  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
$t\rightarrow t_c$, there is a non-analytic behavior in the complexity $c_2(t)$
of the form $|c_2(t) - c_2(t_c)| \sim |t-t_c|^{\nu d}$, up to possible
logarithmic corrections. Here $d$ is the number of spatial dimensions and $\nu$
is the usual critical exponent for the correlation length
$\xi\sim|t-t_c|^{-\nu}$. As a check, for $d=2$ this agrees with the numerical
computation if we use the gaussian critical exponent $\nu=\frac{1}{2}$.
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.
