1. HQC @ Tehran, 4 Jan, 2009 1 Holonomic Quantum Computing Mikio Nakahara Department of Physics, Kinki University, Japan in collaboration with A. Niskanen, M. Salomaa, S. Tanimura, D. Hayashi, Y. Kondo, Y. Ota, M. Bando

2. HQC @ Tehran, 4 Jan, 20092 Plan of the talk 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy 3. Isoholonomic Problem 4. Multi-Partite Construction 5. Physical Realizations 5. Summary and Discussions

3. HQC @ Tehran, 4 Jan, 20093 1. What is holonomy? 1.1 Holonomy holonomy

4. HQC @ Tehran, 4 Jan, 20094 Holonomy in Mechanics 1 Shape = Control Manifold Deformation with Angular Momentum Conserved

5. HQC @ Tehran, 4 Jan, 20095 Holonomy in Mechanics 2

6. HQC @ Tehran, 4 Jan, 20096 1.2 Isoholonomic Problem

7. HQC @ Tehran, 4 Jan, 20097

8. 8 1.3 Holonomic Quantum Computing Change of Hamiltonian adiabatically along a loop in the control manifold produces the Wilczek-Zee holonomy. Unitary matrices necessary for quantum computing is implemented as holonomies in HQC (Zanardi & Rasetti, 1999). Timing does not matter. Only the geometrical image of the loop in the control manifold is important. We are interested in the isoholonomic problem, given a unitary matrix U.

9. HQC @ Tehran, 4 Jan, 20099 Plan of the talk 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy 3. Isoholonomic Problem 4. Multi-Partite Construction 5. Physical Realizations 5. Summary and Discussions

10. HQC @ Tehran, 4 Jan, 200910 2.1 Berry’s Phase

11. HQC @ Tehran, 4 Jan, 200911

12. HQC @ Tehran, 4 Jan, 200912 Wilczek-Zee Holonomy 1

13. HQC @ Tehran, 4 Jan, 200913 Wilczek-Zee Holonomy 2

14. HQC @ Tehran, 4 Jan, 200914 Derivation of WZ holonomy

15. HQC @ Tehran, 4 Jan, 200915

16. HQC @ Tehran, 4 Jan, 200916 Wilczek-Zee Holonomy 3

17. HQC @ Tehran, 4 Jan, 200917 Plan of the talk 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy 3. Isoholonomic Problem 4. Multi-Partite Construction 5. Physical Realizations 5. Summary and Discussions

18. HQC @ Tehran, 4 Jan, 200918 3. Isoholonomic Problem

19. HQC @ Tehran, 4 Jan, 200919 Motivation for Studying the Isoholonomic Problem in HQC Adiabaticity demands that the control parameters be changed as slowly as possible. On the other hand, operation time should be as short as possible to fight against decoherence. Therefore we should employ the shortest path in the control parameter space to compromise between two conflicting requirements.

20. HQC @ Tehran, 4 Jan, 200920 3.1 Geometrical Setting

21. HQC @ Tehran, 4 Jan, 200921

22. HQC @ Tehran, 4 Jan, 200922

23. HQC @ Tehran, 4 Jan, 200923 3.2 Solution of Isoholonomic Problem

24. HQC @ Tehran, 4 Jan, 200924 Solution of Isoholonomic Problem 2

25. HQC @ Tehran, 4 Jan, 200925 Solution of Isoholonomic Problem 3

26. HQC @ Tehran, 4 Jan, 200926 Solution of Isoholonomic Problem 4

27. HQC @ Tehran, 4 Jan, 200927 Inverse Problem

28. HQC @ Tehran, 4 Jan, 200928 Solution to U(g) Isoholonomic Problem

29. HQC @ Tehran, 4 Jan, 200929

30. HQC @ Tehran, 4 Jan, 200930 3.3 Applications to HQC

31. HQC @ Tehran, 4 Jan, 200931

32. HQC @ Tehran, 4 Jan, 200932 CNOT gate

33. HQC @ Tehran, 4 Jan, 200933 DFT2 gate

34. HQC @ Tehran, 4 Jan, 200934 Plan of the talk 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy 3. Isoholonomic Problem 4. Multi-Partite Construction 5. Physical Realizations 5. Summary and Discussions

35. HQC @ Tehran, 4 Jan, 200935 4. Multi-Partite Construction It is not likely to find a quantum system with 2 n -fold degenerate ground state. 2 100 ~10 30 ! Need to find multipartite implementation and make use of entanglement to save computational resources. A. O. Niskanen, M. Nakahara and M. M. Salomaa, Phys. Rev. A 67, 012319 (2003)

36. HQC @ Tehran, 4 Jan, 200936 4.1 Model Hamiltonian

37. HQC @ Tehran, 4 Jan, 200937

38. HQC @ Tehran, 4 Jan, 200938

39. HQC @ Tehran, 4 Jan, 200939

40. HQC @ Tehran, 4 Jan, 200940 Plan of the talk 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy 3. Isoholonomic Problem 4. Multi-Partite Construction 5. Physical Realizations 5. Summary and Discussions

41. HQC @ Tehran, 4 Jan, 200941 5. Physical Implementations of HQC

42. HQC @ Tehran, 4 Jan, 200942 Example; Isospectral deformation of NMR (Ising) Hamiltonian

43. HQC @ Tehran, 4 Jan, 200943 0

44. HQC @ Tehran, 4 Jan, 200944 WZ connection

45. HQC @ Tehran, 4 Jan, 200945 Example: Hadamard Gate

46. HQC @ Tehran, 4 Jan, 200946 2-qubit gates

47. HQC @ Tehran, 4 Jan, 200947

48. HQC @ Tehran, 4 Jan, 200948 Plan of the talk 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy 3. Isoholonomic Problem 4. Multi-Partite Construction 5. Physical Realizations 6. Summary

49. HQC @ Tehran, 4 Jan, 200949 7. Summary

50. HQC @ Tehran, 4 Jan, 200950 Working/studying in my lab. http://alice.math.kindai.ac.jp Postdoc (April, 2009~ ) salary $ 2,800/month must be younger than 35. Senior Scientist (April, 2009~ ) salary $2,600/month for 5 working days/week $3,200/month for 6 working days/week must be older than 35. PhD student, RA salary $13,000/year Tuition fee $10,000/year. (maybe waived) (Almost unlimited research funding for theorists.)

51. HQC @ Tehran, 4 Jan, 200951 JSPS postdoc http://www.jsps.go.jp/english/e-fellow/postdoctoral.html#long Good salary ($3,600/month) and own research funding, but very tough competition (<9%). All applications I have submitted were accepted so far. (Two British, One Iranian (RRD)) JSPS fellow for senior scientists ($5,500/month) up to 2 months. All applications I have submitted were accepted so far. (One Finnish, One Italian)