Fault-Tolerant Quantum Computation With Constant Error Rate

TL;DR

This paper establishes a comprehensive proof that quantum computation can be made fault-tolerant with a constant error threshold, applicable to general noise models and one-dimensional systems, without the need for measurements during computation.

Contribution

It provides a complete, simplified proof of fault-tolerant quantum computation with universal gate sets under broad noise conditions, filling gaps in previous work.

Findings

Fault tolerance achievable below a constant error rate

Simplified proofs for CSS code correctness and universality of two-qubit gates

Fault-tolerant procedures work for general non probabilistic noise models

Abstract

This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, $\eta$, is smaller than a constant threshold, $\eta_c$. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.