Qubit-efficient and gate-efficient encodings of graph partitioning problems for quantum optimization
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Quantum Physics arXiv:2604.21123 (quant-ph) [Submitted on 22 Apr 2026] Title:Qubit-efficient and gate-efficient encodings of graph partitioning problems for quantum optimization Authors:Tristan Zaborniak, Prashanti Priya Angara, Vikram Khipple Mulligan, Hausi Müller, Ulrike Stege View a PDF of the paper titled Qubit-efficient and gate-efficient encodings of graph partitioning problems for quantum optimization, by Tristan Zaborniak and 4 other authors View PDF HTML (experimental) Abstract: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: 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 arXiv-issued DOI via DataCite (pending registration) Submission history From: Tristan Zaborniak [view email] [v1] Wed, 22 Apr 2026 22:26:09 UTC (943 KB) Full-text links: Access Paper: View a PDF of the paper titled Qubit-efficient and gate-efficient encodings of graph partitioning problems for quantum optimization, by Tristan Zaborniak and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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 Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)