Purification complexity in evaporating random circuits: a Page-transition benchmark

We prove early preparation and entropy bounds for an explicit evaporating random-circuit model. Independent Haar two-qubit gates scramble an active register before each emission, and emitted qubits never interact again. Preparation complexity allows arbitrary one- and two-qubit gates and any finite number of initially pure ancillas, at fixed trace-distance error. We specialize the established two-copy purity method to permanent deletion and bound its evolution by a moving hyperbolic-cosine comparison profile. The first scrambling round gives a uniform relative bound on expected subsystem purity; positivity preserves this bound through every subsequent emission, even if later scrambling rounds are shortened or omitted. For the stipulated schedule, radiation before a logarithmic window around half emission has a linear preparation upper bound with an explicit failure bound. The entropy of every radiation prefix differs from its dimension ceiling by at most a logarithmic term, simultaneously with high probability. Numerical recurrence calculations, exact subset identities and small-system state-vector simulations provide reproducible checks. These results settle the early-preparation and entropy clauses of the benchmark. They do not establish its post-Page complexity lower bound at the accumulated gate-count scale. We identify the additional distributional information that such a lower bound requires and explain why the entropy Page law alone is insufficient.

This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, authorized this deposit and is responsible for it.