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arXiv · 0711.2773

Geometric vs. Dynamical Gates in Quantum Computing Implementations Using Zeeman and Heisenberg Hamiltonians

Yu Shi
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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 $π/8$ 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 $π/8$ and Hadamard gates and dynamical $\sqrt{\rm SWAP}$ gate.

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Yu Shi. 2008-09-12. Geometric vs. Dynamical Gates in Quantum Computing Implementations Using Zeeman and Heisenberg Hamiltonians. https://doi.org/10.1209/0295-5075/83/50002

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