HQC @ Tehran, 4 Jan, 2009 1 Holonomic Quantum Computing Mikio Nakahara Department of Physics, Kinki University, Japan in collaboration with A. Niskanen,

HQC @ Tehran, 4 Jan, 2009 1

Holonomic Holonomic QuantumQuantum 　　 ComputinComputin

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Mikio NakaharaMikio NakaharaDepartment of Physics, Kinki University, JapanDepartment of Physics, Kinki University, Japanin collaboration with in collaboration with A. Niskanen, M. Salomaa, S. Tanimura, D. Hayashi, Y. Kondo, A. Niskanen, M. Salomaa, S. Tanimura, D. Hayashi, Y. Kondo, Y. Ota, M. BandoY. Ota, M. Bando


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


1. What is holonomy?1.1 Holonomy

holonomy


Holonomy in Mechanics 1

Shape = Control

Manifold

Deformation with Angular Momentum Conserved　


Holonomy in Mechanics 2


1.2 Isoholonomic Problem1.2 Isoholonomic Problem

28

14



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.




2.1 Berry’s Phase



Wilczek-Zee Holonomy 1




Derivation of WZ holonomy







3. Isoholonomic Problem


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.


3.1 Geometrical Setting3.1 Geometrical Setting




3.2 　 Solution of Isoholonomic Problem


Solution of Isoholonomic Problem 2






Inverse Problem


Solution to U(g) Isoholonomic Problem



3.3 Applications to HQC



CNOT gate


DFT2 gate




4. Multi-Partite Construction It is not likely to find a quantum system wit

h 2n-fold degenerate ground state. 2100~103

0! Need to find multipartite implementation a

nd make use of entanglement to save computational resources.

A. O. Niskanen, M. Nakahara and M. M. Salomaa, Phys. Rev. A 67, 012319

(2003)


4.1 Model Hamiltonian







5. Physical Implementations of HQC


Example; Isospectral deformation of NMR (Ising) Hamiltonian


0


WZ connection


Example: Hadamard Gate


2-qubit gates



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


7. Summary7. Summary


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/mo

nth 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.)


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)

Documents

HQC @ Tehran, 4 Jan, 2009 1 Holonomic Quantum Computing Mikio Nakahara Department of Physics, Kinki University, Japan in collaboration with A. Niskanen,