On the Significance of the Gottesman–Knill Theorem

Abstract

According to the Gottesman-Knill theorem, quantum algorithms which utilise only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this paper that this conclusion is misleading. First, the statement of the theorem (that the particular set of quantum operations in question can be simulated using a classical computer) is, on reflection, already evident when we consider Bell's and related inequalities in the context of a discussion of computational machines. This, in turn, helps us to understand that the appropriate conclusion to draw from the Gottesman-Knill theorem is not that entanglement is insufficient to enable a quantum performance advantage, but rather that if we limit ourselves to the operations referred to in the Gottesman-Knill theorem, we will not have used the resources provided by an entangled quantum system to their full potential.

No-go theorems: What are they good for?

“…Note that P, M, F G, B does not imply P, M, F G ′ , B when G implies G ′ . This is nicely illustrated by two recent papers by Cuffaro (2017Cuffaro ( , 2018. While discussing the Bell inequality and the GHZ equality, he distinguishes between two kinds of context: the theoretical and the practical context.…”

“…The mathematician and polymath John von Neumann claimed to have proven in his classic (Neumann, 1932) the impossibility to complete quantum mechanics by hidden variables. A couple of years later, Grete Hermann (1935/2017 challenges von Neumann, claiming that he "introduces into its formal assumptions, without justification, a statement equivalent to the thesis to be proven". Thirty years later, Jauch and Piron (1963) state, unaware of Hermann's claim, that "[t]he question concerning the existence of such hidden variables received an early and rather decisive answer in the form of von Neumann's proof on the mathematical impossibility of such variables in quantum theory".…”

Interaction signatures and non-Gaussian photon states from a strongly driven atomic ensemble coupled to a nanophotonic waveguide

“…This light can be transported through optical fibers with very small losses for long distances, hence making the findings in this work particularly relevant for optical quantum information and communication. Moreover, it has been shown that a quantum circuit containing operations that generate states represented by a negative Wigner function cannot be efficiently simulated with classical algorithms [31][32][33]. Hence, the atom-nanofiber setup considered in this work is a candidate for generating quantum computational resources by making use of state-of-the-art light-matter technologies.…”

“…This negativity exhibits, as well, a strong particle number dependence, but also shows a high degree of robustness against voids in the atomic chain due to the peculiar nature of the all-to-all interactions. Our study shows that strong driving may enable the controlled realization of non-Gaussian quantum states of light with a potential use as quantum resources [31][32][33].…”

Interaction signatures and non-Gaussian photon states from a strongly driven atomic ensemble coupled to a nanophotonic waveguide

“…These findings are particularly relevant for optical quantum computing and simulation. In fact, it has been shown that a quantum circuit containing operations that generate states represented by a negative Wigner function cannot be efficiently simulated with classical algorithms [31][32][33]. Hence, the atom-nanofiber setup considered in this work is an excellent candidate for generating quantum computational resources by making use of state-of-the-art light-matter technologies.…”