Dissipation as computation: a BQP-hard observable that settles faster than the gap

Dissipation typically destroys quantum coherence and caps quantum computation depth, yet it may serve as a computational resource: a driven system relaxes to a nonequilibrium steady state that can encode computational outcomes, and prior work established that estimating the associated local observable is BQP-hard. Earlier postselection-free readout schemes suffered from suppressed signals and failed to achieve constant margin. This work constructs a dissipative Feynman–Kitaev clock with qubit data and readout registers plus a K-state clock register. Periodic refresh of the readout encodes outputs from arbitrary BQP circuits, evading the system’s slow dynamical modes and lifting the margin to O(1). The steady-state convergence rate outpaces the Lindbladian spectral gap and is independent of the number of data qubits. No sampling conjecture is required; the BQP-hardness proof relies only on primitivity, uniqueness of the steady state, polynomial spectral gap, and the identity relating the readout observable to circuit acceptance probability. Three constructions — register, cell-embedded and one-hot — realize complexity reduction, geometric locality and the spectral-gap result respectively. The finite-dimensional core is machine-checked in Lean 4, and the new proof tools may generalize to hardness arguments for other open quantum systems.