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arXiv:2012.00226 · IEEE Transactions on Communications

Asymmetric Quantum Concatenated and Tensor Product Codes With Large Z-Distances

Jihao Fan, Jun Li, Jianxin Wang, Zhihui Wei, Min-Hsiu Hsieh

Published Mar 9, 2021• 10 citations
arXivHTMLPDF

Abstract

In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs from$\Omega (n_{2}a^{n_{1}})(a>1)$to$O(n_{2})$by considering error degeneracy, where$n_{1}$and$n_{2}$are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly.

DOI
10.1109/tcomm.2021.3064566

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In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs fromΩ(n2an1)(a>1)\Omega (n_{2}a^{n_{1}})(a>1)Ω(n2​an1​)(a>1)toO(n2)O(n_{2})O(n2​)by considering error degeneracy, wheren1n_{1}n1​andn2n_{2}n2​are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly.
In most practical quantum mechanical systems, quantum noise due to decoherence is highly biased towards dephasing. The quantum state suffers from phase flip noise much more seriously than from the bit flip noise. In this work, we construct new families of asymmetric quantum concatenated codes (AQCCs) to deal with such biased quantum noise. Our construction is based on a novel concatenation scheme for constructing AQCCs with large asymmetries, in which classical tensor product codes and concatenated codes are utilized to correct phase flip noise and bit flip noise, respectively. We generalize the original concatenation scheme to a more general case for better correcting degenerate errors. Moreover, we focus on constructing nonbinary AQCCs that are highly degenerate. Compared to previous literatures, AQCCs constructed in this paper show much better parameter performance than existed ones. Furthermore, we design the specific encoding circuit of the AQCCs. It is shown that our codes can be encoded more efficiently than standard quantum codes.
We present a general framework for the construction of quantum tensor product codes (QTPC). In a classical tensor product code (TPC), its parity check matrix is constructed via the tensor product of parity check matrices of the two component codes. We show that by adding some constraints on the component codes, several classes of dual-containing TPCs can be obtained. By selecting different types of component codes, the proposed method enables the construction of a large family of QTPCs and they can provide a wide variety of quantum error control abilities. In particular, if one of the component codes is selected as a burst-error-correction code, then QTPCs have quantum multiple-burst-error-correction abilities, provided these bursts fall in distinct subblocks. Compared with concatenated quantum codes (CQC), the component code selections of QTPCs are much more flexible than those of CQCs since only one of the component codes of QTPCs needs to satisfy the dual-containing restriction. We show that it is possible to construct QTPCs with parameters better than other classes of quantum error-correction codes (QECC), e.g., CQCs and quantum BCH codes. Many QTPCs are obtained with parameters better than previously known quantum codes available in the literature. Several classes of QTPCs that can correct multiple quantum bursts of errors are constructed based on reversible cyclic codes and maximum-distance-separable (MDS) codes.
Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and ℏ\hbarℏ-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and ℏ\hbarℏ-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.
Using the idea of code concatenation in classical coding theory, we make the suggestion of constructing general concatenated quantum codes by choosing certain quantum codes as outer code and inner code. Then by concatenating quantum RS outer codes with a set of special quantum inner codes, one can construct a family of concatenated quantum codes, which is asymptotically good. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum code is constructed using bad classical codes.
Product codes are a class of quantum error correcting codes built from two or more constituent codes. They have recently gained prominence for a breakthrough yielding quantum low-density parity-check (qLDPC) codes with favorable scaling of both code distance and encoding rate. However, despite its powerful algebraic formulation, the physical mechanism for assembling a general product code from its constituents remains unclear. In this letter, we show that the tensor and balanced product codes admit an intuitive coupled-layer construction by taking a stack of one code and condensing a set of excitations in the pattern given by the checks of the other code. We also make a connection to concatenated codes by showing that the tensor product code can be obtained by gauging large-weight logicals in concatenated codes, making them qLDPC. Our framework accommodates both classical or quantum CSS input codes, unifies known physical mechanisms for constructing higher dimensional topological phases via anyon condensation, and naturally extends to non-topological codes.
Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction ofqqq-ary AQCs is extended by removing the$ \BBF _{q}−linearityrequirementaswellasthelimitationonthetypeofinnerproductused.TheproposedconstructionsarecalledCSS−likeconstructionsandutilizepairsofnestedsubfieldlinearcodesunderoneoftheEuclidean,traceEuclidean,Hermitian,andtraceHermitianinnerproducts.Afterestablishingsometheoreticalfoundations,best−performingCSS−likeAQCsareconstructed.Combiningsomeconstructionsofnestedpairsofclassicalcodesandlinearprogramming,manyoptimalandgoodpure-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure−linearityrequirementaswellasthelimitationonthetypeofinnerproductused.TheproposedconstructionsarecalledCSS−likeconstructionsandutilizepairsofnestedsubfieldlinearcodesunderoneoftheEuclidean,traceEuclidean,Hermitian,andtraceHermitianinnerproducts.Afterestablishingsometheoreticalfoundations,best−performingCSS−likeAQCsareconstructed.Combiningsomeconstructionsofnestedpairsofclassicalcodesandlinearprogramming,manyoptimalandgoodpureq−aryCSS−likecodesfor-ary CSS-like codes for−aryCSS−likecodesforq \in \{ 2,3,4,5,7,8,9\}uptoreasonablelengthsarefound.Inmanyinstances,removingtheup to reasonable lengths are found. In many instances, removing theuptoreasonablelengthsarefound.Inmanyinstances,removingthe \BBF _{q}$-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.