Title: Dual-Unitary Classical Shadow Tomography

Abstract: We study operator spreading in random dual-unitary circuits within the context of classical shadow tomography. Primarily, we analyze the dynamics of the Pauli weight in one-dimensional qubit systems evolved by random two-local dual-unitary gates arranged in a brick-wall structure, ending with a final measurement layer. We do this by deriving general constraints on the Pauli weight transfer matrix and specializing to the case of dual-unitarity. We first show that dual-unitaries must have a minimal amount of entropy production. Remarkably, we find that operator spreading in these circuits has a rich structure resembling that of relativistic quantum field theories, with massless chiral excitations that can decay or fuse into each other, which we call left- or right-movers. We develop a mean-field description of the Pauli weight in terms of $\rho(x,t)$, which represents the probability of having nontrivial support at site $x$ and depth $t$ starting from a fixed weight distribution. We develop an equation of state for $\rho(x,t)$, and simulate it numerically using Monte Carlo simulations. Lastly, we demonstrate that the fast-thermalizing properties of dual-unitary circuits make them better at predicting large operators than shallow brick-wall Clifford circuits.