A Quantum-Classical Hybrid Branch & Bound Algorithm

TL;DR

This work introduces a complete quantum-classical hybrid branch-and-bound (QCBB) framework for binary linear programs with equality constraints, integrating a variational quantum optimizer (demonstrated with QAOA) into a branching, bounding, and pruning tree that yields convergence guarantees. The method derives a Master Ising Hamiltonian, performs conflict-driven branching using samples from the quantum subproblems, and computes Goemans–Williamson-based lower bounds to bound subproblems, enabling optimality proofs under a big-$M$ formulation. Experimental results on Set Partitioning Problems show that subproblem reductions reduce quantum circuit complexity and improve the energy progress across the tree, with convergent upper and lower bounds and comparative gains over plain QAOA. The framework presents a principled hybrid approach with potential improvements in both classical and quantum components, offering a pathway toward quantum-assisted MILP solvers with interpretable performance metrics.

Abstract

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.

Figures (8)

• Figure 1: An example for the tree of the branching algorithm. First, the Master Problem is solved. Then, we branch by assigning values to one variable (blue arrows). After that, we try to further reduce the problem by constraint propagation steps, that also assign values to variables (green dashed arrows). In each node, we calculate a bound, and if possible, prune based on that (orange node). We also prune infeasible subproblems (red nodes).
• Figure 2: Steps of a single node evaluation in the tree. The boxes with thick red, orange borders note the prune criteria: checking infeasibility and bounding. The blue box indicates the possible update on the incumbent value.
• Figure 3: Convergence of bounds. A blue line with x-shaped markers shows the upper bound (cost of incumbent solution) as a function of the number of nodes evaluated. The red line with circular markers shows the lower bound as the function of the number of nodes evaluated. A solid black line marks the optimum, aligned to $0$, with dashed black lines representing the cost $F$ of the worst feasible solution, in this case $172$. Between the dashed lines the y axis is scaled linearly while outside the dashed line a logarithmic scale is used.
• Figure 4: Fraction of many-body terms. A blue line with circular markers shows the fraction of many-body terms remaining (compared to \ref{['eq:MH']}) as a function of nodes evaluated.
• Figure 5: Comparison of branch and bound with plain QAOA. The plots show the expected cost associated with the quantum state, as a function of queries made to a quantum simulator, as it evolves during the execution of the VQA in each node. Subplot (a), in blue, uses the branch and bound approach proposed in this work with a limit of $50$ iterations per node. In subfigure (b), in red, only QAOA is used. The cost at the optimum is adjusted to $0$.

Theorems & Definitions (8)

• Theorem 1: Goemans1995, Theorem 3.2.1.
• Theorem 2
• proof
• Theorem 3
• proof
• Remark 1
• proof
• proof