Jeongwan Haah (Apr 22 2025).
Abstract: (i) We point out that every shallow unitary circuit of depth smaller than the linear system size is easily distinguished from a 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 quantum cellular automaton in
D-dimensional lattice of volume~
V can be implemented using only
O(V) local gates in a staircase fashion.
