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Holonomic Quantum Computing

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. Plan of the talk. 1. What is holonomy (anholonomy)? 2. Wilczek-Zee holonomy

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Holonomic Quantum Computing

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  1. Holonomic QuantumComputing 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 HQC @ Tehran, 4 Jan, 2009

  2. 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 HQC @ Tehran, 4 Jan, 2009

  3. 1. What is holonomy?1.1 Holonomy holonomy HQC @ Tehran, 4 Jan, 2009

  4. Holonomy in Mechanics 1 Deformation with Angular Momentum Conserved Shape = Control Manifold HQC @ Tehran, 4 Jan, 2009

  5. Holonomy in Mechanics 2 HQC @ Tehran, 4 Jan, 2009

  6. 1.2 Isoholonomic Problem HQC @ Tehran, 4 Jan, 2009

  7. HQC @ Tehran, 4 Jan, 2009

  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. HQC @ Tehran, 4 Jan, 2009

  9. 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 HQC @ Tehran, 4 Jan, 2009

  10. 2.1 Berry’s Phase HQC @ Tehran, 4 Jan, 2009

  11. HQC @ Tehran, 4 Jan, 2009

  12. Wilczek-Zee Holonomy 1 HQC @ Tehran, 4 Jan, 2009

  13. Wilczek-Zee Holonomy 2 HQC @ Tehran, 4 Jan, 2009

  14. Derivation of WZ holonomy HQC @ Tehran, 4 Jan, 2009

  15. HQC @ Tehran, 4 Jan, 2009

  16. Wilczek-Zee Holonomy 3 HQC @ Tehran, 4 Jan, 2009

  17. 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 HQC @ Tehran, 4 Jan, 2009

  18. 3. Isoholonomic Problem HQC @ Tehran, 4 Jan, 2009

  19. 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. HQC @ Tehran, 4 Jan, 2009

  20. 3.1 Geometrical Setting HQC @ Tehran, 4 Jan, 2009

  21. HQC @ Tehran, 4 Jan, 2009

  22. HQC @ Tehran, 4 Jan, 2009

  23. 3.2Solution of Isoholonomic Problem HQC @ Tehran, 4 Jan, 2009

  24. Solution of Isoholonomic Problem 2 HQC @ Tehran, 4 Jan, 2009

  25. Solution of Isoholonomic Problem 3 HQC @ Tehran, 4 Jan, 2009

  26. Solution of Isoholonomic Problem 4 HQC @ Tehran, 4 Jan, 2009

  27. Inverse Problem HQC @ Tehran, 4 Jan, 2009

  28. Solution to U(g) Isoholonomic Problem HQC @ Tehran, 4 Jan, 2009

  29. HQC @ Tehran, 4 Jan, 2009

  30. 3.3 Applications to HQC HQC @ Tehran, 4 Jan, 2009

  31. HQC @ Tehran, 4 Jan, 2009

  32. CNOT gate HQC @ Tehran, 4 Jan, 2009

  33. DFT2 gate HQC @ Tehran, 4 Jan, 2009

  34. 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 HQC @ Tehran, 4 Jan, 2009

  35. 4. Multi-Partite Construction • It is not likely to find a quantum system with 2n-fold degenerate ground state. 2100~1030! • 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) HQC @ Tehran, 4 Jan, 2009

  36. 4.1 Model Hamiltonian HQC @ Tehran, 4 Jan, 2009

  37. HQC @ Tehran, 4 Jan, 2009

  38. HQC @ Tehran, 4 Jan, 2009

  39. HQC @ Tehran, 4 Jan, 2009

  40. 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 HQC @ Tehran, 4 Jan, 2009

  41. 5. Physical Implementations of HQC HQC @ Tehran, 4 Jan, 2009

  42. Example; Isospectral deformation of NMR (Ising) Hamiltonian HQC @ Tehran, 4 Jan, 2009

  43. 0 HQC @ Tehran, 4 Jan, 2009

  44. WZ connection HQC @ Tehran, 4 Jan, 2009

  45. Example: Hadamard Gate HQC @ Tehran, 4 Jan, 2009

  46. 2-qubit gates HQC @ Tehran, 4 Jan, 2009

  47. HQC @ Tehran, 4 Jan, 2009

  48. 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 HQC @ Tehran, 4 Jan, 2009

  49. 7. Summary HQC @ Tehran, 4 Jan, 2009

  50. 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.) HQC @ Tehran, 4 Jan, 2009

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