Quantum collapse in three and two dimensions (3D and 2D) is induced by attractive potential ~ -1/r^2. It was demonstrated that the mean-field (MF) cubic self-repulsion in the 3D bosonic gas suppresses the collapse and creates the missing ground state (GS). However, the cubic nonlinearity is not strong enough to suppress the 2D collapse. We demonstrate that the Lee-Hung-Yang (LHY) quartic term, induced by quantum fluctuations around the MF state, is sufficient for the stabilization of the 2D gas against the collapse. By means of numerical solution of the Gross-Pitaevskii equation including the LHY term, as well as with the help of analytical methods, such as expansions of the wave function at small and large distances from the center and the Thomas-Fermi approximation, we construct stable GS, with a singular density, ~ 1/r^{4/3}, but convergent integral norm. Counter-intuitively, the stable GS exists even if the external potential is repulsive, with the strength falling below a certain critical value. An explanation to this finding is given. Along with the GS, singular vortex states are produced too, and their stability boundary is found analytically. Unstable vortices spontaneously transform into the stable GS, expelling the vorticity to periphery.
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