1207.5236 (Michael E. Cuffaro)

On the Implications of the Gottesman-Knill Theorem for our Understanding
of the Resources Involved in Quantum Speedup    [PDF]

Michael E. Cuffaro

According to the Gottesman-Knill theorem, any quantum algorithm or protocol which exclusively utilises the elements of a certain restricted set of quantum operations can be efficiently simulated by classical means. Since some of the algorithms and protocols falling into this category involve entangled states, it is usually concluded that entanglement cannot be sufficient to enable quantum speedup. In this short note I argue that this conclusion is misleading. As I explain, the quantum operations to which the Gottesman-Knill theorem applies are precisely those which will never cause a qubit to take on an orientation, with respect to the other subsystems comprising the total system of which it is a part, that yields a violation of the Bell inequalities. Thus, while it is true that more than entanglement is required to realise quantum computational speedup in the sense that a quantum computer implementing an entangled quantum state must utilise more than the relatively small portion of its state space that is accessible from the Gottesman-Knill group of transformations alone if it is to outperform a classical computer; i.e., while it is the case that one must \emph{use} such a state to its full potential, it is nevertheless the case that if one is asked what \emph{physical resources} suffice to enable one to bring about a quantum performance advantage, then one can legitimately answer that entanglement alone is sufficient for this task.

View original: http://arxiv.org/abs/1207.5236