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We conduct a systematic study of quantum circuits composed ofmultiple-control $Z$-rotation (MCZR) gates as primitives, since they arewidely-used components in quantum algorithms and also have attracted muchexperimental interest in recent years. Herein, we establish acircuit-polynomial correspondence to characterize the functionality of quantumcircuits over the MCZR gate set with continuous parameters. An analytic methodfor exactly synthesizing such quantum circuit to implement any given diagonalunitary matrix with an optimal gate count is proposed, which also enables thecircuit depth optimal for specific cases with pairs of complementary gates.Furthermore, we present a gate-exchange strategy together with a flexibleiterative algorithm for effectively optimizing the depth of any MCZR circuit,which can also be applied to quantum circuits over any other commuting gateset.  Besides the theoretical analysis, the practical performances of our circuitsynthesis and optimization techniques are further evaluated by numericalexperiments on two typical examples in quantum computing, including diagonalHermitian operators and Quantum Approximate Optimization Algorithm (QAOA)circuits with tens of qubits, which can demonstrate a reduction in circuitdepth by 33.40\% and 15.55\% on average over relevant prior works,respectively. Therefore, our methods and results provide a pathway forimplementing quantum circuits and algorithms on recently developed devices.

Characterization, synthesis, and optimization of quantum circuits over multiple-control $\textit{Z}$-rotation gates: A systematic study