Absence of localization in two-dimensional Clifford circuits

TL;DR

This paper demonstrates that two-dimensional Floquet Clifford circuits do not exhibit localization, with local operators spreading ballistically, contrasting with one-dimensional cases where localization occurs, supported by analytical and numerical evidence.

Contribution

It provides a rigorous proof of the absence of localization in 2D Clifford circuits using percolation theory and random graphs, complemented by numerical simulations and spectral analysis.

Findings

Two-dimensional Clifford circuits show ballistic operator growth.

One-dimensional Clifford circuits exhibit strong localization.

Spectral form factor in 2D resembles that of chaotic fermionic systems.

Abstract

We analyze a Floquet circuit with random Clifford gates in one and two spatial dimensions. By using random graphs and methods from percolation theory, we prove in the two dimensional setting that some local operators grow at ballistic rate, which implies the absence of localization. In contrast, the one-dimensional model displays a strong form of localization characterized by the emergence of left and right-blocking walls in random locations. We provide additional insights by complementing our analytical results with numerical simulations of operator spreading and entanglement growth, which show the absence (presence) of localization in two-dimension (one-dimension). Furthermore, we unveil that the spectral form factor of the Floquet unitary in two-dimensional circuits behaves like that of quasi-free fermions with chaotic single particle dynamics, with an exponential ramp that persists till times scaling linearly with the size of the system. Our work sheds light on the nature of disordered, Floquet Clifford dynamics and its relationship to fully chaotic quantum dynamics.