Complexity measures in QFT and constrained geometric actions

A bstract We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to circuit associated set of gates. Inhomogeneous cost functions — many examples which have been recently proposed in literature are ruled out by our analysis. Such measures shown be unrelated general and produce severe violations Lloyd’s bound simple situations. Among do bounds, idea is select those tightest possible ones. This establishes hierarchy considerably reduces list candidate measures. In particular, criterion suggests canonical way dealing with penalties, consisting assigning infinite costs directions not belonging gate set. discuss how this can implemented through use Lagrange multipliers. argue that one surviving defines particularly notion sense that: i) it straightforwardly follows from standard Hermitian metric Hilbert space; ii) its functional closely related Kirillov’s coadjoint orbit action, providing an explicit realization “complexity equals action” idea; iii) arises Hamilton-Jacobi analysis “quantum describing dynamics phase space canonically every space. Finally, we explain these structures natural framework for characterizing chaos classical systems on equal footing, find minimal geodesic connecting two nearby trajectories, describe sensitive Lyapunov exponents.