The PRODSAT phase of random quantum satisfiability

TL;DR

This paper investigates the PRODSAT phase in random quantum satisfiability, establishing a geometric threshold for product states and exploring the role of entanglement in the zero energy ground states.

Contribution

It provides a rigorous proof that the PRODSAT phase threshold equals the dimer covering threshold using algebraic and complex analysis methods.

Findings

Product states of zero energy exist with high probability below the threshold.

The threshold is a purely geometric quantity related to dimer coverings.

Numerical experiments suggest entanglement influences the zero energy ground state space.

Abstract

The $k$-QSAT problem is a quantum analog of the famous $k$-SAT constraint satisfaction problem. We must determine the zero energy ground states of a Hamiltonian of $N$ qubits consisting of a sum of $M$ random $k$-local rank-one projectors. It is known that product states of zero energy exist with high probability if and only if the underlying factor graph has a clause-covering dimer configuration. This means that the threshold of the PRODSAT phase is a purely geometric quantity equal to the dimer covering threshold. We revisit and fully prove this result through a combination of complex analysis and algebraic methods based on Buchberger's algorithm for complex polynomial equations with random coefficients. We also discuss numerical experiments investigating the presence of entanglement in the PRODSAT phase in the sense that product states do not span the whole zero energy ground state space.