  We compute the universal conductivity of the (2+1)-dimensional XY
universality class, which is realized for a superfluid-to-Mott insulator
quantum phase transition at constant density. Based on large-scale Monte Carlo
simulations of the classical (2+1)-dimensional $J$-current model and the
two-dimensional Bose-Hubbard model, we can precisely determine the conductivity
on the quantum critical plateau, $\sigma(\infty)=0.359(4)\sigma_Q$ with
$\sigma_Q$ the conductivity quantum. The universal conductivity curve is the
standard example with the lowest number of components where the bottoms-up
AdS/CFT correspondence from string theory can be tested and made to use [R. C.
Myers, S. Sachdev, and A. Singh, Phys. Rev. D83, 066017 (2011)]. For the first
time, the shape of the $\sigma(i\omega_n)- \sigma(\infty)$ function in the
Matsubara representation is accurate enough for a conclusive comparison and
establishes the particle-like nature of charge transport. We find that the
holographic gauge/gravity duality theory for transport properties can be made
compatible with the data if temperature of the horizon of the black brane is
different from the temperature of the conformal field theory. The requirements
for measuring the universal conductivity in a cold gas experiment are also
determined by our calculation.
