Numerical analysis of quantum circuits for state preparation and unitary operator synthesis

We perform optimal-control-theory calculations to determine the minimum number of two-qubit CNOT gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multi-qubit entangling gates instead of two-qubit CNOT gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multi-qubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.