Quantum Universality from Magic States Distillation Applied to CSS Codes

Abstract

Given Clifford group operations and the ability to repeatedly prepare a single-qubit mixed state ρ, can one do universal quantum computation? We show a sharp threshold in the Hadamard “magic” direction of the Bloch sphere between those ρ allowing universal quantum computation, and those for which any calculation can be efficiently classically simulated. As a corollary, the ability to repeatedly prepare any pure state which is not a stabilizer state (e.g., any single-qubit pure state which is not a Pauli eigenstate), together with Clifford group operations, gives quantum universality. As motivation for this question, “magic state” distillation procedures can reduce the general fault-tolerance problem to that of performing fault-tolerant Clifford group circuits.