Absence of localization in two-dimensional Clifford circuits

Unitary circuit models provide a useful lens on the universal aspects of the dynamics of many-body quantum systems. Although interactions lead to spreading of quantum information and thermalization, the presence of strong disorder can prohibit this process, by inducing localization of information. Unrestricted randomness along space-time in circuit models of quantum dynamics gives rise to quantum chaos and fast scrambling of information. On the contrary, time-periodic circuits can be susceptible to localization. In this work, we show the crucial role played by dimensionality of physical space on the existence of localization in such quantum circuits. We analyse 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 highlights that Clifford circuits can play a vital role in quantum simulations of a wide class of novel quantum many-body phenomena.