Approximating random quantum optimization problems

TL;DR

This paper investigates the complexity of approximating ground states in random quantum satisfiability problems, introducing classical optimization techniques and analyzing their effectiveness and the problem's phase structure.

Contribution

It develops bounds, algorithms, and insights into the solution space of random $k$-QSAT, including a classical approximation method and a generalized belief propagation approach.

Findings

Greedy algorithm performs well across parameter space.

Simulated annealing shows metastability in hard regions.

Belief propagation provides approximate solutions and landscape insights.

Abstract

We report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem $k$-QSAT on large random graphs. As an approximation strategy, we optimize the solution space over `classical' product states, which in turn introduces a novel autonomous classical optimization problem, PSAT, over a space of continuous degrees of freedom rather than discrete bits. Our central results are: (i) The derivation of a set of bounds and approximations in various limits of the problem, several of which we believe may be amenable to a rigorous treatment. (ii) A demonstration that an approximation based on a greedy algorithm borrowed from the study of frustrated magnetism performs well over a wide range in parameter space, and its performance reflects structure of the solution space of random $k$-QSAT. Simulated annealing exhibits metastability in similar `hard' regions of parameter space. (iii) A generalization of belief propagation algorithms introduced for classical problems to the case of continuous spins. This yields both approximate solutions, as well as insights into the free energy `landscape' of the approximation problem, including a so-called dynamical transition near the satisfiability threshold. Taken together, these results allow us to elucidate the phase diagram of random $k$-QSAT in a two-dimensional energy-density--clause-density space.