Issue
EPL
Volume 83, Number 5, September 2008
Article Number 50002
Number of page(s) 5
Section General
DOI http://dx.doi.org/10.1209/0295-5075/83/50002
Published online 21 August 2008
EPL, 83 (2008) 50002
DOI: 10.1209/0295-5075/83/50002

Geometric vs. dynamical gates in quantum computing implementations using Zeeman and Heisenberg Hamiltonians

Yu Shi

Department of Physics, Fudan University - Shanghai 200433, China

yushi@fudan.edu.cn

received 4 June 2008; accepted in final form 8 July 2008; published September 2008
published online 21 August 2008

Abstract
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $\pi$/8 and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases, or based on Aharonov-Anandan phases when the gyromagnetic ratios of the two qubits are equal. In implementing a universal set of quantum gates, one may combine geometric $\pi$/8 and Hadamard gates and dynamical $\sqrt{\rm SWAP}$ gate.

PACS
03.67.Lx - Quantum computation architectures and implementations.
03.65.Vf - Phases: geometric; dynamic or topological.
73.21.La - Quantum dots.

© EPLA 2008