Applying quantum algorithms to constraint satisfaction problems

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. Many quantum algorithms are known, for tasks as diverse as integer factorisation [92] and computing Jones polynomials [4]. Indeed, at the time of writing, the Quantum Algorithm Zoo website [62] cites 392 papers on quantum algorithms. However, there Ashley

Practical Implementation of a Quantum Backtracking Algorithm

“…While Campbell, Khurana and Montanaro [7] assumed access to an extremely large number of physical qubits to propose a depth optimized method to implement Montanaro's algorithm, we have presented techniques minimizing the space usage. For that, we especially looked at the implementation of the predicate and the heuristic.…”

“…9 and 10). Note that contrary to what is said in [7], we just need to add one control qubit to three (resp. one) controlled-Z gates in step 6 to control the whole operator R A (resp.…”

“…Thus, Montanaro's algorithm is of high interest in computing science and cryptography. In 2018, Campbell, Khurana and Montanaro [7] reviewed its complexity when applied to the graph coloring problem and SAT, assuming access to a very large amount of qubits, since they aggressively optimized the circuit depth. The aim of this work is to investigate a memory efficient implementation of this backtracking algorithm, in order to be able to validate the algorithm on small instances via classical emulation.…”

A hybrid algorithm framework for small quantum computers with application to finding Hamiltonian cycles

“…First is the fact that we deal with asymptotic speedups, and focus on algorithm performance in the worst-case exponential run-times. For the issue of asymptotic statements, there has lately been increasing interest in analysing performance for finite-size settings [28], which were moderately promising, but required arbitrarily-sized quantum computers. It would be interesting to see if a similar claim could be made for size-limited quantum computers.…”

Quantum speedup of branch-and-bound algorithms

“…Finally, in order to determine whether the quantum branchand-bound algorithm will genuinely outperform the best classical methods for problems of practical interest, once all overheads are taken into account, a more detailed analysis of the algorithm's runtime should be undertaken, extending previous analysis for backtracking [13].…”