2014/03/06 by Niel de Beaudrap, de Beaudrap, Niel
Physics and Astronomy · #F.2 #FOS: Physical sciences #G.2.1 #J.2 #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.1403.1588
25 pages, 2 figures. Fixed errata concerning frustrated figure eights (relating to the junction probability), and the threshold for a decoupled regime on bond-percolated 3D cubic lattices
arxiv created 2014/07/01 · arxiv updated 2014/07/02
The problem 2-quantum-satisfiability (2-QSAT) 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. Similarly to the classical problem 2-SAT, the counting problem #2-QSAT of determining the size (i.e. the dimension) of the set of satisfying states is #P-complete. However, if we consider random instances of #2-QSAT 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 #2-QSAT instances. We consider #2-QSAT involving only discrete distributions over tensor product operators, which interpolates between classical #2-SAT and #2-QSAT involving arbitrary product constraints. We find that such instances of #2-QSAT, defined on Erdos--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 #2-SAT.