An extensive study of the thermodynamics of a two-dimensional periodic array of ultrasmall Josephson junctions with and without a transverse magnetic field is presented. A quantum Monte Carlo algorithm is introduced to study a model that includes the Josephson energy, EJ, as well as the charging energy, Ec, contributions. The superfluid density, internal energy, and specific heat for different lattice sizes and numbers of Monte Carlo simulation sweeps are studied as a function of the ratio α=Ec/EJ, the temperature and the magnitude of the magnetic field. When α0, it is found that as the temperature is lowered the model has two phase transitions. First, a second-order Berezinskii-Kosterlitz-Thouless (BKT) transition renormalized by the quantum fluctuations represented by a finite α. Below this BKT transition the system has long-range phase coherence; thus it is a state with zero resistance. At lower temperatures, a first-order phase transition appears which is entirely due to the quantum fluctuations that nucleate vortex excitations. Below this ''quantum induced transition'' (QUIT) the model still has a finite but diminished superfluid density, thus indicating that the QUIT is between two different zero-resistance states, one dominated by thermal fluctuations and the other by quantum fluctuations. A QUIT is found to be more pronounced in the case where there is a magnetic field. The case studied here corresponds in the classical limit to the fully frustrated state. Finally, we discuss the physical properties of this new low-temperature phase as well as the necessary conditions to test this prediction experimentally.