A method of simulating quantum gates includes shifting a Fock basis for the simulation such that the simulation can be performed in a smaller (e.g. truncated) Hilbert dimension space. To shift the Fock basis, non-orthonormalized basis states are first defined. The defined basis states are then orthonormalized to construct orthonormalized shifted Fock basis state. Matrix elements are determined for an operator in the orthonormalized shifted Fock basis and the operator is used to simulate the quantum gate in the shifted Fock basis.