Variational Quantum Algorithm for Constrained Combinatorial Optimization Problems

TL;DR

This work introduces an alternative VQA whose core innovation lies in a strategically designed loss function that provably guaranteed that its global minimum corresponds uniquely to the optimal feasible solution, as this is achieved by ensuring universally higher loss values for all infeasible solutions.

Abstract

While variational quantum algorithms (VQAs) have demonstrated considerable success in unconstrained optimization, their application to constrained combinatorial problems face a trade-off. Penalty-based methods, despite their circuit simplicity, suffer from a fundamental limitation: inefficient sampling in vast infeasible regions. This often results in suboptimal solutions that violate constraints and impede convergence to high-quality results. In contrast, ansatz-based approaches enforce solution feasibility by design but require complex, problem-specific circuits that are challenging to implement on current noisy intermediate-scale quantum devices. To overcome these limitations, we introduce an alternative VQA whose core innovation lies in a strategically designed loss function. This function offers a dual advantage. First, it is provably guaranteed that its global minimum corresponds uniquely to the optimal feasible solution, as this is achieved by ensuring universally higher loss values for all infeasible solutions. Second, it furnishes distinct computational pathways for feasible versus infeasible regions, thus creating clear and non competing guidance for the optimizer. As a result of these combined features, the algorithm's overall performance is significantly enhanced. Regarding hardware overhead, our design requires adding only an efficient validation oracle module to the penalty-based circuit, resulting in a circuit complexity significantly lower than that of ansatz-based approaches with their custom mixers. To validate the practical efficiency of our method, we empirically demonstrate its effectiveness by solving minimum vertex cover and maximum independent set problems on random graphs of varying small-scale sizes.

Figures (7)

• Figure 1: Quantum circuit designed to implement the oracle $\hat{U}_v$ from the ESOP expression $F=c_1\oplus c_2 \oplus \ldots \oplus c_m$. (a) Global quantum circuit realization. (b) Circuit realization for the operator $\hat{U}_{c_i}$ involved in (a). The $k_i$ qubits (marked by blue lines) denote the negated variables of the cube $c_i$ and $l_i$ qubits (marked by red lines) embody the positive variables of the cube $c_i$.
• Figure 2: Performance of the penalty-based method across problem sizes(a) 3, (b) 4, (c) 5, (d) 6, (e) 7,(f) 8, (g) 9, and (h) 10. Subfigures (a)--(h) show the results for each size. Blue dashed lines represent the mean accuracy for depth 2, while orange solid lines correspond to depth 3. Blue shaded areas indicate the mean plus or minus one standard deviation for depth 2 and orange shaded areas denote the same for depth 3. Penalty factors are uniformly sampled from the interval $(a,b]$.
• Figure 3: Performance comparison between our method and the penalty-based method across problem sizes (a) 3,(b) 4, (c) 5, (d) 6, (e) 7, (f) 8, (g) 9, and (h) 10. Subfigures (a)--(h) show results for each size, evaluated over six sets of randomly generated initial parameters. For the penalty-based method, the penalty factor yielding the best average performance among five randomly generated values is selected. Blue dash-dotted lines represent the mean accuracy of our approach, while orange solid lines denote the penalty-based method. Shaded regions indicate the mean plus or minus one standard deviation (blue for our method and orange for the penalty-based method).
• Figure 4: Optimal accuracy achieved under the respective optimal initial parameters for each method across different problem sizes (a) 3, (b) 4, (c) 5, (d) 6, (e) 7, (f) 8, (g) 9, and (h) 10. Blue dash-dotted lines are for our approach and orange solid lines for the penalty-based method.
• Figure 5: Performance of the penalty-based method across different problem sizes (a) 3, (b) 4, (c) 5, (d) 6, (e) 7,(f) 8, (g) 9, and (h) 10. Blue dashed lines represent the mean accuracy for depth 2, while orange solid lines correspond to depth 3. Blue shaded areas indicate the mean plus or minus one standard deviation for depth 2 and orange shaded areas denote the same for depth 3. Penalty factors are uniformly sampled from the interval $(a,b]$.