The Power of Noisy Fermionic Quantum Computation

TL;DR

This paper investigates the threshold noise levels in fermionic quantum computation with Majorana fermions, showing when it can be efficiently simulated classically based on the noise in ancilla states.

Contribution

It extends entanglement theory to Gaussian fermionic states, providing criteria to distinguish convex mixtures of Gaussian states from non-Gaussian states.

Findings

Classical simulation possible above 89% noise rate.

States below 53% noise are not convex mixtures of Gaussian states.

Developed criteria for Gaussian state convexity using symmetric extensions.

Abstract

We consider the realization of universal quantum computation through braiding of Majorana fermions supplemented by unprotected preparation of noisy ancillae. It has been shown by Bravyi [Phys. Rev. A 73, 042313 (2006)] that under the assumption of perfect braiding operations, universal quantum computation is possible if the noise rate on a particular 4-fermion ancilla is below 40%. We show that beyond a noise rate of 89% on this ancilla the quantum computation can be efficiently simulated classically: we explicitly show that the noisy ancilla is a convex mixture of Gaussian fermionic states in this region, while for noise rates below 53% we prove that the state is not a mixture of Gaussian states. These results were obtained by generalizing concepts in entanglement theory to the setting of Gaussian states and their convex mixtures. In particular we develop a complete set of criteria, namely the existence of a Gaussian-symmetric extension, which determine whether a state is a convex mixture of Gaussian states.