A recent preprint (arXiv:2609.40252) extends the local coordinate-wise linear (LCL) witness framework of Levi, Mosheiff, and Shagrithaya from classical linear codes to CSS quantum codes. The central structural difficulty is that a CSS code is a nested pair of spaces S ⊆ C, so a local witness has two ranks: a physical rank on representatives in C and a logical rank in the quotient C/S. The resulting theory yields a threshold theorem for random CSS codes whose striking corollary is that the per-sector rate threshold equals the classical rate threshold, together with quantum subspace designs and explicit folded constructions that are themselves qLDPC — the first explicit quantum list-decodable and list-recoverable codes with optimal list sizes. This paper is a reconciled analytical companion to that work. We compute the binary entropy H₂(0.1) = 0.468996 and the associated classical threshold R* = 1 − H₂(0.1) = 0.5310, exhibit a concrete random CSS allocation with per-sector rates R_X = R_Z = 0.3 and total rate 0.4 that lies strictly below threshold in both sectors, and quantify the downstream resource consequences: a rate-1/2 good qLDPC code uses 2 physical qubits per logical qubit — a rate statement independent of distance; at matched distance d = 100 this requires block length n ≈ 910 (relative distance δ ≈ 0.11, the largest value consistent with the quantum Hamming bound at R = 1/2), versus 19,999 for a surface code — a factor of ≈10⁴ on locality-permissive architectures, conditional on the distance being achieved at that block length. We also present an alternative threshold convention (R* = 1 − H(p) − 1/L) used in one source draft for fixed list size L, and document the resolution in Appendix A. We identify finite-length behavior, decoder realizability, and check geometry as the principal gaps between the asymptotic theory and deployable systems. ## 1. Introduction The gold standard in coding theory is a code that is explicit, efficient to encode and decode, and Full text and updates: papers.qnfo.org/papers/from-random-quantum-codes-to-explicit-qld-codes-a-reconciled-threshold-analysis-/
A recent preprint (arXiv:2609.40252) extends the local coordinate-wise linear (LCL) witness framework of Levi, Mosheiff, and Shagrithaya from classical linear codes to CSS quantum codes. The central structural difficulty is that a CSS code is a nested pair of spaces S ⊆ C, so a local witness has two ranks: a physical rank on representatives in C and a logical rank in the quotient C/S. The resulting theory yields a threshold theorem for random CSS codes whose striking corollary is that the per-sector rate threshold equals the classical rate threshold, together with quantum subspace designs and explicit folded constructions that are themselves qLDPC — the first explicit quantum list-decodable and list-recoverable codes with optimal list sizes. This paper is a reconciled analytical companion to that work. We compute the binary entropy H₂(0.1) = 0.468996 and the associated classical threshold R* = 1 − H₂(0.1) = 0.5310, exhibit a concrete random CSS allocation with per-sector rates R_X = R_Z = 0.3 and total rate 0.4 that lies strictly below threshold in both sectors, and quantify the downstream resource consequences: a rate-1/2 good qLDPC code uses 2 physical qubits per logical qubit — a rate statement independent of distance; at matched distance d = 100 this requires block length n ≈ 910 (relative distance δ ≈ 0.11, the largest value consistent with the quantum Hamming bound at R = 1/2), versus 19,999 for a surface code — a factor of ≈10⁴ on locality-permissive architectures, conditional on the distance being achieved at that block length. We also present an alternative threshold convention (R* = 1 − H(p) − 1/L) used in one source draft for fixed list size L, and document the resolution in Appendix A. We identify finite-length behavior, decoder realizability, and check geometry as the principal gaps between the asymptotic theory and deployable systems. ## 1. Introduction The gold standard in coding theory is a code that is explicit, efficient to encode and decode, and Full text and updates: papers.qnfo.org/papers/from-random-quantum-codes-to-explicit-qld-codes-a-reconciled-threshold-analysis-/
Constructing explicit codes matching the parameters of random codes has been a central and largely elusive question in coding theory. The quantum setting is even more challenging since it is highly desirable that the quantum code be an LDPC code.
Local coordinate-wise linear (LCL) [Levi, Mosheiff, and Shagrithaya, FOCS 2025] witnesses provide a unifying language for many coding-theoretic properties, from distance to list decoding and list recovery. In particular, it provides a framework to study properties of random linear codes, which achieve optimal parameters for many properties of linear codes.
For CSS quantum codes, however, a local witness has two distinct ranks: its physical rank before quotienting by stabilizers and its logical rank after quotienting. We develop a quantum version of the LCL framework for nested spaces S⊆C, in which local constraints are imposed on physical representatives while independence is measured in the logical quotient C/S. The resulting theory gives a threshold theorem for random CSS codes, and as a consequence shows that the per-sector rate threshold is equal to the classical rate threshold. We also define a quantum analogue of subspace design [Guruswami and Xing, STOC 2013] and show that they can be described in a natural manner within the quantum-LCL framework.
Finally, we give explicit constructions for arbitrary folded quantum-LCL properties, in a manner similar to the LCL derandomization of [Jeronimo and Shagrithaya, STOC 2026]. As a consequence, we obtain the first explicit constructions of quantum list-decodable codes and list-recoverable codes that have optimal list sizes, in addition to explicit quantum subspace design codes. We note that all our explicit constructions are qLDPC codes, an important property for quantum error-correcting codes.
Quantum list recovery (QLR) and quantum list decoding (QLD) seek short lists of logically distinct Pauli corrections consistent with a syndrome and prescribed error constraints. For CSS codes, two issues arise: treating the X- and Z-sectors separately can multiply their output list sizes, while distinct classical candidates can collapse after stabilizer quotienting.
We study combinatorial upper and lower bounds for balanced folded quantum Reed--Solomon (FQRS) codes and balanced random CSS codes, both studied by Bergamaschi, Golowich, and Gunn (STOC 24). Let R∈(0,1) be the quantum rate and R1=(1+R)/2 the common component rate. As the radius ρ=(1−R)/2−γ approaches the quantum Singleton bound, we have, deterministically for FQRS codes and w.h.p. for balanced random CSS codes, LQLR⋆=ℓΘ(R1/γ),LQLD⋆=Θ(γ1−R)(γ↓0). The asymptotically exact QLD radius tradeoff is ρL⋆=L+1L21−R. These conclusions extend to the average-radius setting.
For achievability, building on the work of Brakensiek, Chen, Dhar, and Zhang (STOC 2026), we establish a pairing lemma for joint X/Z candidate lists that preserves the one-sector coefficient and avoids a product loss in list size. For the QLD converse, we prove a quantum generalized Singleton bound based on the classical projection-and-patching argument with stabilizer distinctness. For the QLR lower bounds, we adapt the folded Reed--Solomon construction of Chen and Zhang (STOC 2025) so that the candidates remain stabilizer distinct. For random CSS codes, we show classical bad lists survive the stabilizer quotient with high probability.
Quantum secret sharing (QSS) schemes can be constructed from quantum error-correcting codes, but characterizing their access structures and implementation costs becomes challenging for finite-length, high-rate quantum low-density parity-check (qLDPC) codes. We develop an exact computational framework for QSS induced by CSS qLDPC codes. The framework characterizes authorized, forbidden, and intermediate coalitions using restricted parity-check matrices and binary linear algebra, and introduces the accessible logical dimension κ(A), defined through the symplectic rank of locally supported logical classes, to refine the characterization of intermediate coalitions. We further incorporate physical network constraints through separate Tanner-incidence, Steiner-routing, and scheduling-depth costs, and treat the partitioning of physical qubits into participants as an explicit design variable. Numerical experiments on four CSS qLDPC codes verify the predicted complement duality and monotonicity properties and show substantial variation in access transitions and network costs. For the codes studied, the bivariate bicycle instance exhibits the sharpest transition, while network embedding costs vary by more than an order of magnitude across physical topologies. We also show that participant partitioning can substantially improve block-level security and reconstruction thresholds. These results provide a practical bridge between the algebraic structure of CSS qLDPC codes, the graded accessibility of quantum secrets, and the physical constraints of networked QSS implementations.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group G and two subsets of G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%, approximately matching that of the [[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
We construct quantum LDPC (qLDPC) and high-rate CSS codes from our recently introduced classical fair-density parity-check (FDPC) codes. To this end, we introduce a structured sparsification of FDPC parity-check matrices, which reduces their check weights while preserving the underlying combinatorial structure and distance guarantees. Combined with the hypergraph-product construction, this yields finite-length qLDPC codes with analytically controlled blocklength, dimension, certified distance, and stabilizer weight. For quantum blocklengths N<105, the constructions introduced here span guaranteed rates from approximately 0.35% to 25.8% and certified quantum distances from 12 to 69, with stabilizer weights between 8 and 16. In the large-blocklength regime, allowing the FDPC order and sparsified check weight to scale moderately with blocklength yields a family of high-rate CSS codes, which we refer to as quantum FDPC (qFDPC) codes, with rate RQ and minimum distance D satisfying RQ=1−O(loglogN1),D=Ω(N1/4), and stabilizer weight O(logNloglogN). Finally, the analytically available FDPC weight distribution provides explicit information about the logical operators of the resulting hypergraph-product codes. Over the quantum erasure channel, this structure yields rigorous first-order maximum likelihood (ML) expressions and a higher-order weight-distribution approximation to the ML logical block error probability. This enables estimation of finite-length operating points and the onset of the error-floor regime. To the best of our knowledge, beyond surface-code-type constructions, this is the first finite-rate qLDPC framework to provide both finite-length certified minimum-distance information and an analytical characterization of low-weight logical multiplicities.
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.