Bounds on the Quantum Satisfiability Threshold

Quantum k-SAT is the problem of deciding whether there is a n-qubit state which is perpendicular to a set of vectors, each of which lies in the Hilbert space of k qubits. Equivalently, the problem is to decide whether a particular type of local Hamiltonian has a ground state with zero energy. We consider random quantum k-SAT formulas with n variables and m = αn clauses, and ask at what value of α these formulas cease to be satisfiable. We show that the threshold for random quantum 3-SAT is at most 3.594. For comparison, convincing arguments from statistical physics suggest that the classical 3-SAT threshold is αc ≈ 4.267. For larger k, we show that the quantum threshold is a constant factor smaller than the classical one. Our bounds work by determining the generic rank of the satisfying subspace for certain gadgets, and then using the technique of differential equations to analyze various algorithms that partition the hypergraph into a collection of these gadgets.