Classical simulation of quantum computation, the gottesman-Knill theorem, and slightly beyond

The Gottesman-Knill theorem states that every “Clifford” quantum circuit, i.e., a circuit composed of Hadamard, CNOT and phase gates, can be simulated efficiently on a classical computer. It is was later found that a highly restricted classical computer (using only NOT and CNOT gates) suffices to simulate all Clifford circuits, implying that these circuits are most likely even significantly weaker than classical computation. On the other hand, it is also known that extending the Clifford group with any non-Clifford gate immediately yields universal quantum computation. This last feature makes it nontrivial to extend the GottesmanKnill theorem to larger classes of simulatable quantum circuits. The aim of this paper is to study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We first show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. Moreover, the normal form highlights why Clifford circuits have such limited computational power in spite their high entangling power. Second, using the normal form we show how the Gottesman-Knill theorem can be extended such as to yield simple classes of quantum circuits which are classically simulatable and (contrary to Clifford circuits) capable of performing universal classical computation. These extended circuits can by efficiently simulated by classical sampling techniques (“weak simulation”) even though the problem of exactly computing the outcomes of measurements for these circuits (“strong simulation”) is proved to be #P-complete—thus showing that there is a separation between weak and strong classical simulation of quantum computation. ∗Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany. 1 ar X iv :0 81 1. 08 98 v1 [ qu an tph ] 6 N ov 2 00 8