulty of Prime Number Facotrization Factorization of N=890208368187479079568319892720916003 03613264603794247032637647625631554961 638351 is difficult. It is easy, in principle, to show the product of p=928101320540413151847590244727697333 8969 and q =9591715349237194999547 050068718930514279 is N. This fact is used in RSA (Rivest-Shamir- Adleman) cryptosystem. 40

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

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

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

44. Colloquium @ William & Mary 44

45. 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

46. 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

47. Colloquium @ William & Mary 47

48. 4. 量子暗号鍵配布 三省堂サイエンスカフェ 2009 年 6 月 48

49. 量子暗号鍵配布 1 三省堂サイエンスカフェ 2009 年 6 月 49

50. 量子暗号鍵配布 2 三省堂サイエンスカフェ 2009 年 6 月 50

51. 量子暗号鍵配布 3 三省堂サイエンスカフェ 2009 年 6 月 51

52. 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。

53. 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。

54. 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

55. 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.

56. Colloquium @ William & Mary 56 7.1 Computational path in U(2 n )

57. Colloquium @ William & Mary 57 Map of Kyoto

58. Colloquium @ William & Mary 58 7.2 Optimization of 2-qubit gates

59. Colloquium @ William & Mary 59 NMR Hamiltonian

60. Colloquium @ William & Mary 60 Time-Optimal Path in SU(4)

61. Colloquium @ William & Mary 61 Cartan Decomposition of SU(4) Cartan Decomposition of SU(4)

62. Colloquium @ William & Mary 62 How to find the Cartan Decomposition

63. Colloquium @ William & Mary 63

64. Colloquium @ William & Mary 64 Example: CNOT gate

65. Colloquium @ William & Mary 65

66. 奈良女子大学セミナー 28 Jan. 2005 66 6. Warp-Drive を用いた量子アルゴリズ ムの加速 (quant-ph/0411153)

67. 奈良女子大学セミナー 28 Jan. 2005 67

68. 奈良女子大学セミナー 28 Jan. 2005 68

69. 奈良女子大学セミナー 28 Jan. 2005 69 7. 実験結果 Carbon-13 で置換したクロロフォルム qubit 1 = 13 C, qubit 2 = H 初期状態 出力状態 Qubit 1Qubit 2

70. 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

71. 奈良女子大学セミナー 28 Jan. 2005 71 8. Summary I: Cartan 分解

72. 奈良女子大学セミナー 28 Jan. 2005 72 Summary II: Warp-Drive

73. Colloquium @ William & Mary Power of Entanglement 73