Applying quantum algorithms to constraint satisfaction problems

TL;DR

This paper explores applying quantum algorithms to NP-complete problems like boolean satisfiability and graph coloring, showing potential large quantum speedups but highlighting significant practical challenges with current hardware and fault-tolerance.

Contribution

It demonstrates the potential quantum speedups for solving classical NP-complete problems using general-purpose quantum algorithms, with detailed comparison to classical methods.

Findings

Potential quantum speedup over 10^5 times for certain problem instances.

Quantum advantage diminishes when accounting for classical decoding costs.

Large qubit requirements and fault-tolerance remain major hurdles.

Abstract

Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over $10^5$ relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over $10^3$. However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.