Solving Highly Constrained Search Problems with Quantum Computers

TL;DR

This paper generalizes a quantum search algorithm to solve highly constrained k-SAT problems efficiently, outperforming classical methods and providing guarantees of unsolvability detection in certain cases.

Contribution

It extends a quantum search algorithm to a broader class of k-SAT problems and establishes bounds on problem constraints for constant-time solutions.

Findings

Quantum algorithm solves certain k-SAT problems in constant steps.

Classical algorithms require linear steps, quantum algorithms ignoring structure require exponential.

The method can certify unsolvability for some problems.

Abstract

A previously developed quantum search algorithm for solving 1-SAT problems in a single step is generalized to apply to a range of highly constrained k-SAT problems. We identify a bound on the number of clauses in satisfiability problems for which the generalized algorithm can find a solution in a constant number of steps as the number of variables increases. This performance contrasts with the linear growth in the number of steps required by the best classical algorithms, and the exponential number required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems in fact have no solutions, unlike previously proposed quantum search algorithms.