Encoding Binary Comparison Constraints in QUBO for Quantum Annealing
Philippe Codognet · Proceedings of the Genetic and Evolutionary Computation Conference Companion · 2025
We consider the encoding in Quadratic Unconstrained Binary Optimization (QUBO) of binary comparison constraints, that is, constraints of the form x < y + c, x ≤ y + c, and x = y + c, where x and y are two integer variables and c is an integer constant. For the sake of simplicity, we will consider that x and y are bounded and belong to the same interval {1,..., n}. We will show that such constraints can be encoded without the need of any extra-variable, conversely to the general scheme for linear inequations, which requires between n - 1 and ⌈log2(n - 1)⌉ extra-variables, depending on the encoding chosen for integers. Limiting the number of extra-variables needed in QUBO formulations is important as each Boolean QUBO variable will be implemented as a (logical) qubit on a quantum device and the number of such qubits is limited in quantum annealing systems such as D-Wave machines.