Sparse QUBO Formulation for Efficient Embedding via Network-Based Decomposition of Equality and Inequality Constraints

TL;DR

This paper introduces a network-based decomposition method to create sparser QUBO models for equality and inequality constraints, significantly reducing qubit overhead and improving embedding efficiency on quantum annealers.

Contribution

It presents a novel approach to decompose dense constraints into sparser models using auxiliary variables, enabling more efficient embedding on quantum hardware.

Findings

Reduces quadratic terms from O(N^2) to O(N) for one-hot constraints

Decreases quadratic terms to O(N log N) in worst case for general constraints

Experimental results show fewer qubits, shorter chains, and higher solution rates

Abstract

Quantum annealing is a promising approach for solving combinatorial optimization problems. However, its performance is often limited by the overhead of additional qubits required for embedding logical QUBO models onto quantum annealers. This overhead becomes severe when logical QUBO models have dense connectivity. Such dense structures frequently arise when formulating equality and inequality constraints. To address this issue, we propose a method to construct a significantly sparser logical QUBO model for these constraints. By adding auxiliary variables based on specific network structures, our approach decomposes the original constraint into smaller, more manageable constraints. We demonstrate that this method reduces the number of edges (quadratic terms) from $O(N^2)$ to $O(N)$ for the one-hot constraint and to $O(N\log N)$ in the worst case for general equality constraints, where $N$ is the number of variables. Experimental results on D-Wave's hardware show that our formulation leads to substantial reductions in the number of qubits required for embedding, shorter average chain lengths, lower chain break rates, and higher feasible solution rates compared to conventional methods. This work provides a practical tool for efficiently solving constrained optimization problems on current and future quantum annealers.