Qubit-efficient and gate-efficient encodings of graph partitioning problems for quantum optimization

We introduce a qubit- and gate-efficient higher-order unconstrained binary optimization (HUBO) encoding for graph partitioning problems requiring label-count minimization. This widely applicable class of problems includes minimum graph coloring, minimum $k$-cut, and community detection. To the best of our knowledge, this is the first work to address the optimization versions of these problems in a quantum setting, rather than only their decision counterparts. Our construction encodes each $k$-valued vertex variable using $\lceil \log_2 k \rceil$ bits and employs a novel lexicographic penalty system that implicitly minimizes partition count without requiring dedicated indicator variables. We derive provably sufficient conditions on all penalty coefficients, including those arising from Rosenberg quadratization, guaranteeing feasibility and optimality of the lowest-energy solution. Analogous conditions are derived for a one-hot encoding to enable controlled comparison. We also show that our encoding reduces two-qubit gate count per QAOA layer from $\Theta(|V||k|^2 + |E||k|)$ for the one-hot encoding to $\Theta(|E| \cdot |k| \lceil\log_2 |k|\rceil)$. Benchmarking on a quantum annealer demonstrates that our logarithmic encoding significantly improves solution quality and time-to-solution for minimum graph coloring relative to one-hot encoding, with greater advantage as problem size increases.