Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization

Quantum Physics [Submitted on 2 Apr 2026] Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization Jie Wang, David Jansen, Irénée Frerot, Marc-Olivier Renou, Victor Magron, Antonio Acín A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard approaches, such as variational calculations, consist of writing down a wave function ansatz and minimizing over the possible states expressible by this ansatz, one can alternatively formulate the problem as a noncommutative polynomial optimization problem. This optimization problem can then be addressed using a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, the semidefinite program can provide lower bounds for ground state energies and upper and lower bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that leveraging the inherent structures of the system can significantly mitigate these scalability challenges and thus allows us to compute meaningful bounds for quantum spin systems on up to 16\times16 square lattices. Comments: 37 pages, 9 figures Subjects: Quantum Physics (quant-ph); Optimization and Control (math.OC) MSC classes: 90C23, 81-08, 47N10 Cite as: arXiv:2604.01555 [quant-ph]   (or arXiv:2604.01555v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.01555 Focus to learn more Submission history From: Jie Wang [view email] [v1] Thu, 2 Apr 2026 03:00:01 UTC (59 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: math math.OC 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?)