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CRC Press eBooks

Quantum Mechanics

Frank Gaitan

Published Feb 7, 2008• 0 citations
Publisher

Abstract

In Chapters 2-5 we gave a detailed presentation of the theory of quantum error correction and fault-tolerant quantum computing. The stage is finally set for a proof of the accuracy threshold theorem [2-9]. As pointed out in Chapter 1, this theorem spells out the conditions under which quantum computing can be done reliably using imperfect quantum gates and in the presence of noise. The proof presented in this chapter follows Ref. [6]. Section 6.1 addresses a number of preliminary topics. It explains why a concatenated quantum error correcting code (QECC) should be used to protect the computational data when fault-tolerance is an issue; it states the principal assumptions underlying the threshold calculations presented in Section 6.2; and finally, it closes with a statement of the accuracy threshold theorem. The proof of this theorem is taken up in Section 6.2. For the error model introduced in Section 6.1, the accuracy threshold is calculated for gates in the Clifford group N(Qn), storage registers, and the Toffoli gate. The calculations make explicit use of the recursive structure of a concatenated QECC and the fault-tolerant procedures introduced in Chapter 5. The theorem is proved by showing that all three cases yield non-zero threshold values.

DOI
10.1201/b15868-10

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It has been known that error-correction via concatenated codes can be done with exponentially small failure rate if the error rate for physical qubits is below a certain accuracy threshold (probably ∼ 10 −3 –10 −6). Other, un-concatenated codes with their own attractive features—e.g., an accuracy threshold ∼ 10 −2 —have also been studied. A method to obtain universal computation is presented here which does not rely on any concatenated structure within the code itself, but instead emulates this structure with logical qubits in order to construct an encoded Toffoli gate. This realizes ∼ 10 −2 as a threshold for fault-tolerant quantum computation. 1 QEC codes and universal computation In the “space ” of all possible quantum error-correcting codes, much recent work has focused on a relatively small class, namely concatenated codes [1]–[7]. The basic idea behind these is to improve the results of a given few-qubit code by

•May 7, 1999•2 citations
In Chapters 2-5 we gave a detailed presentation of the theory of quantum error correction and fault-tolerant quantum computing. The stage is finally set for a proof of the accuracy threshold theorem [2-9]. As pointed out in Chapter 1, this theorem spells out the conditions under which quantum computing can be done reliably using imperfect quantum gates and in the presence of noise. The proof presented in this chapter follows Ref. [6]. Section 6.1 addresses a number of preliminary topics. It explains why a concatenated quantum error correcting code (QECC) should be used to protect the computational data when fault-tolerance is an issue; it states the principal assumptions underlying the threshold calculations presented in Section 6.2; and finally, it closes with a statement of the accuracy threshold theorem. The proof of this theorem is taken up in Section 6.2. For the error model introduced in Section 6.1, the accuracy threshold is calculated for gates in the Clifford group N(Qn), storage registers, and the Toffoli gate. The calculations make explicit use of the recursive structure of a concatenated QECC and the fault-tolerant procedures introduced in Chapter 5. The theorem is proved by showing that all three cases yield non-zero threshold values.
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.
In this chapter we begin to examine quantum stabilizer codes in detail. Sec­ tion 3.1 introduces the theoretical framework that underlies this class of quan­ tum error correcting codes, and some examples are given in Section 3.2. Sec­ tion 3.3 shows how these codes can be described in terms of a binary vector space with symplectic inner product, and finally, concatenated quantum er­ ror correcting codes are introduced in Section 3.4. The latter discussion will focus on concatenated codes that are put together using quantum stabilizer codes. We will see in Chapter 6 that concatenated codes play an important role in establishing the accuracy threshold theorem for fault-tolerant quantum computing.
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.
This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, ηηη, is smaller than a constant threshold, ηcη_cηc​. 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.
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.
It has been known that error-correction via concatenated codes can be done with exponentially small failure rate if the error rate for physical qubits is below a certain accuracy threshold (probably ∼ 10 −3 –10 −6). Other, un-concatenated codes with their own attractive features—e.g., an accuracy threshold ∼ 10 −2 —have also been studied. A method to obtain universal computation is presented here which does not rely on any concatenated structure within the code itself, but instead emulates this structure with logical qubits in order to construct an encoded Toffoli gate. This realizes ∼ 10 −2 as a threshold for fault-tolerant quantum computation. 1 QEC codes and universal computation In the “space ” of all possible quantum error-correcting codes, much recent work has focused on a relatively small class, namely concatenated codes [1]–[7]. The basic idea behind these is to improve the results of a given few-qubit code by