Theory of fault-tolerant quantum computation

TLDR

Fault‑tolerant operations on encoded states are required to harness quantum error‑correcting codes for improved quantum computer performance. The paper presents a theory of fault‑tolerant operations on stabilizer codes grounded in stabilizer symmetries. The theory derives fault‑tolerant operations from stabilizer symmetries and illustrates the approach with examples such as the five‑qubit code. The theory enables straightforward identification of fault‑tolerant operations for a code and shows that universal fault‑tolerant computation is achievable with any stabilizer code.

Abstract

In order to use quantum error-correcting codes to improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a theory of fault-tolerant operations on stabilizer codes based on symmetries of the code stabilizer. This allows a straightforward determination of which operations can be performed fault-tolerantly on a given code. I demonstrate that fault-tolerant universal computation is possible for any stabilizer code. I discuss a number of examples in more detail, including the five-quantum-bit code.