Tehran, 4 Jan, 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

Tehran, 4 Jan, 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

Tehran, 4 Jan, What is holonomy? 1.1 Holonomy holonomy

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

Tehran, 4 Jan, Holonomy in Mechanics 2

Tehran, 4 Jan, Isoholonomic Problem

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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.

Tehran, 4 Jan, 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

Tehran, 4 Jan, Berry’s Phase

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Tehran, 4 Jan, Wilczek-Zee Holonomy 1

Tehran, 4 Jan, Wilczek-Zee Holonomy 2

Tehran, 4 Jan, Derivation of WZ holonomy

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Tehran, 4 Jan, Wilczek-Zee Holonomy 3

Tehran, 4 Jan, 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

Tehran, 4 Jan, Isoholonomic Problem

Tehran, 4 Jan, 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.

Tehran, 4 Jan, Geometrical Setting

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Tehran, 4 Jan, Solution of Isoholonomic Problem

Tehran, 4 Jan, Solution of Isoholonomic Problem 2

Tehran, 4 Jan, Solution of Isoholonomic Problem 3

Tehran, 4 Jan, Solution of Isoholonomic Problem 4

Tehran, 4 Jan, Inverse Problem

Tehran, 4 Jan, Solution to U(g) Isoholonomic Problem

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Tehran, 4 Jan, Applications to HQC

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Tehran, 4 Jan, CNOT gate

Tehran, 4 Jan, DFT2 gate

Tehran, 4 Jan, 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

Tehran, 4 Jan, Multi-Partite Construction It is not likely to find a quantum system with 2 n -fold degenerate ground state ~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, (2003)

Tehran, 4 Jan, Model Hamiltonian

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Tehran, 4 Jan, 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

Tehran, 4 Jan, Physical Implementations of HQC

Tehran, 4 Jan, Example; Isospectral deformation of NMR (Ising) Hamiltonian

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Tehran, 4 Jan, WZ connection

Tehran, 4 Jan, Example: Hadamard Gate

Tehran, 4 Jan, qubit gates

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Tehran, 4 Jan, 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

Tehran, 4 Jan, Summary

Tehran, 4 Jan, Working/studying in my lab. 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.)

Tehran, 4 Jan, JSPS postdoc 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)