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

Quantum Physics [Submitted on 22 Apr 2026] Qubit-efficient and gate-efficient encodings of graph partitioning problems for quantum optimization Tristan Zaborniak, Prashanti Priya Angara, Vikram Khipple Mulligan, Hausi Müller, Ulrike Stege 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. Comments: 12 pages, 2 figures Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.21123 [quant-ph]   (or arXiv:2604.21123v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.21123 Focus to learn more Submission history From: Tristan Zaborniak [view email] [v1] Wed, 22 Apr 2026 22:26:09 UTC (943 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)