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Recent works have shown that quantum computers can polynomially speed upcertain SAT-solving algorithms even when the number of available qubits issignificantly smaller than the number of variables. Here we generalise thisapproach. We present a framework for hybrid quantum-classical algorithms whichutilise quantum computers significantly smaller than the problem size. Given anarbitrarily small ratio of the quantum computer to the instance size, weachieve polynomial speedups for classical divide-and-conquer algorithms,provided that certain criteria on the time- and space-efficiency are met. Wedemonstrate how this approach can be used to enhance Eppstein's algorithm forthe cubic Hamiltonian cycle problem, and achieve a polynomial speedup for anyratio of the number of qubits to the size of the graph.

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