  Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum
information processing. In the HQC strategy information is encoded in
degenerate eigen-spaces of a parametric family of Hamiltonians. The
computational network of unitary quantum gates is realized by driving
adiabatically the Hamiltonian parameters along loops in a control manifold. By
properly designing such loops the non-trivial curvature of the underlying
bundle geometry gives rise to unitary transformations i.e., holonomies that
implement the desired unitary transformations. Conditions necessary for
universal QC are stated in terms of the curvature associated to the non-abelian
gauge potential (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 gates are performed makes in principle HQC an appealing way towards
universal fault-tolerant QC.
