An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation

TL;DR

This paper introduces quantum error correction and fault-tolerance, explaining how quantum codes protect against errors and how fault-tolerant procedures enable reliable quantum computation below certain error thresholds.

Contribution

It provides a comprehensive overview of quantum error-correcting codes, stabilizer formalism, and fault-tolerance theory, highlighting their connections to classical coding theory and the threshold theorem.

Findings

Quantum error correction is essential for reliable quantum computing.

Stabilizer formalism simplifies the characterization of quantum codes.

Fault-tolerance enables quantum computations below an error threshold.

Abstract

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.