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

TL;DR

This paper compares geometric and dynamical quantum gates using Zeeman and Heisenberg Hamiltonians, highlighting limitations in geometric phase-based two-qubit gates and proposing a hybrid approach for universal quantum computation.

Contribution

It introduces a novel physical representation for qubit states and analyzes the feasibility of geometric phase gates in specific Hamiltonian systems, revealing fundamental constraints.

Findings

Impossible to construct two-qubit Berry phase gates with equal gyromagnetic ratios.

A hybrid gate set combining geometric and dynamical gates enables universal quantum computation.

Geometric phases provide partial fault tolerance but face implementation limitations.

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.