Establish sublinear-depth convergence to Porter–Thomas

Establish whether the output distribution of brickwork random quantum circuits becomes close to the Porter–Thomas distribution in total variation distance at some sublinear circuit depth.

Background

The main theorem establishes inverse-polynomial convergence to the Porter–Thomas distribution for brickwork random circuits at polynomial depth. The paper notes that certain random-circuit quantities approach their Haar values at logarithmic depth and that some circuit ensembles form relative-error designs at logarithmic depth.

These observations motivate the unresolved question of whether full closeness of the output-probability distribution to Porter–Thomas, specifically in total variation distance, can occur at a depth that is sublinear in the number of qubits.

References

Thus we leave open the question of whether closeness to Porter-Thomas occurs at some sublinear depth.

— 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

Another key open question is whether a similar result can be established in the presence of (sufficiently weak) noise. For random circuits with local depolarizing noise (or any local unital noise), for what depth and noise rate $$ is the output distribution still close to Porter-Thomas? There is a recently understood phase transition in the noise rate . As local noise rapidly drives the output distribution to uniform, a sufficient condition for the noise rate should be $\ll 1/n\ell $ . If $\ell_{\rm PT}$ is the depth at which the noiseless circuit first becomes approximately Porter-Thomas, is there some depth $\ell_{\rm PT}\leq \ell \ll 1/ n$ at which the noisy circuit output is close in TV distance to Porter-Thomas?

— 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