The quantum XY model was first proposed [Matsubara:56a] in 1956 to study the lattice quantum fluids. Later, high-temperature series studies raised the possibility of a divergent susceptibility for the two-dimensional model. For the classical planar model, the remarkable theory of Kosterlitz and Thouless [Kosterlitz:73a] provided a clear physical picture and correctly predicted a number of important properties. However, much less is known about the quantum model. In fact, it has been controversial. Using a large-order high-temperature expansion, Rogiers, et al. [Rogiers:79a] suggested a second-order transition at for spin-1/2. Later, real-space renormalization group analysis was applied to the model with contradictory and inconclusive results. DeRaedt, et al. [DeRaedt:84a] then presented an exact solution and Monte Carlo simulation, both based on the Suzuki-Trotter transformation with small Trotter number m. Their results, both analytical and numerical, supported an Ising-like (second-order) transition at the Ising point , with a logarithmically divergent specific heat. Loh, et al. [Loh:85a] simulated the system with an improved technique. They found that specific peak remains finite and argued that a phase transition occurs at -0.5 by measuring the change of the ``twist energy'' from the lattice to the lattice. The dispute between DeRaedt, et al., and Loh, et al., centered on the importance of using a large Trotter number m and the global updates in small-size systems, which move the system from one subspace to another. Recent attempts to solve this problem still add fuel to the controversy.