Quantum circuits include quantum gates that require a certain amount of physical resources. It is desirable to reduce the number of certain quantum gates in order to improve the hardware efficiency of a quantum circuit. A system and a method are provided for synthesizing a quantum circuit that includes the operations of determining an ordered set or vector of phase coefficients and specifying a linear permutation. The operations also include determining a sequence of CNOT and Rz gates using the set of phase coefficients and the linear permutation, such that the number of Rz gates is reduced.