Hierarchical divide and conquer quantum approach to combinatorial optimization problems with tunable reduction

TL;DR

This paper presents a hierarchical divide and conquer quantum algorithm that partitions large combinatorial optimization problems into smaller subproblems, enabling solutions with fewer qubits while maintaining high accuracy.

Contribution

It introduces a novel iterative reduction method that leverages energy ranges to minimize qubit requirements in quantum optimization.

Findings

Solved problems with 40 variables using about 10 qubits.

Achieved an approximation ratio of approximately 99.9%.

Observed increased reduction efficiency with larger problem sizes.

Abstract

Combinatorial optimization is considered a promising class of problems in which quantum computers can show significant advantages. However, problems of practical relevance typically have more variables than current or foreseeable quantum computers have qubits. Here we introduce a divide and conquer approach that partitions the optimization problem into subgraphs that can be represented on smaller quantum processors. We then find all states of the subgraphs that can possibly be part of the solution to the entire problem by determining the cost or energy ranges in which the local subgraph energies of these states must be contained. This allows us to reduce the problem by only considering the subspace spanned by these states. We then recombine the system using a binary encoding for each subgraph with a local energy ordering. This process can be iterated until no further reduction is possible. We also find that the number of necessary qubits can be reduced further when only retaining states in a fraction of the relevant energy range at very little expense in terms of approximation ratio to the global ground state. In numerical simulations, we find that our approach allows us to solve combinatorial optimization problems on weighted random 3-regular graphs with $|\mathcal{V}|=40$ discrete variables on $\sim |\mathcal{V}| / 4$ qubits while retaining a possible approximation ratio of $\sim99.9\%$. We also observe an increasing reduction with larger system sizes.