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arXiv:0904.2557

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

Daniel Gottesman

Published Apr 16, 2009• 129 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.0904.2557

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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.
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.
This thesis is concerned with certain theoretical problems that arise naturally in the context of fault-tolerant quantum computation. Fault-tolerance can be defined as the art of building reliable devices from unreliable components, and is of particular importance for quantum computers that aim to precisely control the dynamics of extremely sensitive quantum systems. A model of quantum computation is a specification of the basic building blocks by which a quantum computation is implemented. The best known model is the quantum circuit model, where computations are implemented by means of unitary quantum gates that are applied to two-level quantum systems known as qubits. In a physical implementation of a quantum circuit, the gates and qubits will inevitably be affected by noise. Fault-tolerant quantum circuits are designed to be resilient against the effects of this noise, provided that it is not too strong. Fault-tolerance in the quantum circuit model is well developed thanks to the theory of quantum error-correcting codes. These codes allow for the correction of small numbers of errors introduced by a variety of noise processes. In a fault-tolerant quantum circuit, qubits are replaced with encoded qubits, and quantum gates with encoded gates that are immediately followed by special quantum circuits for error correction. Provided the rate at which errors occur is below a constant threshold value, the accumulation of errors can be checked so that the correct output of the computation correctly determined. The threshold acts as both a measure of how good the design of a quantum circuit is, and also as a target for experimenters aiming to implement quantum circuits. Much current research in quantum computation is aimed at designing quantum circuits that increase the noise threshold, hopefully to the point where it becomes within the reach of experimenters. A fault-tolerant encoded quantum gate must limit the propagation of errors so that the code’s corrective capabilities are not overwhelmed. It is not so easy to design encoded gates that satisfy this property, and to date only a finite handful of such gates are known. In a faulttolerant quantum circuit, all quantum gates must be decomposed or compiled in terms of those that may be implemented fault-tolerantly. In the first part of this thesis, we present two results that may be applied to this problem of gate compilation. The first is a generic method based on the Solovay-Kitaev theorem that may be applied to all quantum gates, but is most effective for those that act on single qubits. We present the Solovay-Kitaev theorem in its simples known form as an algorithm, together with novel constructions that can be used to implement it. Following this we give a two specialized methods for two-qubit gate decomposition, based on the Cartan decomposition of the Lie group SU(4). The cluster-state model of computation is an alternative to the quantum circuit model, and makes use of quantum measurements and highly entangled cluster-states to implement a quantum computation. The fault-tolerant techniques developed for quantum circuits are not immediately applicable in this model, so in order for it to be a realistic candidate for performing computations we prove that such techniques are possible. In the second part of the thesis we prove that constant fault-tolerance thresholds may be achieved in the cluster-state model, and in particular in an adaption to an optical implementation. Following this we design a complete error correction scheme for optical cluster-state computation, and numerically determine the threshold of this model in the face of the dominant noise models likely to affect such an implementation.
In its short history, the field of quantum computing has produced a num­ ber of truly startling revelations about the computational power inherent in quantum systems. These discoveries generated a great deal of excitement, though this excitement was tempered by the realization that the quantum correlations responsible for this power are extremely fragile. It became clear that to properly assess the performance of a quantum computer, the effects of noise and imperfect quantum gates would have to be included in the analysis. It was thought by many that the virtues of a quantum computer would not survive the presence of these effects. Remarkably, in three years (1995-1997), theoretical tools were developed that stood this expectation on its head. Be­ ginning with the groundbreaking work of P. W. Shor and A. M. Steane, a verygeneral framework for constructing quantum error correcting codes was estab­ lished, and protocols were developed that allowed quantum computations to be done in a fault-tolerant manner. This line of development culminated in a demonstration of the accuracy threshold theorem. This theorem showed that fault-tolerant protocols for quantum computation, in conjunction with concatenated quantum error correcting codes, would allow a quantum compu­ tation of arbitrary duration to be done with arbitrarily small error probability, provided all quantum gates used in the computation had error probabilities that fell below a value known as the accuracy threshold Pa. This threshold was calculated for a number of simple error models yielding values in the range 10-6 < p a < io -3 . The good news is that Pa ^ 0 so that reliable quantum computing in the presence of noise and imperfect quantum gates is possible. The bad news is that Pa is small so that building quantum gates that satisfy the tolerances of the threshold theorem will be technically challenging.
The so-called "threshold" theorem says that, once the error rate per qubit per gate is below a certain value, indefinitely long quantum computation becomes feasible, even if all of the qubits involved are subject to relaxation processes, and all the manipulations with qubits are not exact. The purpose of this article, intended for physicists, is to outline the ideas of quantum error correction and to take a look at the proposed technical instruction for fault-tolerant quantum computation. It seems that the mathematics behind the threshold theorem is somewhat detached from the physical reality, and that some ideal elements are always present in the construction. This raises serious doubts about the possibility of large scale quantum computations, even as a matter of principle.
It was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists. Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to date—quantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code. Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible.