Polynomial relaxation time for state t-designs

Establish that for n-qubit pure states forming a unitary state t-design (i.e., states whose first t moments match the Haar measure), the Markov chain on the Boolean hypercube defined by the transition probabilities P(x,y) induced from the measurement distribution π(x)=|⟨x|ψ⟩|^2 has relaxation time τ bounded by a polynomial in n, thereby enabling efficient certification via the shadow-overlap protocol.

Background

The paper proves that for Haar-random n-qubit states the relaxation time of the Markov chain derived from the measurement distribution is τ=O(n^2), which in turn yields efficient certification from few single-qubit measurements. Extending this guarantee beyond Haar randomness would broaden the applicability of the method, particularly to ensembles that approximate Haar randomness in low moments, such as state t-designs.

State t-designs can be generated by random quantum circuits of size poly(n,t), and their first t moments match those of the Haar measure. Demonstrating polynomial relaxation for the Markov chain associated with these states would provide theoretical support for efficient certification of physically realizable ensembles prepared by random circuits.