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 105 relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over 103. 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.
Highlights
Many quantum algorithms are known, for tasks as diverse as integer factorisation [92] and computing Jones polynomials [4]
We focus on problems in the general area of constraint satisfaction and optimisation – an area critical to many different industry sectors and applications – and aim to quantify the likely advantage that could be achieved by quantum computers
3. (Performance bounds) We should compute the performance of the quantum and classical algorithms explicitly for particular problem instances, including all relevant overheads
Summary
Many quantum algorithms are known, for tasks as diverse as integer factorisation [92] and computing Jones polynomials [4]. We focus on problems in the general area of constraint satisfaction and optimisation – an area critical to many different industry sectors and applications – and aim to quantify the likely advantage that could be achieved by quantum computers. (Rigour) There should exist a quantum algorithm which solves the problem with provable correctness and rigorous performance bounds. 2. (Broad utility) The abstract problem solved by the algorithm should be broadly useful across many different applications. 3. (Performance bounds) We should compute the performance of the quantum and classical algorithms explicitly for particular problem instances, including all relevant overheads. 4. (Runtime) The problem instance used for comparison should be one that can be solved by the quantum computer within a reasonable time
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