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arXiv:1202.3338

New constructions of CSS codes obtained by moving to higher alphabets

Iryna Andriyanova, Denise Maurice, Jean–Pierre Tillich

Published Feb 15, 2012• 9 citations
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Abstract

We generalize a construction of non-binary quantum LDPC codes over $\F_{2^m}$ due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, the new codes obtained in this fashion inherit the distance properties of the underlying toric codes and have therefore a minimum distance which grows as the square root of the length of the code for fixed $m$.

DOI
10.48550/arxiv.1202.3338

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We generalize a construction of non-binary quantum LDPC codes over \F2m\F_{2^m}\F2m​ due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, the new codes obtained in this fashion inherit the distance properties of the underlying toric codes and have therefore a minimum distance which grows as the square root of the length of the code for fixed mmm.
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Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large X-distance but a much smaller Z-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance n1/2logn, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n1/2log1/2n. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.
Existing quantum stabilizer codes constructed from the classic binary codes exclusively belong to the special subclass of Calderbank-Shor-Steane (CSS) codes. This paper fills in the gap by proposing five systematic constructions for non-CSS stabilizer codes, the first four of which are based on classic binary quasi-cyclic low-density parity-check (LDPC) codes and last on classic binary LDPC-convolutional codes. These new constructions exploit structured sparse graphs, make essential use of simple and powerful coding techniques including concatenation, rotation and scrambling, and generate rich classes of codes with a wide range of lengths and rates. We derive the sufficient, and in some cases also the necessary, conditions for each construction to satisfy the general symplectic inner product (SIP) condition, and develop practical decoder algorithms for these codes. The resulting codes are the first classes of non-CSS quantum LDPC codes and non-CSS quantum convolutional codes rooted from classic binary codes (rather than codes in GF(4)), and some of them perform as well as or better than the existing quantum codes.
In classical coding theory, the lift of a code is a practical and natural method for generatingfamilies of LDPC codes from a single chosen input code. This strategy has proven to bevery successful in producing LDPC codes of various blocklengths, reaching good parametersand approaching the capacity of certain channels in the asymptotic regime. It is also nowapplied in the 5G New Radio [RK18].It is widely believed that a quantum computer will integrate a quantum error correctingscheme, and the last three decades have seen the emergence of a variety of code-basedquantum error correction known as CSS codes. It is natural to ask whether we can performlifts of quantum codes, with the hope of having the same benefits as for classical codes.However, the method to lift linear codes cannot be directly applied to quantum CSS codes,as the constituent linear codes usually lose their orthogonality. This dissertation presentsthree new results on lifts of quantum CSS codes.Firstly, we propose a general notion of lift for quantum CSS codes. It is based on theexistence of a canonical complex associated to any CSS code, that we call the Tanner cone-complex, and over which we generate covering spaces. We show that our notion generalizesclassical code lifting and generate important CSS code families. Our main application ofthe lift is to devise new types of quantum CSS codes. So far, most seriously consideredconstructions rely on topology or some kind of product operations. Here, we intend toexplore beyond these paradigmatic ideas and for that we construct new non-topologicaland non-product CSS codes. We then apply the prescription to generate their lifts andshow that this procedure can improve their relative parameters, reaching parameter regionsnot explored prior to this work.Secondly, we introduce new families of quantum Tanner codes, a class of quantum codeswhich first appeared in the work of Leverrier and Zémor [LZ22]. We build these codesby generalizing the lift of classical Tanner codes, based on graph coverings, to the case ofCSS codes, based on coverings of 2D geometrical complexes. We present several explicitfamilies, and identify instances of moderate length quantum codes which are degenerate,have low check weight, and for which the distance is greater than the square root of thecode length.Lastly, we present a topological approach to lifting a quantum CSS code. Our originaldefinition of CSS code lifting was inspired by the work of Freedman and Hastings [FH21],which associates CSS codes with handlebodies. This thesis is a good opportunity to showhow the handlebody realization of a code can also be used to perform code lifting. Weprovide a detailed discussion of why in many cases this is essentially equivalent to ourprevious approach.