Quantum error correction codes

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 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 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. 

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. 

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) ... 

We conclude this section by noting that much more efficient quantum error correcting codes can be constructed with a capacity of more than one qubit and an error correction capability of more than single errors (Steane 1996, Calderbank and Shor 1995). These codes rely heavily on the theory of classical error correcting codes (Sloane and MacWilliams 1977). However, the discussion of these codes would exceed the scope of this chapter. 

Calderbank, A. R., Shor, P. W. (1995) Good Quantum Error-correcting Codes Exist, preprint available on the LANL preprint archive http //xxx. lanl. gov, manuscript quant-ph/9512032. 

Keywords Decoherence, quantum information processing, errors correction codes, quan-... 

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. 

Noise is obviously not a characteristic of quantum information, it also concerns classical devices. Indeed, if, on the one hand, components in classical computers are extremely reliable, and can almost be regarded as noiseless, systems like modems and CD players, on the contrary, do suffer from noise. To remedy this parasitic process, error-correcting codes have been well developed and are currently widely used in such classical devices. 

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. 

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). ... 

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

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. 

It is an open question, to what extend the coupling to a heat bath will destroy the conclusions of this and other work about dissipation-free quantum computing. The problem is that dissipation destroys the coherence of quantum states. This coherence is essential for quantum computation. Since it is not easily possible to avoid dissipation entirely, a careful investigation is necessary. One of the few results on the influence of dissipation should be mentioned For the (nonlocal) quantum computer that is able to factor numbers in polynomial time [9,10], Unruh has shown [46,47] that totally coherent computation has to be performed within the thermal time scale (i.e. h/kT), before dissipation becomes a problem. Beside the decrease of temperature, the use of error correcting codes has been proposed to reduce the influence of dissipation [19,27]. A possible way of avoiding decoherence... 

In the theory of classical error correcting codes the strategy is to introduce redundancy (Sloane and MacWilliams 1977). The most trivial example was already given in Eq, (6.23). The triple repetition code has the capacity to store one classical bit and can correct for a single error. This simple strategy does not work in the general quantum case. 

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... 

In any of the new methods of computing, errors can of course not be avoided. This requires new, fast error-correcting codes for parallel computers and the qubits on which quantum computers feed (Calderbank, Rains, Shor and Sloane, IEEE Trans. Inform. Theory, to appear). 

Other simple codes exist such as the phase flip code, which protects information against phase flip (see below for the definition of the phase flip) and can be simply derived from the bit flip code. Merging these two codes, Peter Shor proposed a code which protects one qubit of information against the action of arbitrary single qubit errors (bit and phase flips) this code involves nine physical qubits and shows the same schematic structure as the previous example. Its publication renewed the interest of physicists for the domain and gave hope that quantum errors are correctable. 

Actually, it is possible to extend the correction conditions (7), and show that, for a subspace C to be a correcting code for the set of errors j E/.., it is necessary and sufficient that there exists a complex Hermitian matrix [aki] such that, for any pair E/c, E/j of quantum errors,... 

As we can see at the end of this short review of the main practical error-correction methods, it seems that all the ways of explicitly building quantum codes, or more generally that all the existing protection schemes require special features from the errors they combat. In the following, we address the problem of unitary errors and show that information can be protected from their action through a generalization of the quantum Zeno effect, without making any symmetry assumption about them. 

Fig. 6.14 Error correction of single bit flips. A qubit is encoded redundantly in three. An error is detected by computing the error syndrome in a ancillary quantum system. The syndrome is measured and an appropriate quantum gate is applied to the code word to reconstruct the original state. A similar procedure is applied for correcting general errors.

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