Monte Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces

Abstract

Recent studies into the properties of quantum statistical ensembles in high-dimensional Hilbert spaces have encountered difficulties associated with the Monte-Carlo sampling of quantum superpositions constrained by the energy expectation value. A straightforward Monte-Carlo routine would enclose the energy constrained manifold within a larger manifold, which is easy to sample, for example, a hypercube. The efficiency of such a sampling routine decreases exponentially with the increase of the dimension of the Hilbert space, because the volume of the enclosing manifold becomes exponentially larger than the volume of the manifold of interest. The present paper explores the ways to optimise the above routine by varying the shapes of the manifolds enclosing the energy-constrained manifold. The resulting improvement in the sampling efficiency is about a factor of five for a 14-dimensional Hilbert space. The advantage of the above algorithm is that it does not compromise on the rigorous statistical nature of the sampling outcome and hence can be used to test other more sophisticated Monte-Carlo routines. The present attempts to optimise the enclosing manifolds also bring insights into the geometrical properties of the energy constrained manifold itself.