A generalization of the stabilizer formalism for simulating arbitrary quantum circuits

We present a new approach to simulate arbitrary quantum circuits on a classical computer. Our technique generalizes the stabilizer formalism, the underpinnings of the Gottesman-Knill theorem, to include arbitrary states and arbitrary quantum operations. The core of our approach is a novel state representation combining the density matrix and stabilizer representations. Obviously, not all simulations are efficient within our formalism, but we find special cases, wherein the input state and number of non-Clifford gates are restricted, which are. Importantly, the stabilizer circuits remain efficiently simulatable by our techniques, as a special case. The non-stabilizer quantum circuits for which we prove efficient simulations are just mildly non-stabilizer as judged by a certain heuristic, in terms of which our formalism’s time complexities are naturally expressed. Interestingly, it seems that our approach is related to an interaction picture of quantum circuits, where stabilizer operations take the role of the unperturbed evolution.