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arXiv:2609.40262

Quantum List Recovery and Decoding: Achievability and Limitations

Fernando Granha Jeronimo, Xiaojuan Ma

quant-phcs.ITSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

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 \(R\in(0,1)\) be the quantum rate and \(R_1=(1+R)/2\) 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, \[ L^\star_{\rm QLR} = \ell^{Θ(R_1/γ)}, \qquad L^\star_{\rm QLD} = Θ\!\left(\frac{1-R}γ\right) \qquad (γ\downarrow0). \] The asymptotically exact QLD radius tradeoff is \[ ρ_L^\star = \frac{L}{L+1}\frac{1-R}{2}. \] 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.

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From Random Quantum Codes to Explicit qLD Codes: A Reconciled Threshold Analysis of the Quantum-LCL Framework

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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-/

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From Random Quantum Codes to Explicit qLDPC Codes via Local Properties

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In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.

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CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

Evagoras Stylianou, Vinayak Ramkumar, Holger Boche +1 more

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Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime

Andrey Boris Khesin, Jonathan Z. Lu

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.

quant-pharXiv:2603.05496•Mar 6, 2026•0 citations
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\in(0,1)) be the quantum rate and (R_1=(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, [
L^\star_{\rm QLR}
=
\ell^{Θ(R_1/γ)},
\qquad
L^\star_{\rm QLD}
=
Θ!\left(\frac{1-R}γ\right)
\qquad (γ\downarrow0). ] The asymptotically exact QLD radius tradeoff is [
ρ_L^\star
=
\frac{L}{L+1}\frac{1-R}{2}. ] 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.
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⊆CS \subseteq CS⊆C, in which local constraints are imposed on physical representatives while independence is measured in the logical quotient C/SC/SC/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.
In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.
Folded Reed-Solomon codes, introduced by Guruswami and Rudra in 2007, have been shown to achieve the information-theoretically best possible trade-off between the rate of a code and the error-correction radius. In 2024, Bergamaschi, Golowich and Gunn extended this framework by constructing folded quantum Reed-Solomon codes (CSS codes obtained by folding) demonstrating that these codes tolerate errors up to the quantum Singleton bound. In this paper, we construct folded quantum Hermitian codes using the CSS framework and show that these codes are also list-decodable, tolerating errors up to the quantum Singleton bound. Compared to Reed-Solomon codes, Hermitian codes admit comparable lengths over smaller alphabets, enabling more efficient implementations.
In this work, we study (r,t,x)(r,t,x)(r,t,x) quantum locally recoverable codes (qLRCs) with locality rrr, ttt recovery sets per qudit, and intersection parameter xxx. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an (r,t,x)(r,t,x)(r,t,x)-qLRC if and only if the underlying classical codes are (r,t,x)(r,t,x)(r,t,x) classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing (r,t,x)(r,t,x)(r,t,x)-cLRCs, which yield binary (r,t,x)(r,t,x)(r,t,x)-qLRCs via the CSS construction. For CSS (r,t,x)(r,t,x)(r,t,x)-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.
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 GGG and two subsets of GGG 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]][[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]][[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]][[ 85, 8, 9 ]][[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>nkd > nkd>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%0.2\%0.2%, approximately matching that of the [[144,12,12]][[ 144, 12, 12 ]][[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.