Short remarks on shallow unitary circuits

(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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> -dimensional lattice of volume <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> can be implemented using only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> local gates in a staircase fashion using invertible subalgebra pumping.