Quantum List Decoding

• Quantum List Decoding is a framework where decoders return a bounded list of candidate messages or error cosets from quantumly corrupted codewords.
• It leverages varied approaches such as presence-based decoding, syndrome-based techniques for CSS/stabilizer codes, and POVM measurements in pure-state channels.
• Recent developments integrate efficient algorithmic constructions like QLDPC and folded codes to enhance decoding performance and address inherent quantum limitations.

Searching arXiv for recent and foundational papers on quantum list decoding and closely related decoding frameworks. Quantum list decoding denotes a family of decoding tasks in which the decoder is permitted to return a bounded list of candidates rather than a unique answer. In one formulation, introduced for classical block codes accessed through a quantumly corrupted codeword, the goal is to output a short list containing all messages whose codewords have sufficiently high “presence” in the corrupted quantum object [0610200]. In quantum error correction, especially for CSS and stabilizer codes, the corresponding task is to output a bounded list of error cosets consistent with the measured syndrome and a decoding radius, thereby accommodating degeneracy and logically equivalent error patterns (Bergamaschi et al., 2024). In pure-state classical-quantum channels, zero-error list decoding is formulated through a POVM whose outcomes are lists that must contain the transmitted message with probability $1$ (Dalai et al., 14 Jan 2026).

1. Scope of the term and basic decoding objects

Across the cited literature, “quantum list decoding” refers to several related but non-identical problems. The common structural feature is controlled ambiguity: a decoder replaces unique reconstruction by a short list whose size is bounded independently of the particular received object.

Setting	Input object	Output guarantee
Quantumly corrupted classical codewords	A quantumly corrupted codeword of a classical block code	All messages with high codeword “presence” appear in a short list
CSS/stabilizer quantum codes	Syndrome pair, or a noisy syndrome in fault-tolerant decoding	A bounded list of syndrome-consistent error cosets or candidates
Classical-quantum channels	Pure-state channel outputs over $n$ uses	A POVM outcome list containing the transmitted message with probability $1$

The first setting is representation-theoretic: a classical codeword is accessed through a quantumly corrupted oracle-like object, and the decoder must recover every message whose codeword is sufficiently present in that object [0610200]. The second is syndrome-theoretic: list decoding operates on Pauli errors, cosets modulo stabilizers, and logical equivalence classes, so the object being listed is typically not a single physical error but a syndrome-consistent class of errors (Bergamaschi et al., 2024). The third is channel-theoretic: the list is a direct output of the receiver’s measurement, and the central quantity is zero-error capacity under fixed list size (Dalai et al., 14 Jan 2026).

This diversity of formulations is not merely terminological. It reflects three distinct motivations that recur throughout the literature: handling quantum access to classical codewords, exploiting degeneracy in quantum error correction, and replacing unique recovery by bounded ambiguity in regimes where unique decoding is too restrictive.

2. CSS and stabilizer formulations

For CSS codes, the basic classical data are two $\mathbb F_q$-linear subspaces $C_X,C_Z\subseteq \mathbb F_q^n$ satisfying $C_Z^\perp\subseteq C_X$, equivalently $C_X^\perp\subseteq C_Z$. The associated rate and distance are given by

$$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$

with $\|e\|$ denoting Hamming weight (Srivastava, 2024). In this setting, list decoding receives measured syndromes and must output error cosets of the form $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$ containing every actual error pair of weight at most $n$0 (Srivastava, 2024).

A closely related formulation for CSS codes defines, for a syndrome pair $n$1, the sets $n$2 and $n$3 of nearby cosets, and then the error-list

$n$4

A quantum CSS code is $n$5-list-decodable if this list has size at most $n$6 for every syndrome, and efficiently so if a superset can be produced in polynomial time (Bergamaschi et al., 2024). The formulation makes degeneracy explicit: the fundamental listed objects are cosets modulo $n$7 and $n$8, not individual errors.

The principal combinatorial radius in this framework is the quantum Johnson bound,

$n$9

For CSS codes of relative distance $1$0, the cited covering-lemma argument shows that up to $1$1 the list size is $1$2, and for any radius $1$3 only $1$4 distinct cosets can lie in the ball (Bergamaschi et al., 2024). This bound is the quantum analogue of the classical Johnson regime and serves as the target radius for several recent QLDPC constructions.

In stabilizer language, list decoding can be expressed operationally through a CPTP recovery map. A code $1$5 is $1$6-list-decodable for an error set $1$7 if there exists $1$8 such that for every $1$9 and every logical input $\mathbb F_q$0,

$\mathbb F_q$1

where $\mathbb F_q$2 contains at most $\mathbb F_q$3 Pauli errors of the same syndrome as $\mathbb F_q$4 (Arvind et al., 10 Sep 2025). This formulation isolates the recovery guarantee without committing to a specific algorithmic implementation.

3. Presence-based quantum decoding of classical block codes

A 2006 formulation studied quantum list decoding for classical error-correcting block codes in the presence model. There, the task is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high “presence” in the corrupted codeword [0610200]. The work notes that efficient quantum list decoders had already been used to prove a quantum hardcore property of classical codes, but that all previously known efficiently quantum list-decodable families had code rates too small for broader practical use [0610200].

The main positive result concerns a specific code family of polynomially small code rate over a fixed code alphabet, obtained by concatenating generalized Reed-Solomon codes as outer codes with Hadamard codes as inner codes. For this family, an efficient quantum list-decoding algorithm exists when the relevant codewords have relatively high codeword presence in the given quantumly corrupted codeword [0610200]. As an immediate application, the same work uses this quantum list decodability to solve a certain form of quantum search problems in polynomial time [0610200].

The paper also identifies a sharp change in behavior as the admissible presence threshold decreases. When codeword presence becomes smaller, the quantum list decodability of generalized Reed-Solomon codes with high confidence becomes closely related to efficient solvability of the noisy polynomial interpolation problem and the bounded distance vector problem [0610200]. Under the assumption that $\mathbb F_q$5, it further proves that no efficient quantum list decoder exists for generalized Reed-Solomon codes [0610200]. This establishes an early complexity-theoretic obstruction: efficient quantum list decoding is not a generic property of algebraically strong code families, and threshold-sensitive behavior can be computationally decisive.

4. Explicit constructions and asymptotic theory

Recent work has shifted the emphasis from isolated families to asymptotic constructions with simultaneously strong rate, distance, sparsity, and algorithmic guarantees. A central development is the construction of QLDPC codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time (Bergamaschi et al., 2024). The construction relies on new algorithmic results for codes obtained via the quantum analogue of the Alon-Edmonds-Luby distance amplification scheme, together with convex relaxations from the Sum-of-Squares hierarchy that reduce list decoding of the amplified code to unique decoding of the base code (Bergamaschi et al., 2024).

This line of work is explicitly positioned against two earlier limitations: previous constructions of list-decodable good-distance quantum codes either required access to a classical side channel or relied on algebraic constructions that precluded the LDPC property (Bergamaschi et al., 2024). By choosing as base codes recent asymptotically good QLDPC families with efficient unique decoders, including the constructions of Panteleev-Kalachev and Leverrier-Zémor, the amplified construction yields efficiently list-decodable QLDPC codes (Bergamaschi et al., 2024). A related 2024 thesis presents the broader list-decoding framework, its proof-system viewpoint, and its extension to the quantum version of AEL distance amplification, emphasizing polynomial-time list decodability for quantum LDPC codes (Srivastava, 2024).

The same framework formalizes a quantum AEL distance theorem. If $\mathbb F_q$6 and $\mathbb F_q$7 are outer and inner CSS codes and $\mathbb F_q$8 is an $\mathbb F_q$9-expander, then the amplified code has relative distance satisfying

$C_X,C_Z\subseteq \mathbb F_q^n$0

and remains LDPC when the constituent codes are LDPC (Bergamaschi et al., 2024). The resulting list decoder operates via a bounded-degree SoS SDP, correlation rounding, and repeated invocation of the outer unique decoder; the stated runtime is $C_X,C_Z\subseteq \mathbb F_q^n$1 for decoding up to radius $C_X,C_Z\subseteq \mathbb F_q^n$2 (Bergamaschi et al., 2024).

A separate algebraic route is provided by folded quantum Hermitian codes. These are CSS codes built from folded Hermitian one-point codes over $C_X,C_Z\subseteq \mathbb F_q^n$3. For fold parameter $C_X,C_Z\subseteq \mathbb F_q^n$4, the folded code has length $C_X,C_Z\subseteq \mathbb F_q^n$5, unchanged rate, and list-decoding radius

$C_X,C_Z\subseteq \mathbb F_q^n$6

with list size $C_X,C_Z\subseteq \mathbb F_q^n$7 and decoding time $C_X,C_Z\subseteq \mathbb F_q^n$8 (Matthews et al., 11 May 2026). The construction is stated to meet the quantum Singleton bound asymptotically, and, compared with folded quantum Reed-Solomon codes, to achieve comparable lengths over smaller alphabets (Matthews et al., 11 May 2026). The parameter comparison in the cited exposition gives alphabet size $C_X,C_Z\subseteq \mathbb F_q^n$9 for folded quantum Hermitian codes versus $C_Z^\perp\subseteq C_X$0 for the folded quantum Reed-Solomon construction after distance amplification (Matthews et al., 11 May 2026).

Taken together, these constructions show that efficient quantum list decoding is no longer confined to either extremely low-rate families or dense algebraic objects. The current theory includes QLDPC constructions reaching the Johnson regime and folded algebraic CSS families reaching the quantum Singleton tradeoff.

5. Decoder architectures for quantum polar, surface, and LDPC codes

At the algorithmic level, recent work treats list decoding as a practical decoding primitive rather than only a combinatorial existence statement. Several decoder families instantiate this shift in distinct coding architectures.

Decoder family	Code class	Main property
SCL-E / SCL-C	Quantum polar CSS codes	$C_Z^\perp\subseteq C_X$1; class-oriented aggregation improves logical error rate (Gong et al., 2023)
Extended BP-OSD	Surface codes with erroneous syndrome	Joint qubit-and-syndrome recovery without extra measurements (Liang et al., 2024)
MBBP-LD	QLDPC and bicycle codes	Multiple augmented parity-check bases with BP-like latency (Rabeti et al., 13 May 2026)
BF-OSD	QLDPC codes	Best-first coset search in decreasing likelihood order (Banfi et al., 25 May 2026)
RL-LS BP	QLDPC codes	Learned sequential scheduling with list branching (Moradi et al., 18 Jun 2026)

Gong and Renes adapt successive-cancellation list decoding to quantum polar codes and show that the resulting decoders inherit the low complexity of the classical case, approximating the quantum maximum-likelihood decoder for certain channels (Gong et al., 2023). Their polarization-weight construction avoids the need for the small amounts of entanglement assistance that appeared in previous quantum polar code constructions, and yields CSS codes with parameters $C_Z^\perp\subseteq C_X$2 (Gong et al., 2023). Two variants are emphasized. SCL-E chooses the most likely single error pattern, while SCL-C aggregates posterior mass over error equivalence classes and then chooses the most likely class. Both run in $C_Z^\perp\subseteq C_X$3 time and $C_Z^\perp\subseteq C_X$4 memory, and SCL-C is reported to improve the logical error rate over SCL-E by factors up to $C_Z^\perp\subseteq C_X$5 in the tabulated regime (Gong et al., 2023).

For surface codes with noisy syndrome measurements, list decoding has been integrated with syndrome-soft-information processing. The cited algorithm first performs belief propagation with soft syndrome information and then applies ordered statistics decoding to a virtual code of length $C_Z^\perp\subseteq C_X$6 with parity-check matrix $C_Z^\perp\subseteq C_X$7, thereby generating a list of joint qubit-and-syndrome candidates (Liang et al., 2024). On the $C_Z^\perp\subseteq C_X$8 surface code under depolarizing data noise and syndrome bit-flip noise, the reported extended BP-OSD decoder reduces logical error rates by approximately $C_Z^\perp\subseteq C_X$9–$C_X^\perp\subseteq C_Z$0 across $C_X^\perp\subseteq C_Z$1 at $C_X^\perp\subseteq C_Z$2, while avoiding extra measurements (Liang et al., 2024).

For QLDPC codes, list decoding is increasingly implemented by generating decoding diversity across multiple parity-check realizations. The Multiple-Bases Belief-Propagation List Decoder constructs redundant parity-check matrices from cycle-free subtree decompositions of the Tanner graph and runs BP in parallel across these representations (Rabeti et al., 13 May 2026). The 2026 study reports up to $C_X^\perp\subseteq C_Z$3 reduction in error rate compared to BPGD and up to $C_X^\perp\subseteq C_Z$4 compared to BP-OSD for bivariate bicycle codes in low- and moderate-error regimes, while retaining BP-like latency under parallel implementation (Rabeti et al., 13 May 2026). A closely related 2025 study introduces the same decoder name together with the Frequency-Weighted Scoring rule for final selection and reports up to $C_X^\perp\subseteq C_Z$5 lower logical error rate than BP-OSD on the $C_X^\perp\subseteq C_Z$6 bivariate bicycle code, while also proposing univariate bicycle codes that reduce the search space from $C_X^\perp\subseteq C_Z$7 to $C_X^\perp\subseteq C_Z$8 for the specified construction method (Rabeti et al., 4 Nov 2025).

Ordered-statistics decoding itself is now treated explicitly as a quantum list-decoding method because it enumerates a subset of the affine coset $C_X^\perp\subseteq C_Z$9 and returns the minimum-cost candidate. Best-First OSD replaces preselected brute-force enumeration by a best-first traversal of that coset in order of decreasing likelihood (Banfi et al., 25 May 2026). The method always invokes OSD after a fixed small number of BP iterations, rather than only after BP failure, and Monte Carlo simulations on bivariate bicycle codes under full circuit-level noise show that it matches the BP+OSD baseline while using roughly $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$0 of the original query budget (Banfi et al., 25 May 2026).

A further variant incorporates reinforcement learning. The RL-LS BP decoder extends reinforcement-learning-based sequential variable-node scheduling by maintaining a list of trajectories and branching to a “second-symbol” alternative when local posterior scores are nearly tied (Moradi et al., 18 Jun 2026). Paths are ranked by a cumulative path metric equal to the sum of local log-likelihood penalties incurred by second-best branch choices. On the benchmark codes reported in the paper, RL-LS improves the block error rate of the underlying learned decoder and often reduces average iterations substantially; for example, on the BB$$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$1 code, RL-LS with $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$2 and $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$3 matches the BLER of RL-S with $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$4 while using about $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$5 iterations on average rather than $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$6 (Moradi et al., 18 Jun 2026).

These algorithmic developments indicate that, in practical QEC, list decoding is tightly connected to degeneracy-aware post-processing, syndrome-consistent candidate enumeration, and the reuse of soft information produced by BP-like front ends.

6. Zero-error channels, adversarial regimes, and boundary phenomena

In pure-state classical-quantum channels, zero-error list decoding is defined directly at the level of the measurement. For a channel $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$7, an $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$8-list decoder is a POVM $$R=\frac{\dim(C_X)-\dim(C_Z^\perp)}{n}, \qquad \delta=\min\!\Bigl\{\frac{\|e\|}{n}\,\Bigm|\,e\in(C_X\setminus C_Z^\perp)\cup(C_Z\setminus C_X^\perp)\Bigr\},$$9 such that, when message $\|e\|$0 is sent, the outcome list contains $\|e\|$1 with probability $\|e\|$2 (Dalai et al., 14 Jan 2026). Dalai, Girardi, and Lami derive an achievability bound for list size $\|e\|$3 and a converse bound for every fixed list size. Writing

$\|e\|$4

they prove

$\|e\|$5

and also

$\|e\|$6

for general fixed $\|e\|$7 (Dalai et al., 14 Jan 2026). When the absolute-overlap matrix $\|e\|$8 is positive semidefinite, the bounds coincide and

$\|e\|$9

(Dalai et al., 14 Jan 2026). The trine channel exhibits a specifically quantum boundary phenomenon: the paper gives $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$0 for every $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$1, while the sphere-packing quantity is $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$2, so fixed but arbitrarily large list size does not recover the divergence rate that would be approached in the classical case (Dalai et al., 14 Jan 2026). This directly contradicts the classical intuition that increasing a fixed list size should eventually saturate the sphere-packing threshold.

Adversarial quantum error correction introduces a different boundary phenomenon: list decoding can surpass the half-distance limitation of unique decoding, but only if one accepts ambiguity and then resolves it by an additional mechanism. In this setting, generalized Knill-Laflamme conditions characterize when a stabilizer code is $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$3-list-decodable for a chosen error set $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$4. If two errors have different syndromes then $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$5; if they share a syndrome, the projected products must decompose through a syndrome-dependent list of size at most $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$6 and associated unitary mixing matrices (Arvind et al., 10 Sep 2025). The same work states that, for rate $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$7, list-decodable stabilizer codes can achieve $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$8 with $\{e+C_Z^\perp\}\times\{f+C_X^\perp\}$9, whereas standard unique decoding is described by $n$00 (Arvind et al., 10 Sep 2025). For $n$01, the reported comparison is $n$02 versus $n$03 (Arvind et al., 10 Sep 2025).

The cited adversarial protocol resolves the residual ambiguity cryptographically. Starting from an $n$04-list-decodable code $n$05, it applies a family of pseudorandom unitaries before stabilizer encoding and then tests list elements by inverse unitaries and ancilla checks (Arvind et al., 10 Sep 2025). Under the stated assumptions, Protocol 1 uniquely recovers the plaintext with fidelity at least $n$06 and achieves trace-distance security $n$07 against any quantum polynomial-time adversary, even with polynomially many decoding attempts and key reuse (Arvind et al., 10 Sep 2025). This places quantum list decoding at the interface of coding theory and complexity-based quantum cryptography.

The combined picture is technically nuanced. In some settings, such as QLDPC constructions up to the Johnson bound, list decoding extends algorithmic reach while preserving sparsity and polynomial-time decodability (Bergamaschi et al., 2024). In others, such as generalized Reed-Solomon presence decoding or zero-error pure-state channels, the literature identifies explicit complexity barriers or intrinsic quantum gaps that prevent the naive transfer of classical expectations 0610200.