Difficult Instances of the Counting Problem for 2-quantum-SAT are Very Atypical

The problem 2-QUANTUM-SATISFIABILITY (QSAT[2]) is the generalisation of the 2-CNF-SAT problem to quantum bits, and is equivalent to determining whether or not a spin-1/2 Hamiltonian with two-body terms is frustration-free. imilarly to the classical problem #SAT[2], the counting problem #QSAT[2] of determining the size (i.e. the dimension) of the set of satisfying states is #P-complete. However, if we consider random instances of QSAT[2] in which constraints are sampled from the Haar measure, intractible instances have measure zero. An apparent reason for this is that almost all two-qubit constraints are entangled, which more readily give rise to long-range constraints. We investigate under which conditions product constraints also give rise to efficiently solvable families of #QSAT[2] instances. We consider #QSAT[2] involving only discrete distributions over tensor product operators, which interpolates between classical #SAT[2] and #QSAT[2] involving arbitrary product constraints. We find that such instances of #QSAT[2], defined on Erdös-Renyi graphs or bond-percolated lattices, are asymptotically almost surely efficiently solvable except to the extent that they are biased to resemble monotone instances of #SAT[2].