Constraint-preserving quantum algorithm for the multi-frequency antenna placement problem

TL;DR

The paper tackles constrained combinatorial optimization in telecommunications by designing a constraint-preserving quantum adiabatic algorithm (QAA) for the multi-frequency antenna placement problem (mAPP). It introduces an initial-state preparation and a mixer that exactly preserve one-hot site encoding and the cardinality constraint, allowing the problem Hamiltonian to encode only the objective without soft penalties. Extending to large-scale instances, the authors integrate the SPLIT decomposition to partition problems into subproblems solved by QAA-APP or classical solvers, achieving competitive performance with branch-and-bound and simulated annealing baselines. The results demonstrate that constraint-aware quantum methods can match or exceed classical methods on large, industrially relevant instances, and that reformulating to QUBO with penalties can be substantially less effective than constraint-preserving approaches. Overall, the work argues for problem-centric quantum algorithm design to enable practical quantum advantages in constrained industrial problems, with forward-looking potential for broader applications bearing similar constraint structures.

Abstract

Quantum algorithms for combinatorial optimization typically encode constraints as soft penalties within the objective function, which can reduce efficiency and scalability compared to state-of-the-art classical methods that instead exploit constraints to guide the search toward high-quality solutions. Although solving this issue for an arbitrary problem is inherently a hard task, we address this challenge for a specific problem in the field of telecommunications, the multi-frequency antenna placement problem, by introducing a constraint-preserving quantum adiabatic algorithm (QAA). To this aim, we construct a quantum circuit that prepares an initial state comprising an equal superposition of all feasible solutions, and define a custom mixer that preserves both the one-hot encoding constraint for vertex coloring and the cardinality constraint on the number of antennas. This scheme can be extended to a broader range of applications characterized by similar constraints. We first benchmark the performance of this quantum algorithm against a basic version of QAA, demonstrating superior performance in terms of feasibility and success probability. We then apply this algorithm to large problem sizes with hundreds of variables using a constraint-aware decomposition method based on the SPLIT framework. Our results indicate competitive performance against other large-scale classical approaches, such as branch-and-bound and simulated annealing. This work supports previous claims that constraint-aware algorithms are crucial for the practical and efficient application of quantum methods in industrial settings.

Figures (5)

• Figure 1: Top: schematic illustration of the multi-frequency antenna placement problem. Grey areas represent empty sites ($p=0$), while green/purple areas represent antennas operating at different frequencies ($p=1,2$). The red area indicates interfering signals. Bottom: solution of a mAPP instance on the Liguria region (Italy).
• Figure 2: Example of the state preparation circuit for the case of two sites with $F=3$, which creates the initial mAPP state of Eq. \ref{['eq:feas_sup']}. The block indicated with $D^N_{N-k}$ creates the Dicke state with $N-k$ Hamming weight on the first $N=2$ qubits. The $W_R$ blocks prepare the corresponding W states on the target qubits in the $R_1$ and $R_2$ registers. The empty circle indicates an inversely controlled gate.
• Figure 3: Average percentage of feasibility fraction ($p_{\rm feasible}$) of the various quantum methods tested on small instances (triangles and lines), superimposed to a boxplot showing the success probability ($p_{\rm success}$), in percentage. Each box shows the median and interquartile range, with dots indicating outliers. Instances with different problem sizes $N(F+1)$ are considered, with $F=3$ frequencies and $N=3,4,5,6,7$ sites. The number of antennas is set to $k=\lfloor \frac{N}{2} \rfloor$.
• Figure 4: Boxplots of the normalized difference $\Delta \alpha$ [Eq.\ref{['eq:deltaalpha']}] for different optimization methods (indicated in the legend) across intermediate problem sizes using hybrid quantum-classical and purely classical approaches. Each box shows the median and interquartile range, with dots indicating outliers. Instances with different problem sizes $N(F+1)$ are considered, with $F=3$ frequencies and $N=15,20,25,30,35,40,45,50$ sites. The number of antennas is set to $k=\lfloor \frac{F}{F+1}(N+1) \rfloor$.
• Figure 5: Boxplots of the normalized difference $\Delta \alpha$ for different classical optimization methods (indicated in the legend) across large problem sizes. Each box shows the median and interquartile range, with dots indicating outliers. The methods shown here always find a feasible solution. Instances with different problem sizes $N(F+1)$ are considered, with $F=4$ frequencies and $N=20,40,60,80,100,120,140,160$ sites. The number of antennas is set to $k=\lfloor \frac{F}{F+1}(N+1) \rfloor$.