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On product, generic and random generic quantum satisfiability
We report a cluster of results on k-QSAT, the problem of quantum
satisfiability for k-qubit projectors which generalizes classical
satisfiability with k-bit clauses to the quantum setting. First we define the
NP-complete problem of product satisfiability and give a geometrical criterion
for deciding when a QSAT interaction graph is product satisfiable with positive
probability. We show that the same criterion suffices to establish quantum
satisfiability for all projectors. Second, we apply these results to the random
graph ensemble with generic projectors and obtain improved lower bounds on the
location of the SAT--unSAT transition. Third, we present numerical results on
random, generic satisfiability which provide estimates for the location of the
transition for k=3 and k=4 and mild evidence for the existence of a phase which
is satisfiable by entangled states alone.Comment: 9 pages, 5 figures, 1 table. Updated to more closely match published
version. New proof in appendi
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