A quantum-classical hybrid branch & bound algorithm

We propose a complete quantum-classical hybrid branch-and-bound algorithm (QCBB) to solve binary linear programs with equality constraints. That includes bound calculation, convergence metrics and optimality guarantee to the quantum optimization based algorithm, which makes our method directly comparable to classical methods. Key aspects of the proposed algorithm are (i) encapsulation of the quantum optimization method, (ii) utilization of noisy samples for problem reduction, (iii) classical approximation based bound calculation, (iv) branch and bound traits like gap-based stopping criterion and monotonic increase in solution quality, (v) integrated composition of many different solutions that can be improved individually. We show numerical results on set partitioning problem instances and provide many details about the characteristics of the different steps of the algorithm.

💡 Research Summary

The paper introduces a Quantum‑Classical Hybrid Branch‑and‑Bound (QCBB) algorithm designed to solve binary linear programs with equality constraints. The authors embed a variational quantum algorithm (specifically QAOA, though any VQA would work) inside a classical branch‑and‑bound framework, thereby achieving a method that can be directly compared with traditional solvers while still leveraging quantum sampling. The workflow begins by converting the original integer linear program into a Quadratic Unconstrained Binary Optimization (QUBO) and then into an Ising Hamiltonian (the “Master Hamiltonian”). A large penalty constant M enforces the equality constraints as energy penalties.

Quantum sampling is performed on the Hamiltonian, producing a set of binary strings (samples) together with their frequencies. From these samples the authors construct a violation matrix V that records whether each constraint is satisfied in each sample. By averaging over samples they obtain a normalized violation vector ν, which reflects the proportion of samples that violate each constraint. Multiplying ν by a variable‑to‑constraint incidence matrix P yields a “conflict vector” γ; each entry γ_i quantifies how much variable i contributes to overall constraint violations. The variable with the highest γ value is selected as the branching variable.

Branching consists of fixing the chosen variable to 0 or 1, updating the right‑hand side, coefficient matrix, and objective vector accordingly, and removing the corresponding column from the problem. The authors also apply constraint propagation in the spirit of DPLL‑based SAT solvers: fixing one variable may force the values of others, and any inconsistency discovered at this stage leads to immediate pruning of the entire subtree.

Node selection follows a simple greedy rule: at each iteration the frontier node with the lowest lower bound is evaluated next, which tends to close the optimality gap quickly because node evaluation is the most expensive operation.

Bounding is a critical component because a conventional LP‑relaxation is not readily available for the reduced subproblems. Instead, the reduced Hamiltonian is transformed into a Max‑Cut instance, and the Goemans‑Williamson (GW) algorithm is applied to obtain an approximate cut value z_GW. Using the known worst‑case approximation ratio α≈0.87856, the authors compute a rigorous lower bound:
LB_GW = –2α z_GW + (2α – 2) W⁻ + W,
where W⁻ is the sum of all negative couplings and W is the sum of all couplings in the Hamiltonian. This bound is then corrected by adding back the constant terms that were omitted during the Ising transformation, yielding a valid lower bound for the original binary program. If the bound for a child node is looser than its parent’s bound, the parent’s bound is retained.

Pruning occurs in two ways. First, if a node’s lower bound exceeds the best incumbent solution found so far (the best feasible solution obtained from quantum sampling), the node and its entire subtree are discarded. Second, if constraint propagation reveals infeasibility, the subtree is cut immediately.

The authors evaluate QCBB on a set‑partitioning benchmark. They report that conflict‑based branching dramatically reduces the number of nodes needed compared with random branching, and that the GW‑based lower bound is sufficiently tight to achieve aggressive pruning. Despite the presence of quantum noise, the conflict values remain informative enough to guide the search effectively. While the current implementation is slower than state‑of‑the‑art classical solvers on the tested instances, the authors argue that improvements in quantum hardware (more qubits, lower error rates) and algorithmic refinements (better VQAs, smarter sampling strategies) could make QCBB competitive for larger, industrial‑scale problems.

In conclusion, QCBB demonstrates a concrete pathway for integrating quantum variational algorithms into a rigorous, globally optimal branch‑and‑bound scheme. By systematically converting quantum sample statistics into branching decisions and by employing a provably sound approximation‑based lower bound, the method retains the optimality guarantees of classical branch‑and‑bound while exploiting quantum sampling to reduce problem size at each step. The paper outlines several avenues for future work, including hybrid lower‑bound strategies that combine GW with LP relaxations, adaptive sample allocation, and the use of more advanced quantum optimizers. The presented framework thus constitutes a promising foundation for future quantum‑classical hybrid optimization technologies.