Circuit optimization of informationally complete positive operator–valued qubit measurements for shadow estimation

In quantum information processing, using positive operator--valued measurements (POVMs) enables the design of general measurement schemes and efficient estimation methods with fixed circuits, but the CNOT-gate count during compilation becomes a bottleneck. This study shows that any single-qubit minimal informationally complete POVM (IC POVM) can be realized with at most two CNOT gates---and a symmetric case (SIC POVM) requires only one. This is achieved by virtually inserting gates to adjust unrelated parameters in the compilation circuit. The authors also provide a concise compilation circuit for single-qubit SIC POVMs, paving the way for practical application.