Quantum Error Correction

Abstract The protection of the coherence of open quantum systems against the influence of their environment is a very topical issue. The main features of quantum error-correction are reviewed here. Moreover, an original scheme is proposed which protects a general quantum system from the action of a set of arbitrary uncontrolled unitary evolutions. This method draws its inspiration from ideas of standard error-correction (ancilla adding, coding and decoding) and the Quantum Zeno Effect. A pedagogical demonstration of our method on a simple atomic system, namely a Rubidium isotope, is proposed. 

In the context of quantum information, the effects of interactions with the environment, known as "quantum errors", may render information storage and processing unreliable [Preskill 1998 (c) Nielsen 2000]. Since Shor s demonstration that error-correcting schemes exist in quantum computation [Shor 1995], a general framework of quantum error-correction has been built upon the formalism of quantum operations. The idea of quantum error-cor-... 

The paper is organized as follows. In Sec. 1, we introduce the main features of quantum error-correction, and, particularly, we present the already well-developed theory of quantum error-correcting codes. In Sec. 2, we present a multidimensional generalization of the quantum Zeno effect and its application to the protection of the information contained in compound systems. Moreover, we suggest a universal physical implementation of the coding and decoding steps through the non-holonomic control. Finally, in Sec. 3, we focus on the application of our method to a rubidium isotope. 

Inspired by the existing classical error-correcting techniques, P. Shor built a code in the quantum domain [Shor 1995], which was able to protect one qubit of information against arbitrary single qubit errors. Following this important step, a general theory of quantum error-correcting codes has been set up, in the framework of quantum operations [Knill 1997],... 

In this part, we propose an overview of the field of quantum error-correction. We shall first introduce the basic concepts of error-correcting codes in the classical as well as in the quantum case. Then we shall deal with the general theory of quantum error-correction in particular, we will present the general mathematical correction conditions, as well as the main existing technical methods to build codes explicitly. 

At the end of this brief introduction, quantum codes seem to be much alike their classical counterparts. Indeed, they are based on the same idea of redundancy, resulting from the addition of extra physical qubits. Moreover, quantum error-correcting schemes have the same frame as classical ones after encoding the information on well chosen codewords, one sends the system through a noisy channel then one measures the syndrome, which tells us exactly which error occurred and thus allows us to recover the original information. 

In the following section, we shall set these features in the broader context of the theory of quantum error-correcting codes. 

In this section, we briefly present the general formal framework of quantum error-correction. First, we shall introduce quantum errors in the operator-sum formalism as the operator elements of the quantum operation describing the interaction of the computer with its environment. Then we shall review the main concepts and results of the already well developed theory of quantum error-correcting codes. Finally we will briefly present some of the most important explicit constructive methods to build quantum codes. 

The First idea of quantum error-correction, which we have already employed in the bit flip code, is to "give space" to the system by adding extra qubits, which play the role of ancillary qubits this ancilla adding procedure is highly related to the notion of redundancy in classical error-correction. Then, one encodes the information onto a well-chosen subspace C, the code space, of the extended Hilbert space of the system comprising the initial plus extra qubits. In other words, one applies a well-chosen unitary transformation C, the coding matrix, which "delocalizes" information on all the qubits of the system. That is exactly what we did in the bit flip code, when encoding information onto the subspace spanned by 0l) = 000), 1 l) = 111). ... 

B. We may compare our conditions (15b) with the general conditions (10) of standard quantum error-correction. First, let us introduce Q j, the... 

In this paper, we have reviewed the main methods currently used to protect quantum information from the effects of the environment we have presented the general idea of quantum error-correction as well as the formal theory which has recently emerged. Moreover, we have briefly introduced the most impor-... 

Gottesman D., Stabilizer Codes and Quantum Error Correction, (PhD thesis, California Institue of Technology, Pasadena CA, 1997). 

The first prototype of quantum cryptographic apparatus came into existence around 1990 [147]. In the meantime, quantum cryptography has become a well-known technique of communication in a provably secure way, and together with an intensive research in the held of quantum computers it has given rise to a whole new branch of science quantum information theory [148]. Viewed from this perspective, quantum cryptography today is only a subset of a broad held of quantum communications that also include quantum teleportation, quantum dense coding, quantum error-correcting codes, and quantum data compression. 

Recent theoretical developments in quantum error correction have addressed these points, and it has been shown that quantum computing can be fault-tolerant. 

Devitt SJ, Munro WJ, Naemoto K (2013) Quantum error correction for beginners. Rep Prog Phys 76 076001... 

A major surprise in the early days of quantum computing theory was that quantum error correction was possible at all it has been shown that if a qubit of quantum information is redundantly coded into several qubits, earors in quantum computation can be reduced just as they can be corrected in classical communications channels (Nielsen and Chuang, 2000). One certainty is that the operation of scalable quantum computers will rely heavily on earor correction. There is a threshold for error corrected continuous quantum computation. When errors at the single-qubit-level quantum operations are reduced below this threshold, quantum computation becomes possible. 

Efficient on-chip quantum communication will be essential for the development of large-scale solid-state quantum computing. Communication between devices is also important in conventional computers, but the need for quantum error correction necessitates the continuous transfer of redundant qubits throughout the computer. Rapid flow of quantum information will thus be essential for scalable quantum computing. 

The state after the phase error is also a valid but different superposition state spanned by the two logical basis states 0)l and 1)l. There is no way to finding out if the right hand side of Equation (6.26) is an erroneous state or not. Therefore the triple repetition code (6.25) is not a general quantum error correcting code. It turns out, however, that this code can correct bit flip errors. This is the simplest case of quantum error correction and I will discuss this case first to introduce some basic ideas. 

The necessary steps for quantum error correction in this case are schematically depicted in Fig. 6.14. Initially we prepare the three encoded qubits in the desired... 

In general bit flip errors, phase errors and any combination of those can take place. Quantum error correction in this case is significantly more difficult. As pointed out earlier in this Sect. 6.8.5 the simple triple repetition code (6.25) can not cope with phase errors. On the other hand, 3-bit codes do exist that can correct for phase... 

However, the above new encoding does not have to capability to correct for bit-flip errors in the old basis. What is needed is a quantum error correcting code that can cope with phase errors and amplitude errors simultaneously. It turns out that this is not possible by encoding one qubit redundantly in three. The minimum number of qubits needed to protect a single qubit against one error has been shown to be five (Knill and Laflamme 1996). An example for a quantum error correcting code with five qubits is given by (Laflamme et al. 1996) ... 

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