"Geometry Day", University of Patras - Department of Mathematics, Monday 12 October 2015
Title: On the Geometry of Quantum Computation: the case of Grassmann Manifolds
Abstract: We first review the methodology of the error-avoiding paradigm for the generation of gauge theoretic non abelian connections a.k.a holonomies, over a control manifold that effectively generates a minimal universal set of gates capable of simulating any quantum computation circuit. A family of iso-spectral hamiltonians on N dimensional Hilbert spaces are considered over an associated set of control parameters geometrically identified with a Grassmann manifold and adiabatically traced loops in the manifold are considered. Loops can give rise to u(N) Lie algebra valued 1-forms connection that can generate efficiently 1 and 2 qubit universal quantum gates. Further the 3 qubit generic state vector in Hilbert space C^3 is shown to admit a Gr(4,2) Grassmann manifold parametrization that is associated with points on a Klein quartic algebraic variety. Special lower dimensional restrictions of the variety and their unitary symmetries are investigated.