# On product, generic and random generic quantum satisfiability

## TL;DR

This paper investigates quantum satisfiability (k-QSAT), establishing criteria for product satisfiability, analyzing phase transitions in random ensembles, and providing numerical estimates for the satisfiability transition points.

## Contribution

It introduces a geometrical criterion for product satisfiability, extends it to quantum satisfiability, and improves bounds on the SAT--UNSAT transition in random quantum graph models.

## Key findings

- Product satisfiability can be decided by a geometric criterion.
- Improved lower bounds on the SAT--UNSAT transition in random k-QSAT.
- Evidence for a phase where solutions are only entangled states for k=3 and 4.

## Abstract

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.