1. A method for maintaining global phases during simulation of at least one quantum gate of a quantum computer using a classical computer, the method comprising: receiving, at a processor of the classical computer, a quantum state that is a superposition of a plurality of stabilizer states, wherein the quantum state is represented by a stabilizer matrix associated with the plurality of stabilizer states, a plurality of phase vectors representing each of the stabilizer states, and an amplitude vector, wherein each entry in the amplitude vector represents a global phase associated with one of the plurality of phase vectors; receiving, at a processor of the classical computer, a matrix representation of the at least one quantum gate; and determining, by a processor of the classical computer, the effect of the at least one quantum gate on the quantum state in a plurality of iterations, each iteration including: applying, by a processor of the classical computer, one of the plurality of phase vectors to the stabilizer matrix; determining, by a processor of the classical computer, an input basis state associated with the one phase vector applied to the stabilizer matrix; determining, by a processor of the classical computer, an input non-zero amplitude associated with the input basis state; determining, by a processor of the classical computer, a first output non-zero amplitude associated with an output basis state by applying the matrix representation of the at least one quantum gate to the input non-zero amplitude and the input basis state; determining, by a processor of the classical computer, a second output non-zero amplitude of the output basis state using the stabilizer matrix and the matrix representation of the at least one quantum gate; and adjusting, by a processor of the classical computer, the entry in the amplitude vector associated with the one phase vector applied to the stabilizer matrix, wherein the entry is adjusted proportionally to the first output non-zero amplitude and the second output non-zero amplitude.