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(n2) 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.

References

Further extending the reach of our certification protocol based on the shadow overlap raises many interesting open questions. More generally, can we show that states prepared with (random) quantum circuits of arbitrary depth satisfy a relaxation time τ ≤ poly(n)?

— Certifying almost all quantum states with few single-qubit measurements  (2404.07281 - Huang et al., 2024) in Outlook

From a theoretical perspective, the numerical evidence presented here suggests that the $\tau$ reducing layer is able to bring $\tau$ down to $\mathcal{O}(n)$ for a broad range of structured states. If this could be established analytically, it would imply that the shadow overlap method is an efficient certification procedure for arbitrary target states, resolving a central open question of and significantly broadening the practical applicability of shadow overlap certification.

— Efficient quantum state preparation on Quantinuum hardware  (2609.08414 - Butterworth et al., 8 Sep 2026) in Section Conclusions