Quantum-Inspired Hierarchy for Rank-Constrained Optimization

TL;DR

This paper introduces a quantum-inspired hierarchy of semidefinite programs to efficiently solve rank-constrained optimization problems in both quantum and classical information theory, often achieving exact solutions at low levels.

Contribution

It establishes a novel connection between rank-constrained optimization and quantum entanglement, providing a hierarchy of convex relaxations with certified bounds and practical accuracy.

Findings

Hierarchy yields often exact solutions at low levels.

Applicable to quantum information problems like entanglement and channels.

Effective for classical problems such as max cut and graph representations.

Abstract

Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained. We establish a link between rank-constrained optimization and the theory of quantum entanglement. More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem. This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered. We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the optimization over pure states, the characterization of mixed unitary channels and faithful entanglement, and quantum contextuality, as well as in classical information theory including the maximum cut problem, pseudo-Boolean optimization, and the orthonormal representation of graphs. Finally, we show that our ideas can be extended to rank-constrained quadratic and higher-order programming.