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arXiv · 2512.11006

Undecidability of Hitting Times in Computably Described Quantum Dynamics: A No-Go Theorem for Universal Time Selection

Katsufumi Matsuura
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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. Revised version. The main result has been strengthened to undecidability under a single fixed periodically driven unitary dynamics. The discussion of related work and the computability assumptions have also been expanded.

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Katsufumi Matsuura. 2025-12-11. Undecidability of Hitting Times in Computably Described Quantum Dynamics: A No-Go Theorem for Universal Time Selection. https://arxiv.org/abs/2512.11006

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