Circuit complexity near critical points

We consider the Bose-Hubbard model in two and three spatial dimensions and numerically compute the quantum circuit complexity of the ground state in the Mott insulator and superfluid phases using a mean field approximation with additional quadratic fluctuations. After mapping to a qubit system, the result is given by the complexity associated with a Bogoliubov transformation applied to the reference state taken to be the mean field ground state. In particular, the complexity has peaks at the O(2) critical points where the system can be described by a relativistic quantum field theory. Given that we use a Gaussian approximation, near criticality the numerical results agree with a free field theory calculation. To go beyond the Gaussian approximation we use general scaling arguments that imply that, as we approach the critical point t -> t(c), there is a non-analytic behavior in the complexity c(2)(t) of the form vertical bar c(2)(t) - c(2)(t(c))vertical bar similar to vertical bar t - t(c)vertical bar(nu)d, up to possible logarithmic corrections. Here d is the number of spatial dimensions and nu is the usual critical exponent for the correlation length xi similar to vertical bar t - t(c)vertical bar(-nu) As a check, for d = 2 this agrees with the numerical computation if we use the Gaussian critical exponent nu = 1/2. Finally, using AdS/CFT methods, we study higher dimensional examples and confirm this scaling argument with non-Gaussian exponent nu for strongly interacting theories that have a gravity dual.