Practical implementation of a quantum backtracking algorithm

TL;DR

This paper presents a space-efficient quantum backtracking algorithm implementation for solving CSPs, including graph coloring and SAT, with detailed resource analysis and simulation results.

Contribution

It introduces a space-efficient implementation of Montanaro's quantum backtracking algorithm, reducing qubit requirements and detailing practical aspects for CSPs.

Findings

Implementation uses O(n log d) qubits for CSPs.

Detailed implementation for graph coloring and SAT.

Simulation results demonstrating feasibility.

Abstract

In previous work, Montanaro presented a method to obtain quantum speedups for backtracking algorithms, a general meta-algorithm to solve constraint satisfaction problems (CSPs). In this work, we derive a space efficient implementation of this method. Assume that we want to solve a CSP with $m$ constraints on $n$ variables and that the union of the domains in which these variables take their value is of cardinality $d$. Then, we show that the implementation of Montanaro's backtracking algorithm can be done by using $O(n \log d)$ data qubits. We detail an implementation of the predicate associated to the CSP with an additional register of $O(\log m)$ qubits. We explicit our implementation for graph coloring and SAT problems, and present simulation results. Finally, we discuss the impact of the usage of static and dynamic variable ordering heuristics in the quantum setting.