The power of noisy fermionic quantum computation

Abstract

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