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arXiv:2502.20297

Moderate-length lifted quantum Tanner codes

Virgile Guémard, Gilles Zémor

Published Feb 27, 2025• 0 citations
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Abstract

We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Zémor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index $t$ of an $[[n,k,d]]$-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by $k$, and its distance is upper bounded by $t\cdot d$. Moreover, when the dimension of the lifted code is equal to $k$, its distance is lower bounded by $d$. These parameter bounds also apply to the previous methods of code lifting of Guémard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a $[[96,2,12]]$-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.

DOI
10.48550/arxiv.2502.20297

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We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Zémor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index ttt of an [[n,k,d]][[n,k,d]][[n,k,d]]-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by kkk, and its distance is upper bounded by t⋅dt\cdot dt⋅d. Moreover, when the dimension of the lifted code is equal to kkk, its distance is lower bounded by ddd. These parameter bounds also apply to the previous methods of code lifting of Guémard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a [[96,2,12]][[96,2,12]][[96,2,12]]-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.
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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.
Quantum Tanner codes are a class of quantum low-density parity-check codes that provably display a linear minimum distance and a constant encoding rate in the asymptotic limit. When built from left--right Cayley complexes, they can be described through a lifting procedure and a base code, which we characterize. We also compute the dimension of quantum Tanner codes when the right degree of the complex is 2. Finally, we perform an extensive search over small groups and identify instances of quantum Tanner codes with parameters [[144,12,11]][[144,12,11]][[144,12,11]], [[432,20,≤22]][[432,20,\leq 22]][[432,20,≤22]] and [[576,28,≤24]][[576,28,\leq 24]][[576,28,≤24]] for generators of weight 9.
We introduce sequential and parallel decoders for quantum Tanner codes. When the Tanner code construction is applied to a sufficiently expanding square complex with robust local codes, we obtain a family of asymptotically good quantum low-density parity-check codes. In this case, our decoders provably correct arbitrary errors of weight linear in the code length, respectively in linear or logarithmic time. The same decoders are easily adapted to the expander lifted product codes of Panteleev and Kalachev. Along the way, we exploit recently established bounds on the robustness of random tensor codes to give a tighter bound on the minimum distance of quantum Tanner codes.
Lifted product (LP) codes form an important class of quantum error correcting codes with favorable code parameters. We introduce a systematic construction of LP code families based on group extensions and graph lifts. The construction increases the code size while preserving the local structure of the Tanner graph, and relates code parameters, logical operators, and fault-tolerant logical-operation gadgets within the families through chain and cochain maps. As a first application, we obtain LP codes with better code parameters than previously reported ones. We then demonstrate that code-surgery gadgets can be transferred across the selected finite lifts through chain maps and, in several cases, implemented with lower space overhead. We also develop parallel product surgery for lifted clustered cyclic codes. Finally, we propose lifting as a systematic first step toward defining thermodynamic families for algebraically defined qLDPC codes without an underlying Euclidean lattice. For several base codes and selected lifts, coherent information exhibits finite-size crossings, while our results also indicate that additional conditions are needed to determine a unique family.
In this work, we present a generalization of the recently proposed quantum Tanner codes by Leverrier and Zémor, which contains a construction of asymptotically good quantum low-density parity-check codes. Quantum Tanner codes have so far been constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. We show how to enlarge this to graphs with labeled local views and a family of square complexes, which is the largest possible in a certain sense. We show that the proposed generalization contains a family of asymptotically good quantum codes that are based on non-Cayley Schreier graphs, i.e., a new family of (generalized) quantum Tanner codes is provided. Moreover, we evaluate the performance of the generalized codes and compare with those based on Cayley graphs both in terms of minimum distance and logical error rate on the depolarizing channel, demonstrating that the proposed generalized codes based on Schreier graphs outperform those based on Cayley graphs.