Polynomial relaxation time for arbitrary-depth random circuits

Show that for n-qubit states prepared by random quantum circuits of arbitrary depth, the Markov chain on {0,1}^n with transition probabilities P(x,y) defined from π(x)=|⟨x|ψ⟩|^2 has relaxation time τ ≤ poly(n), ensuring that the shadow-overlap certification protocol remains efficient for these circuit-generated states.

Background

The certification protocol’s efficiency hinges on the relaxation time τ of a Markov chain whose stationary distribution is π(x)=|⟨x|ψ⟩|^2. The authors show τ=O(n^2) for Haar-random states and seek to extend such guarantees to more structured and experimentally relevant classes.

Random quantum circuits are a natural source of complex many-body states. Proving polynomial relaxation time for states prepared by arbitrary-depth random circuits would establish that the proposed certification via shadow overlap is feasible for a broad set of highly entangled, circuit-generated states.