Determine whether the total-variation bounds are tight

Determine whether the inverse-polynomial total-variation bounds for convergence of polynomial-depth brickwork random quantum circuits to the Porter–Thomas distribution are tight, given numerical evidence suggesting substantially faster convergence and the authors’ conjecture that the proven bounds are not tight.

Background

The paper proves that brickwork random quantum circuits converge to the Porter–Thomas distribution in total variation distance at an inverse-polynomial rate when the circuit depth is polynomial in the number of qubits. The proof relies on approximate-design moment bounds, Fourier analysis, mollification, and inverse-moment anticoncentration.

The authors explain that their techniques do not appear to yield inverse-exponential total-variation convergence in polynomial depth. They nevertheless cite numerical evidence indicating much faster convergence, leaving unresolved whether the stated bounds reflect the actual convergence rate.

References

We note that our techniques appear to not extend to obtaining inverse-exponential bounds on TV distance in polynomial depth, even under stronger smoothness assumptions about the distribution of random circuits. However, we conjecture that our bounds are not tight due to numerical evidence suggesting extremely fast convergence of the empirical distribution to the Porter-Thomas distribution .

— Local random quantum circuits converge to the Porter-Thomas distribution in polynomial depth  (2610.02125 - Sen et al., 1 Oct 2026) in Section 1, Technical Overview