Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles

Abstract

Under mild conditions the experimentally accessible Hamiltonians in ensembles of N coupled, nondegenerate spins-½ are shown to generate the entire Lie group SU(2 _N ). This is the case e.g., in nuclear magnetic resonance (NMR) spectroscopy [1], where quantum computing gates [2] are implemented by unitary propagators, viz the ‘rf-pulse sequences’.

The maximum achievable coherence transfer amplitude under unitary Hamiltonian dynamics of ensembles translates into a minimum Euclidean distance: the minimal distance between the unitary orbit of some given initial state represented by a density operator (or its signal-relevant components collected in a matrix A) and a given final state or observable (or its components C _†). This distance relates to the C-numerical range of A, W C(A) := {tr(UAU ^-1 C)|U ∈ SU(2^N)}, and its largest absolute value, the C-numerical radius of A, $r_C \left( A \right) : = \mathop {{\rm{max}}}\limits_U |tr\left\{ {U AU^{ - 1} C} \right\}|\left[ {\rm{3}} \right]$ [3]. The geometry of the C-numerical range includes as limiting cases the notions of ensembles comprising entangled as well as classically mixed components. In the latter, entropy and relative entropy relate to Euclidean distances.

Given two arbitrary matrices A,C ∈ Mat _n (C), the first gradient-flow-based computer algorithm [4-6] for minimising the Euclidean distance between the unitary orbit of A and C _† (or rather C) in the general case is presented.