Quantum Holonomies for Quantum Computing

Holonomic Quantum Computation (HQC) is an all-geometrical approach to quant um information processing. In the HQC strategy information is encoded in de generate eigenspaces of a parametric family of Hamiltonians. The computatio nal network of unitary quantum gates is realized by driving adiabatically t he Hamiltonian parameters along loops in a control manifold. By properly de signing such loops the nontrivial curvature of the underlying bundle geomet ry gives rise to unitary transformations i.e., holonomies that implement th e desired unitary transformations. Conditions necessary for universal QC ar e stated in terms of the curvature associated to the non-abelian gauge pote ntial (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gat es are performed makes in principle HQC an appealing way towards universal fault-tolerant BC.