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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
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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

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1. What is holonomy?
1.1 Holonomy
holonomy
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Holonomy in Mechanics 1
Deformation
with Angular
Momentum
Shape =
Control
Manifold
Conserved
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Holonomy in Mechanics 2
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1.2 Isoholonomic Problem
1 
4  
8 2
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1.3 Holonomic Quantum Computing
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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.
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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

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2.1 Berry’s Phase
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Wilczek-Zee Holonomy 1
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Wilczek-Zee Holonomy 2
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Derivation of WZ holonomy
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Wilczek-Zee Holonomy 3
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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

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3. Isoholonomic Problem
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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.

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3.1 Geometrical Setting
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3.2 Solution of Isoholonomic Problem
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Solution of Isoholonomic Problem 2
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Solution of Isoholonomic Problem 3
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Solution of Isoholonomic Problem 4
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Inverse Problem
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Solution to U(g) Isoholonomic Problem
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3.3 Applications to HQC
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CNOT gate
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DFT2 gate
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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

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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)

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4.1 Model Hamiltonian
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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

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5. Physical Implementations of HQC
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Example; Isospectral deformation
of NMR (Ising) Hamiltonian
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0
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WZ connection
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Example: Hadamard Gate
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2-qubit gates
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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

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7. Summary
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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.)

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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)
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