<p>Identifying the precise boundary between quantum and classical computational power is a central challenge in quantum computing. We introduce <i>polyhedral classical simulators</i>, a framework for classical simulation grounded in polyhedral geometry. This framework encompasses well-known methods such as the Gottesman–Knill algorithm, while extending naturally to more recent models including quantum computation with magic states and measurement-based quantum computation. The framework is compositional: The correctness of a simulation reduces to verifying a preservation property for the basic instruments of the model, from which the full adaptive simulation is assembled. Beyond unifying existing simulation methods, this provides a geometric roadmap for pushing the boundary of efficient classical simulation further.</p>