A Quantum Constraint Generation Framework for Binary Linear Programs

We propose a new approach to utilize quantum computers for binary linear programming (BLP), which can be extended to general integer linear programs (ILP). Quantum optimization algorithms, hybrid or quantum-only, are currently general purpose, standalone solvers for ILP. However, to consider them practically useful, we expect them to overperform the current state of the art classical solvers. That expectation is unfair to quantum algorithms: in classical ILP solvers, after many decades of evolution, many different algorithms work together as a robust machine to get the best result. This is the approach we would like to follow now with our quantum ‘solver’ solutions. In this study we wrap any suitable quantum optimization algorithm into a quantum informed classical constraint generation framework. First we relax our problem by dropping all constraints and encode it into an Ising Hamiltonian for the quantum optimization subroutine. Then, by sampling from the solution state of the subroutine, we obtain information about constraint violations in the initial problem, from which we decide which coupling terms we need to introduce to the Hamiltonian. The coupling terms correspond to the constraints of the initial binary linear program. Then we optimize over the new Hamiltonian again, until we reach a feasible solution, or other stopping conditions hold. Since one can decide how many constraints they add to the Hamiltonian in a single step, our algorithm is at least as efficient as the (hybrid) quantum optimization algorithm it wraps. We support our claim with results on small scale minimum cost exact cover problem instances.

💡 Research Summary

The paper introduces a hybrid framework that augments any existing quantum optimization routine with a classical constraint‑generation loop, aiming to solve binary linear programs (BLPs) and, by reduction, general integer linear programs (ILPs). The authors begin by converting the BLP into a quadratic unconstrained binary optimization (QUBO) problem. All linear constraints Ax = b are initially dropped, and a large penalty parameter M is introduced so that the relaxed objective becomes a simple quadratic form. This relaxed QUBO is then mapped to an Ising Hamiltonian H(σ) = −∑{i<j}J{ij}σ_iσ_j − μ∑_i h_iσ_i, where the coupling matrix J and local fields h are derived directly from the penalty matrix \hat A and vector \hat b. In the first iteration \hat A and \hat b are zero, so the Hamiltonian contains only single‑qubit terms and no couplings, making it trivially easy for a quantum optimizer.

A chosen quantum subroutine A (e.g., QAOA, a variational quantum algorithm) is then used to prepare an approximate ground‑state |ψ*⟩ of the current Hamiltonian. From |ψ*⟩ a batch of q samples is drawn; each sample is interpreted as a binary vector X_ℓ. The algorithm checks whether any sampled vector satisfies the original constraints. If a feasible solution is found, the best one (lowest objective value) may be returned immediately, or the loop may continue to seek improvement.

If no feasible solution is present, the algorithm computes a violation matrix V where V_{jℓ}=1 if constraint j is violated by sample ℓ and 0 otherwise. Normalising by the sample weights yields a violation score vector ν∈