Conjecture C Still Stands

Abstract

More than ten years ago the author described a parameter $K(\rho )$ for the complexity of $n$-qubit quantum state $\rho$ and raised the conjecture (referred to as "Conjecture C") that when this parameter is superpolynomial in $n$, the state $\rho$ is not experimentally feasible (and will not be experimentally achieved without quantum fault-tolerance). Shortly afterward [6] (arXiv:1204.3404), Steve Flammia and Aram Harrow claimed that the simple easy-to-construct $W$ states are counterexamples to "Conjecture C." We point out that Flammia and Harrow's argument regarding $W$-states is incomplete. Moreover, the emergent picture from experimental progress of the past decade on noisy intermediate scale quantum (NISQ) computers suggests that $W$-states, as simple as they appear, cannot be achieved experimentally by NISQ computers, and can not be constructed without quantum fault-tolerance.