Abstract

One of the most promising types of algorithms to run on noisy intermediate-scale quantum computers are variational optimization algorithms (VQAs). In those algorithms one deals with a parametrized quantum circuit whose outputs are then a parametrized family of n-particle quantum states. One common problem to solve using VQAs is, given an n-body observable H that can be efficiently implemented (e.g. a locally interacting Hamiltonian), obtain an approximation of the ground state and its associated energy.

The aim of our work is to study the continuous Langevin dynamics in U(n). Proving convergence results in such a setting may potentially lead to poly-time algorithms to solve the problem introduced in the previous paragraph, when considering depth-2 quantum circuits with gates acting on a logarithmic number of sites. Moreover, our results should be applicable to other circuits, under certain assumptions on their structure and the Hamiltonian considered.

We generalize some of the results from [1] to the Lie group U(n); proving that the Gibbs distribution associated to the dynamics does indeed “find” the ground state of H. Furthermore, we prove that our setting satisfies a logarithmic Sobolev inequality, which guarantees exponential convergence of the process to its associated Gibbs distribution.

References: [1] M.B. Li, and M.A. Erdogdu, arXiv:2010.11176, (2020).