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