Hard combinatorial problems and minor embeddings on lattice graphs

TL;DR

This paper introduces new strategies for constructing QUBOs and minor embeddings on lattice graphs, improving the efficiency and effectiveness of quantum adiabatic optimization for NP-hard problems.

Contribution

The authors develop novel embedding techniques that improve asymptotic performance and reduce computational effort for several NP-hard problems on lattice graphs.

Findings

Asymptotically improved embeddings for multiple NP-hard problems

Reduced computational effort in finding embeddings

Potential for more effective hardware implementations

Abstract

Today, hardware constraints are an important limitation on quantum adiabatic optimization algorithms. Firstly, computational problems must be formulated as quadratic unconstrained binary optimization (QUBO) in the presence of noisy coupling constants. Secondly, the interaction graph of the QUBO must have an effective minor embedding into a two-dimensional nonplanar lattice graph. We describe new strategies for constructing QUBOs for NP-complete/hard combinatorial problems that address both of these challenges. Our results include asymptotically improved embeddings for number partitioning, filling knapsacks, graph coloring, and finding Hamiltonian cycles. These embeddings can be also be found with reduced computational effort. Our new embedding for number partitioning may be more effective on next-generation hardware.