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We present a quantum Monte Carlo study of the "quantum glass" phase of the 2DBose-Hubbard model with random potentials at filling $\rho=1$. In the narrowregion between the Mott and superfluid phases the compressibility has the form$\kappa \sim {\rm exp}(-b/T^\alpha)+c$ with $\alpha <1$ and $c$ vanishing orvery small. Thus, at $T=0$ the system is either incompressible (a Mott glass)or nearly incompressible (a Mott-glass-like anomalous Bose glass). At strongerdisorder, where a glass reappears from the superfluid, we find a conventionalhighly compressible Bose glass. On a path connecting these states, away fromthe superfluid at larger Hubbard repulsion, a change of the disorder strengthby only $10\%$ changes the low-temperature compressibility by more than fourorders of magnitude, lending support to two types of glass states separated bya phase transition or a sharp cross-over.

Anomalous quantum glass of bosons in a random potential in two dimensions