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arXiv:0807.3566 · International Symposium on Turbo Codes and Iterative Information Processing

Stabilizer quantum codes: A unified view based on Forney-style factor graphs

Pascal O. Vontobel

Published Sep 1, 2008• 3 citations
arXivHTMLPDF

Abstract

Quantum error-correction codes (QECCs) are a vital ingredient of quantum computation and communication systems. In that context it is highly desirable to design QECCs that can be represented by graphical models which possess a structure that enables efficient and close-to-optimal iterative decoding. In this paper we focus on stabilizer QECCs, a class of QECCs whose construction is rendered non-trivial by the fact that the stabilizer label code, a code that is associated with a stabilizer QECC, has to satisfy a certain self-orthogonality condition. In order to design graphical models of stabilizer label codes that satisfy this condition, we extend a duality result for Forney-style factor graphs (FFGs) to the stabilizer label code framework. This allows us to formulate a simple FFG design rule for constructing stabilizer label codes, a design rule that unifies several earlier stabilizer label code constructions.

DOI
10.1109/turbocoding.2008.4658700

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Quantum error-correction codes (QECCs) are a vital ingredient of quantum computation and communication systems. In that context it is highly desirable to design QECCs that can be represented by graphical models which possess a structure that enables efficient and close-to-optimal iterative decoding. In this paper we focus on stabilizer QECCs, a class of QECCs whose construction is rendered non-trivial by the fact that the stabilizer label code, a code that is associated with a stabilizer QECC, has to satisfy a certain self-orthogonality condition. In order to design graphical models of stabilizer label codes that satisfy this condition, we extend a duality result for Forney-style factor graphs (FFGs) to the stabilizer label code framework. This allows us to formulate a simple FFG design rule for constructing stabilizer label codes, a design rule that unifies several earlier stabilizer label code constructions.
We study normal factor graph (NFG) representations of stabilizer quantum error-correction codes (QECCs), in particular NFG representations of the stabilizer label code and the normalizer label code associated with a stabilizer QECC. The structure of the NFGs we are using is such that the (symplectic) self-orthogonality constraint that stabilizer label codes have to satisfy can be proven rather straightforwardly by applying certain NFG reformulations. We show that a variety of well-known stabilizer QECCs can be expressed in this framework: (tail-biting) convolutional stabilizer QECCs, the toric stabilizer QECCs by Kitaev, and a class of stabilizer QECCs that was recently introduced by Tillich and Zémor. Our approach not only gives new insights into these stabilizer QECCs, but will ultimately help to formulate new classes of stabilizer QECCs and low-complexity (approximate) decoding algorithms.
We introduce a purely graph-theoretical object, namely the coding clique, to construct quantum errorcorrecting codes. Almost all quantum codes constructed so far are stabilizer (additive) codes and the construction of nonadditive codes, which are potentially more efficient, is not as well understood as that of stabilizer codes. Our graphical approach provides a unified and classical way to construct both stabilizer and nonadditive codes. In particular we have explicitly constructed the optimal ((10,24,3)) code and a family of 1-error detecting nonadditive codes with the highest encoding rate so far. In the case of stabilizer codes a thorough search becomes tangible and we have classified all the extremal stabilizer codes up to 8 qubits.
It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.
Quantum Error-Correcting Codes (QECCs) play a crucial role in enhancing the robustness of quantum computing and communication systems against errors. Within the realm of QECCs, stabilizer codes, and specifically graph codes, stand out for their distinct attributes and promising utility in quantum technologies. This study underscores the significance of devising expansive QECCs and adopts the ZX-calculus a graphical language adept at quantum computational reasoning-to depict the encoders of graph codes effectively. Through the integration of ZX-calculus with established encoder frameworks, we present a nuanced approach that leverages this graphical representation to facilitate the construction of large-scale QECCs. Our methodology is rigorously applied to examine the intricacies of concatenated graph codes and the development of holographic codes, thus demonstrating the practicality of our graphical approach in addressing complex quantum error correction challenges. This research contributes to the theoretical understanding of quantum error correction and offers practical tools for its application, providing objective advancements in the field of quantum computing.
We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or graphical models in classical information theory and machine learning. It enables us to formulate the description of the recently-introduced ‘coherent parity check’ quantum error correction codes entirely within the language of classical information theory. This makes our construction accessible without requiring background in quantum error correction or even quantum mechanics in general. More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes.
The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.