List-Decodable Folded Quantum Hermitian Codes

Abstract: 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.

• The paper introduces folded quantum Hermitian codes as a new construction using the CSS framework for enhanced quantum error correction.
• It provides explicit constructions based on Hermitian curves and automorphism-based folding, yielding polynomial-time list decoding with concrete parameter guarantees.
• The paper demonstrates near-optimal trade-offs between rate, distance, and alphabet size without needing distance amplification, benefiting practical quantum implementations.

List-Decodable Folded Quantum Hermitian Codes: Technical Analysis

Background and Motivation

List decoding and quantum error correction (QEC) represent key concurrent developments in coding theory and quantum information. Classical folded Reed-Solomon codes—introduced by Guruswami and Rudra—established that folding can optimize the rate vs. error-correction trade-off, enabling polynomial-time list decoding up to the Singleton bound. In quantum settings, the Calderbank-Shor-Steane (CSS) construction permits CSS codes derived from classical codes, but extending folding and list decoding to quantum algebraic-geometric (AG) codes while maintaining manageable alphabet sizes remains a central challenge.

Hermitian codes, as important AG codes, provide superior code lengths for a given alphabet compared to Reed-Solomon codes, making them attractive for quantum error correction. This paper systematically develops folded quantum Hermitian codes (FQHCs), analyzes their list decodability, and provides explicit constructions and decoding guarantees.

Classical and Quantum List Decoding Framework

Let $C \subseteq \mathbb{F}_q^n$ be an $[n,k,d]$ linear code. The folding process partitions $C$ into consecutive blocks of $m$ symbols, yielding a folded code $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$. For AG codes, automorphisms determine folding, particularly in the Hermitian code context where the automorphism group is large and well-structured. The minimum distance scales as $\lceil d/m \rceil$ in the folded domain, and the rate is preserved.

In list decoding, $(\tau, L)$-list-decodability requires that for any received word, there are at most $L$ codewords within relative Hamming distance $\tau$. For folded AG codes, and in particular Hermitian codes, list decodability can approach the Singleton bound with list size polynomial in the code length.

Quantumly, a CSS code $\operatorname{CSS}(C_1, C_2)$, with $[n,k,d]$0, admits a quantum dimension tied to the dimensions of $[n,k,d]$1 and $[n,k,d]$2 and inherits list-decodability from its classical constituents, with the quantum list size at most $[n,k,d]$3 if both bases are $[n,k,d]$4-list-decodable [(2605.10534), Lemma 1.5 from related literature].

Explicit Construction of Folded Quantum Hermitian Codes

Algebraic and Geometric Setup

Given a Hermitian curve $[n,k,d]$5 defined by $[n,k,d]$6, there are $[n,k,d]$7 $[n,k,d]$8-rational points. The construction considers one-point Hermitian codes $[n,k,d]$9 as ingredients for the CSS quantum code. Automorphisms $C$0 of $C$1 satisfying certain algebraic relations are carefully selected so the evaluation set $C$2 is partitioned into orbits of size $C$3 compatible with folding.

Given parameters $C$4, $C$5 and $C$6, with relevant inclusions for the CSS code, the folded codes $C$7 and $C$8 are defined. The resulting FQHC is the CSS code: $C$9 The corresponding quantum code parameters $m$0 derive from the folded classical code parameters.

List Decoding Properties

Folded Hermitian codes are known to be $m$1-list-decodable for block length $m$2 and rate $m$3 [see, e.g., 10.5555/2634074.2634208]. The quantum version inherits this property with radius

$m$4

where $m$5 is the rate of the FQHC, based on symmetric choices for $m$6 and $m$7. The list size remains polynomial for constant $m$8, and crucially, the alphabet size scales only as $m$9—substantially smaller than for quantum Reed-Solomon code constructions augmented by distance amplification.

An explicit deterministic quantum list-decoding algorithm is described: the syndrome is measured quantumly, then classical list-decoding algorithms for $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$0 and $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$1 are invoked, and solutions are paired. The decoding complexity is polynomial in $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$2 for fixed $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$3.

Parameter Summary

The construction yields quantum codes with

• Rate $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$4,
• Relative distance $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$5,
• Alphabet size $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$6,
• $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$7 list-decodability.

This matches the quantum Singleton bound within $C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$8.

Comparison to Folded Quantum Reed-Solomon and Other AG Constructions

Compared to analogous folded quantum Reed-Solomon (FQRS) codes—where the folding parameter is tied to the cyclic evaluation set—the Hermitian setting admits folding based on richer automorphism orbits. The alphabet size required for FQHC is asymptotically smaller than that of FQRS with distance amplification, as

$C^{(m)} \subseteq (\mathbb{F}_q^m)^{n/m}$9

This result is especially relevant for large-scale or hardware-constrained quantum devices, where qudit dimension is a limiting factor.

The construction generalizes to the entanglement-assisted setting, removing stringent inclusion constraints between $\lceil d/m \rceil$0 and their duals, at the expense of requiring shared ebits, with explicit formulas provided for the quantum dimension and minimum distance.

Theoretical and Practical Implications

The presented construction achieves quantum codes that are efficiently list-decodable up to the Singleton bound, with practical alphabet size over fields of moderate size and explicit decoding algorithms. The elimination of distance amplification simplifies code design and potential hardware implementations for moderate $\lceil d/m \rceil$1.

The central implication is that for realistic, resource-constrained quantum error correction, folded quantum Hermitian codes provide near-optimal decoding guarantees at reduced alphabet cost relative to Reed-Solomon-based analogues. This improves the prospects for high-rate, high-distance quantum codes suitable for future fault-tolerant quantum computation and storage.

On the theoretical side, the results reinforce the universal benefit of folding for enabling efficient list decoding in both classical and quantum AG code families. Open directions include extending explicit constructions to other positive-genus AG codes and optimizing trade-offs between alphabet size, decoding radius, and rate in arbitrary genus settings.

Conclusion

The paper introduces and analyzes folded quantum Hermitian codes, providing explicit constructions, proving list-decodability up to the quantum Singleton bound, and demonstrating that distance amplification is unnecessary. The resulting codes achieve superior field size-rate-distance trade-offs compared to distance-amplified Reed-Solomon constructions, with polynomial-time decoding and polynomial list size for constant $\lceil d/m \rceil$2. The work sets a precedent for further systematic exploitation of AG code structure in quantum error correction, with clear implications for both theory and implementation (2605.10534).