Optimization of universal quantum gates with higher-dimensional spaces

Minimizing the number of necessary two-qubit gates is an important task in quantum information processing. By introducing non-computational quantum states, we construct effective circuits for the controlled-NOT (CNOT) gate and the n-qubit Toffoli gate with (2n-3) qubit-qudit gates and (2n-4) single-qudit gates. The cost of a three-qubit Toffoli gate is reduced from the minimum limit of five two-qubit gates to three nearest-neighbor qubit-qudit gates. By operating with the spatial-mode degree of freedom, polarization CNOT and Toffoli gates are designed with linear optics. Our optical schemes can be achieved with a higher success probability and no extra auxiliary photons are needed in the scheme.