Undecidability of the Unitary Hitting Time Problem: No Universal Time-Step Selector and an Operational No-Go for Finite-Time Decisions

Quantum Physics arXiv:2512.11006 (quant-ph) [Submitted on 11 Dec 2025] Title:Undecidability of the Unitary Hitting Time Problem: No Universal Time-Step Selector and an Operational No-Go for Finite-Time Decisions Authors:Katsufumi Matsuura View a PDF of the paper titled Undecidability of the Unitary Hitting Time Problem: No Universal Time-Step Selector and an Operational No-Go for Finite-Time Decisions, by Katsufumi Matsuura View PDF HTML (experimental) Abstract:We study the Unitary Hitting Time Problem (UHTP) in quantum dynamics. Given computably described pure states |a>, |b> and a time-dependent unitary U(t), define the hitting time as the infimum of t > 0 such that the fidelity between U(t)|a> and |b> reaches a fixed threshold (with infinity if the threshold is never reached). We prove that there is no total algorithm that outputs this hitting time for all inputs; equivalently, the total UHTP is undecidable via a reduction from the halting problem. Operationally, we show a no-go theorem: for any fixed accuracy parameters, there is no universal finite-resource protocol that, for all computably described inputs, correctly outputs the hitting time while obeying uniform finite upper bounds on observation time and on dissipation/work. The proofs use reversible computation embedded into unitary dynamics, a fixed-target beacon construction, and a continuous-time lifting via piecewise-constant Hamiltonians. Our results target systems capable of embedding universal computation and complement prior undecidability results such as spectral-gap and quantum-control reachability. We distinguish logical time (inside the equations) from physical/operational time (of preparation, evolution, measurement), and show that universal time-step selection is impossible in both senses. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2512.11006 [quant-ph] (or arXiv:2512.11006v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.11006 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Katsufumi Matsuura [view email] [v1] Thu, 11 Dec 2025 08:17:18 UTC (9 KB) Full-text links: Access Paper: View a PDF of the paper titled Undecidability of the Unitary Hitting Time Problem: No Universal Time-Step Selector and an Operational No-Go for Finite-Time Decisions, by Katsufumi MatsuuraView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)