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

## TL;DR

This paper introduces a hybrid quantum-classical framework that enables polynomial speedups for solving large combinatorial problems, even with quantum computers significantly smaller than the problem size.

## Contribution

It generalizes previous quantum speedup approaches to a broader class of divide-and-conquer algorithms with small quantum devices.

## Key findings

- Achieves polynomial speedup for Hamiltonian cycle problem
- Framework applicable to quantum computers with arbitrarily small size
- Enhances classical algorithms with quantum subroutines

## Abstract

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.