We present explicit constructions of quantum Tanner (QT) codes with good rate and distance, obtained through two complementary approaches: the left-right Cayley complex (LRCC) description and the "lifting" perspective, in which a seed Calderbank-Shor-Steane (CSS) code is lifted by commuting left-right regular actions of a finite group
G. Through an extensive search over non-abelian groups from GAP's SmallGrp library, we investigate the moderate-blocklength regime (
n∈[500,1000]) and identify several new code instances with distance upper bounds exceeding
20. These include
[[480,8,(≤21,≤21)]],
[[504,4,(≤36,≤27)]],
[[672,4,(≤48,≤28)]],
[[720,6,(≤30,≤30)]], and
[[864,8,(≤39,≤31)]], with these bounds obtained using up to
350 million trials of sQetch, a randomized distance estimator. The code instances presented have check weights ranging from
9 to
20. Using the Tesseract decoder, we estimate pseudo-thresholds of
3.6%-
4.6% under phenomenological noise and
0.14%-
0.27% under circuit-level noise, comparable to prior results at shorter code lengths. We also provide QuantumExpanders
.jl, an open-source Julia library for constructing QT codes and explicit constructions of Ramanujan graphs.
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⋅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.
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.
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.
Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c3locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.
Low check weight is a crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we initiate the theory of weight-constrained stabilizer codes from various foundational perspectives including the complexity of computing code weight and the explicit boundary of feasible low-weight codes in both theoretical and practical settings. We first prove that computing the optimal generator weight of a stabilizer code is
NP-hard, motivating efficiently computable bounds. We derive analytical lower bounds on check weight in terms of code rate and distance, identifying the minimum weights needed for single-qubit error detection and correction, as well as the sharp distance and rate limits of weight-three error-detecting codes. Matching constructions show that these bounds are tight in several regimes. To establish refined finite-size constraints, we develop a linear programming framework based on quantum weight enumerators subject to generator-weight constraints, yielding exact optimal weights for all parameter combinations with
n≤9. Finally, we show that the same framework can incorporate architecture-dependent constraints, using the 127-qubit IBM Eagle chip as a concrete example. Our study brings the weight as a crucial parameter into coding theory and provides guidance for code design and utility in practical scenarios.