A Formalization of Measurement-Commuting Unitaries

Hybrid classical-quantum algorithms promise efficient use of near-term quantum devices by strategically partitioning computation between classical and quantum resources. A central challenge in this paradigm is identifying quantum operations that can be executed independently of specific measurement outcomes, thereby reducing quantum-classical feedback and enabling circuit cutting, early measurement, and simulation optimizations. This paper introduces a formal framework for analyzing when unitary operations commute with projective measurements. We characterize a class of such unitaries, termed eigenvalue permutations, that preserve measurement outcomes and thus commute with measurement in a well-defined sense. These unitaries allow for early measurement without additional computational overhead, offering a new primitive for optimizing hybrid circuits. We discuss theoretical foundations, provide structural results, and outline implications for hybrid execution strategies, circuit cutting, and classical simulation.