We present a complete optimization procedure for hybrid quantum-classical circuits with classical parity logic. While common optimization techniques for quantum algorithms focus on rewriting solely the pure quantum segments, there is interest in applying a global optimization process for applications such as quantum error correction and quantum assertions.

This work, based on the pure-quantum circuit optimization procedure by Duncan et al., uses an extension of the formal graphical ZX-calculus called ZX⏚ as an intermediary representation of the hybrid circuits to allow for granular optimizations below the quantum-gate level. We define a translation from hybrid circuits into diagrams that admit the graph-theoretical focused-gFlow property, needed for the final extraction back into a circuit. We then derive a number of gFlow-preserving optimization rules for ZX⏚ diagrams that reduce the size of the graph, and devise an strategy to find optimization opportunities by rewriting the diagram guided by a Gauss elimination process. Then, after extracting the circuit, we present a general procedure for detecting segments of circuit-like ZX⏚ diagrams which can be implemented with classical gates in the extracted circuit. We have implemented our optimization procedure as an extension to the open-source python library PyZX.