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

TL;DR

This paper proves the undecidability of the Unitary Hitting Time Problem in quantum dynamics and establishes a no-go theorem for universal finite-time decision protocols.

Contribution

It demonstrates that no universal algorithm or finite-resource protocol can determine hitting times for all quantum states, highlighting fundamental limits in quantum control.

Findings

No total algorithm exists for the UHTP due to undecidability.

Universal finite-resource protocols cannot correctly determine hitting times for all inputs.

The results extend undecidability to operational time-step selection in quantum systems.

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.