An Almost Sudden Jump in Quantum Complexity

The Quantum Satisfiability problem (QSAT) is the generalization of the canonical NP-complete problem - Boolean Satisfiability. (k,s)-QSAT is the following variant of the problem: given a set of projectors of rank 1, acting non-trivially on k qubits out of n qubits, such that each qubit appears in at most s projectors, decide whether there exists a quantum state in the null space of all the projectors. Let f(k) be the maximal integer s such that every (k,s)-QSAT instance is satisfiable. Deciding (k,f(k))-QSAT is computationally easy: by definition the answer is "satisfiable". But, by relaxing the conditions slightly, we show that (k,f(k)+2)-QSAT is QMA_1-hard, for k >=15. This is a quantum analogue of a classical result by Kratochv\'il et al. [KST93]. We use the term "an almost sudden jump" to stress that the complexity of (k,f(k)+1)-QSAT is open, where the jump in the classical complexity is known to be sudden. We present an implication of this finding to the quantum PCP conjecture, arguably one of the most important open problems in the field of Hamiltonian complexity. Our implications impose constraints on one possible way to refute the quantum PCP.