Holonomic Quantum Computing

From Qwiki

Holonomic Quantum Computing (HQC) is an approach to implementing quantum logic through quantum holonomy. In this approach, the system Hamiltonian $H (λ i (t))$ is varied adiabatically through a loop in parameter space such that $λ i (t = 0) = λ i (t = T)$ . This evolution, given suitable stucture of the system Hamiltonian (see the references) can generate arbitrary unitary operations on the computational subspace of the Hamiltonian Eigenspace. This operation is anlagous to the geometric phase which is aquired when travelling loops on the surface of a sphere, the classic example of which is the [wikipedia:Foucault_pendulum:Foucault pendulum]. Unlike the traditional approach to implementing quantum logic through Hamiltonian dynamics, the computational subspace remains degenerate throughout the computation. The quantum gate which results is generated entirely by the geometric phase or quantum holonomy generated by the excursion in parameter space. This phase is an immediate generalization of Berry's phase to fiber bundles with a more complicated group structure than $U(1)$ . This method of implementing quantum logic has the advantage that the gates the fidelity is insensitive to the time taken to implement a gate (provided the evolution is adiabatic) and to fluctuatioins which maintain the area of the loop in parameter space.