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Quantum Error Correction and Fault Tolerant Quantum Computing

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Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

Basics of Classical Error Correction

Basic Ideas of Quantum Error Correction

A Simple Example and Shor's Nine-Qubit Code

Conditions for Quantum Error Correction

Quantum Codes from Classical Codes

Techniques for Fault-Tolerant Quantum Computing

The Threshold Theorem

Further Aspects

Bibliography

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Abbreviations

Ancilla qubits:

Refers to qubits that are used to facilitate quantum computations, but serve neither as input nor as output of the computation. The two main uses of ancilla qubits are a) for the purpose of a scratch pad during a quantum computation, i. e., states that are initialized in a known state and can be used during a computation such that they are returned to the initial state at the end, and b) for the purpose of quantum error correction where they serve as space for holding the error syndrome.

Bit-flip error:

An error that affects a single qubit and interchanges the basis states of that qubit.

CSS code:

A special class of quantum codes named after their inventors Calderbank, Steane, and Shor. CSS codes allow to construct quantum codes from certain classical error-correcting codes.

CNOT gate:

The controlled-NOT gate is an example of a quantum gate. It is a two qubit gate and – together with single qubit gates – can be shown to be a universal gate for quantum computation.

Error syndrome:

A classical bit-vector that describes a quantum error that has affected a quantum code. Contrary to the classical case where there is a one-to-one correspondence between syndromes and errors, in the quantum case a syndrome does in general not uniquely identify a quantum error.

Fault-tolerant quantum computing:

The discipline that studies how to perform reliable quantum computations with imperfect hardware.

Phase-flip error:

An error that affects a single qubit and gives a relative phase of −1 to the basis states of that qubit. Contrary to the bit-flip error, this error has no classical analog.

Quantum channel:

A term describing a general physical operation that can affect the state of a quantum-mechanical system. Another name for quantum channels are completely-positive trace preserving maps.

Quantum error-correcting code (QECC):

A method to introduce redundancy to a quantum mechanical system in such a way that certain errors that are affecting the system can be detected or corrected.

Quantum circuits:

Operations that a quantum computer can carry out. Quantum circuits are composed of quantum gates and, when acting on an n qubit system, correspond to unitary matrices of size \( { 2^n \times 2^n } \).

Quantum gate:

A basic operation that a quantum computer can carry out. Common choices for quantum gates are certain unitary matrices of size \( { 2\times 2 } \) (single qubit gates) or \( { 4 \times 4 } \) (two qubit gates).

Qubit:

Shortened form of quantum bit. A physical object that can support states in a two-dimensional complex Hilbert space. Contrary to the classical case where a bit refers to both, the physical object and the states it can hold, a qubit refers to the physical object only. Qubits are the basic units of a quantum computer's memory.

Stabilizer code:

A class of quantum codes that is defined as the joint eigenspace of a group of commuting operators. Stabilizer codes give rise to many examples of quantum codes, comprise the class of CSS codes as a special case, and are equivalent to additive codes over GF(4) that are self-orthogonal with respect to the Hermitian inner product.

Threshold theorem:

An important result of the theory of fault-tolerant quantum computing, stating that there is a threshold value for this noise level such that arbitrarily long quantum computations become possible if the gates have a noise level that is under the threshold.

Transversal gates:

Special class of operations acting on encoded quantum data. Transversal gates have the important property to exhibit benign behavior in case errors happen during the application of the gate.

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Author information

Authors and Affiliations

  1. Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria

    Markus Grassl

  2. NEC Laboratories America, Inc., Princeton, USA

    Martin Rötteler

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  1. Markus Grassl
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Grassl, M., Rötteler, M. (2012). Quantum Error Correction and Fault Tolerant Quantum Computing. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_152

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