Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs

Quantum Physics arXiv:2608.07986 (quant-ph) [Submitted on 8 Aug 2026] Title:Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs Authors:Masayuki Miyamoto View a PDF of the paper titled Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs, by Masayuki Miyamoto View PDF HTML (experimental) Abstract:We study gap amplification of the class $\mathsf{QMA}^{+}(2)$ characterized by unentangled quantum proofs whose amplitudes are nonnegative in the computational basis. This class was recently introduced by Jeronimo and Wu (STOC 2023), and its behavior depends sharply on the completeness-soundness gap: although it captures the power of $\mathsf{NEXP}$ for some small constant gap, it is equal to $\mathsf{QMA}(2)$ for larger constant gap. This is in stark contrast to $\mathsf{QMA}(2)$ where strong gap amplification is known due to the product test by Harrow and Montanaro (FOCS 2010, JACM 2013). In this paper, we prove for every completeness $c$ and soundness $s$ with $c-s=1/\mathrm{poly}(n)$, \[ \mathsf{NEXP} = \mathsf{QMA}^{+}(2,c,s) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14+\frac1{\mathrm{poly}(n)}\right). \] Our result gives a clean complexity phase transition for $\mathsf{QMA}^{+}(2)$ since we have \[ \mathsf{QMA}^{\mathbb R}(2) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14-\frac1{\mathrm{poly}(n)}\right), \] where $\mathsf{QMA}^{\mathbb R}(2)$ denotes $\mathsf{QMA}(2)$ with witnesses restricted to real amplitudes. Our amplification is thus optimal in the sense that a slight improvement of our soundness would have the collapse \[ \mathsf{QMA}^{\mathbb R}(2)=\mathsf{NEXP}. \] Our proof combines symmetric-subspace projections with the relation $\mathsf{QMA}^{+}(1)=\mathsf{NEXP}$ of Bassirian, Fefferman, and Marwaha (ITCS 2024). The main technical ingredient is a dimension-independent de Finetti theorem in Hilbert-Schmidt norm that applies when the number of registers under consideration grows logarithmically. Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) Cite as: arXiv:2608.07986 [quant-ph] (or arXiv:2608.07986v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.07986 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Masayuki Miyamoto [view email] [v1] Sat, 8 Aug 2026 07:45:56 UTC (38 KB) Full-text links: Access Paper: View a PDF of the paper titled Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs, by Masayuki MiyamotoView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-08 Change to browse by: cs cs.CC 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?)