Explicit Capacity-Achieving Quantum LDPC Codes with Near-Linear List Decoding

The story in one sentence

Researchers William Gay, Fernando Granha Jeronimo and Abhi Shukul have posted a paper titled “Explicit Capacity-Achieving Quantum LDPC Codes List Decodable in Near-linear Time,” submitted to arXiv on September 30, 2026, and listed among new quantum-physics submissions on October 1, 2026 [1][3]. The paper’s central claim is that explicit quantum LDPC codes can be constructed to approach the quantum Singleton/list-decoding capacity limit, with constant list sizes and randomized list-decoding algorithms running in near-linear time in the block length [1][2].

Why this matters

Quantum error correction is the practical language in which fragile quantum information is made usable. A code protects a logical quantum state by spreading it across many physical qubits or qudits, so that errors can be detected and corrected from syndrome information rather than by measuring the encoded state directly. The LDPC condition — low-density parity check — matters because it says the checks are sparse: each stabilizer check touches only a bounded number of coordinates, and each coordinate participates in only a bounded number of checks [2].

That sparsity is more than a mathematical convenience. In quantum architectures, local and low-weight checks are central to realistic syndrome extraction, because high-weight measurements are expensive and noisy. The new paper therefore focuses on a particularly demanding combination: not merely good quantum codes, not merely LDPC codes, and not merely list-decodable quantum codes, but explicit quantum LDPC codes that approach the information-theoretic list-decoding limit and come with fast algorithms [1][2].

The authors frame the gap sharply. For classical codes, capacity-achieving list-decodable constructions have been known since the folded Reed–Solomon breakthrough, while quantum coding adds two complications: the quantum Singleton bound cuts the distance scale to roughly half the classical one, and decoding must be phrased modulo stabilizers because errors differing by a stabilizer act the same on the encoded information [2].

What the paper claims

The main theorem states that for every target rate R between 0 and 1, and for every positive slack parameters zeta and xi, there is an explicit infinite family of binary vector-space CSS codes on folded blocks with rate at least R [2]. The same theorem asserts bounded row and column weights for both X- and Z-check matrices, meaning the family is LDPC on the underlying binary coordinates [2].

The distance guarantee is the capacity-facing part. The authors give an explicit lower bound delta_N on the relative distance satisfying delta_N at least (1 minus R_N) divided by 2, up to the slack zeta [2]. That expression is the quantum analogue of approaching the Singleton frontier: because quantum codes pay a factor-of-two penalty relative to the classical Singleton distance scale, the target is not 1 minus R, but roughly (1 minus R) over 2 [2].

The list-decoding guarantee then says that for some radius tau_N at least delta_N minus xi, the code is list decodable with a constant list size ell depending on R, zeta and xi, but not on the growing block length [2]. In operational terms, for every X- or Z-syndrome, only boundedly many stabilizer cosets contain a representative error of weight up to the decoding radius [2].

Finally, the decoder is algorithmic. The theorem gives a randomized syndrome-input decoder running in near-linear time, written as O-tilde with parameters R, zeta and xi fixed, which outputs a list of at most ell cosets containing every low-weight error class with high probability [2]. The paper notes that the returned representatives are verified to have the prescribed syndrome, and that the output may include extra syndrome-consistent cosets beyond the nearby ones [2].

The technical architecture: quantum AEL plus weak regularity

The construction is built from a quantum analogue of Alon–Edmonds–Luby distance amplification, often called qAEL in the paper’s discussion [1][2]. The components are a high-rate outer CSS code, a constant-size inner CSS code, and a regular bipartite expander graph [2]. The outer symbols are encoded locally, placed on expander edges, and then folded into blocks indexed by one side of the graph [2].

The expander is what helps preserve sparse, LDPC structure. In the paper’s formulation, qAEL distance amplification lets the folded code inherit the inner code’s distance, with a controlled loss depending on the graph expansion and the outer code’s distance [2]. This is the combinatorial route that distinguishes the result from earlier algebraic approaches that may achieve capacity-like properties but do not necessarily preserve LDPC structure [1][2].

The decoding side is equally important. A naive syndrome-decoding approach might use global Gaussian elimination, but the authors point out that even sparse parity-check matrices can create dense intermediate objects under elimination, destroying near-linear time [2]. Their workaround is local: use the qAEL check structure to solve constant-size systems around each left vertex of the expander, converting the problem into a collection of local inner-code candidates plus an outer affine syndrome that never needs to be converted into a dense global word [2].

From there, the algorithm recasts candidate generation as an agreement constraint-satisfaction problem on the expander [2]. Each left vertex chooses a label from a constant-size local list, and edge constraints test whether the chosen local word agrees with the received local information along the appropriate port [2]. A genuine low-weight error gives rise to an assignment that satisfies many constraints on a large expander rectangle, so enumerating a small family of representative assignments becomes the decoding bottleneck [2].

The near-linear step comes from weak regularity ideas for expanding CSPs. The paper describes a decomposition of the left side of the expander into a constant number of atoms, depending on the slack and local list size rather than the block length [2]. Assignments that matter can be approximated by assignments that are constant on those atoms, so the decoder can enumerate a blocklength-independent set of candidates and then stitch them through the outer quantum code [2].

The quantum difference: cosets, stabilizers, and stitching

The quantum version is not a direct copy of classical list decoding. In a CSS code, an X-type error is decoded only up to the Z-stabilizer space, and a Z-type error is decoded only up to the X-stabilizer space [2]. The paper formalizes list decoding as producing lists of stabilizer cosets rather than lists of individual error vectors [2].

This coset viewpoint creates an extra rigidity issue. In a classical local list, two different local words are separated by ordinary Hamming distance. Quantumly, two candidates may differ by a stabilizer and therefore represent the same logical effect, so the authors require not only quotient-distance separation between logical cosets but also a lower bound on the weight of nonzero inner stabilizers [2]. That distinction is one of the paper’s key technical adaptations from classical expander-list decoding to the quantum LDPC setting [2].

The final step is “outer stitching.” After candidate enumeration, each surviving local assignment is projected to a logical component of the inner CSS code and interpreted as a noisy outer word [2]. The outer decoder corrects only up to an outer stabilizer, but that is sufficient because, after re-encoding and folding, changing the outer word by such a stabilizer changes the qAEL word only by a global stabilizer [2]. In other words, the decoder does not need a canonical physical error; it needs the correct error class, and the construction is arranged so that this is exactly what stitching recovers [2].

What is new, and what is not yet practical

The novelty is the combination. The paper positions earlier exact list decoding of quantum LDPC-type codes as reaching only the Johnson radius with polynomial-time sum-of-squares methods; for relative distance near one half, the paper notes the Johnson radius is about 0.293, far from the capacity target near one half [2]. The new result claims decoding up to the distance scale itself, with near-linear time, for explicit LDPC families [1][2].

There are caveats. This is a theoretical coding result, not a hardware threshold estimate, a circuit-level fault-tolerance protocol, or a ready-to-run decoder for a superconducting or trapped-ion device. The list size is constant in block length, but the paper states that its bound scales like a double exponential in a polynomial of the inverse slack, and the authors explicitly say they did not optimize it [2]. The decoder is randomized and succeeds with probability tending to one as block length grows, under fixed parameters [2].

Even with those caveats, the conceptual advance is clear. If the result withstands scrutiny, it shows that the three goals often treated as mutually hard — explicitness, quantum LDPC sparsity, and capacity-level list decoding with near-linear algorithms — can be made compatible in one construction [1][2][3]. For quantum error correction theory, that is a significant new waypoint: it narrows the gap between optimal information-theoretic protection and algorithmically usable sparse quantum codes.

Sources from the last 72 hours

1. [1][2609.40313] Explicit Capacity-Achieving Quantum LDPC Codes List Decodable in Near-linear TimeSep 30, 2026, 7:53 PM
2. [2]Explicit Capacity-Achieving Quantum LDPC Codes List Decodable in Near-linear TimeSep 30, 2026, 7:53 PM
3. [3]Quantum Physics: New submissions for Thursday, 1 October 2026Oct 1, 2026, 2:00 AM

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