  Quasicrystals remain among the most intriguing materials in physics and
chemistry. Their structure results in many unusual properties including
anomalously low friction as well as poor electrical and thermal conductivity
but it also supports superconductivity, which shows that quantum effects in
quasicrystals can be quite unique. Here we theoretically study superfluidity in
a model quantum cluster quasicrystal. Using path-integral Monte Carlo
simulations, we explore a 2D ensemble of bosons with the
Lifshitz-Petrich-Gaussian pair potential, finding that moderate quantum
fluctuations do not destroy the dodecagonal quasicrystalline order. This
quasicrystal is characterized by a small yet finite superfluidity,
demonstrating that particle clustering combined with the local cogwheel
structure can underpin superfluidity even in the almost classical regime. This
type of distributed superfluidity may also be expected in certain open
crystalline lattices. Large quantum fluctuations are shown to induce
transitions to cluster solids, supersolids and superfluids, which we
characterize fully quantum-mechanically.
