Strong Refutation for Random Quantum 3-SAT at Constant Density

Abstract

We give a classical polynomial-time algorithm that strongly refutes random quantum $3$-SAT at sufficiently large constant constraint density, thereby disproving the quantum analogue of Feige's random $3$-SAT hypothesis. This stands in sharp contrast to classical random $3$-SAT, for which polynomial-time strong refutation is known only at constraint density $\Delta \gtrsim n^{1/2}$. Although quantum $3$-SAT shares the pairwise-independence barrier of its classical counterpart, our SDP-based refutation overcomes this barrier by exploiting the noncommutativity of quantum constraints.