Understanding the boundary between quantum and classical computation is a fundamental problem in quantum information theory. Quantum computers are expected to outperform classical computers on certain computational tasks, yet a comprehensive mathematical framework that identifies the physical and structural resources responsible for this advantage is still lacking. One direct approach to this problem is to characterize families of quantum circuits that admit efficient classical simulation. Classical simulation methods come in many forms, including state-vector methods, tensor-network algorithms, and more algebraic approaches based on stabilizer theory. A canonical result in the latter direction is the Gottesman--Knill theorem, which shows that stabilizer circuits can be efficiently simulated classically using the stabilizer tableau formalism.

In this talk, I will introduce a framework for classical simulation based on the geometry of polytopes in spaces of quantum operators. From this perspective, stabilizer simulation appears as a particular instance of a broader geometric construction, as do later quasiprobability approaches based on discrete phase-space representations such as discrete Wigner functions. The resulting polyhedral simulators extend the known boundary of efficiently classically simulatable quantum circuits beyond the stabilizer regime. I will describe the geometric principles underlying these simulators and discuss what this perspective suggests about the boundary between classical and quantum computation.