Short remarks on shallow unitary circuits

Abstract: (i) We point out that every local unitary circuit of depth smaller than the linear system size is easily distinguished from a global Haar random unitary if there is a conserved quantity that is a sum of local operators. This is always the case with a continuous onsite symmetry or with a local energy conservation law. (ii) We explain a simple algorithm for a formulation of the shallow unitary circuit learning problem and relate it to an open question on strictly locality-preserving unitaries (quantum cellular automata). (iii) We show that any translation-invariant quantum cellular automaton in $D$-dimensional lattice of volume $V$ can be implemented using only $O(V)$ local gates in a staircase fashion.