Quantum Locally Recoverable Codes

• qLRCs are quantum error-correcting codes that incorporate a locality structure, enabling recovery of erased qudits using a limited number of neighboring qudits.
• They extend classical locally recoverable codes using stabilizer, CSS, and Hermitian frameworks to achieve sharp Singleton-like bounds and optimal recovery tradeoffs.
• Recent advances include explicit constructions, hierarchical and intersecting recovery set designs, and algebraic methods that demonstrate optimal locality and decoding performance.

Searching arXiv for papers on quantum locally recoverable codes to ground the article in the current literature. Quantum locally recoverable codes (qLRCs) are quantum error-correcting codes endowed with a locality structure: a lost qudit can be reconstructed by a recovery operation acting only on a small subset of other qudits. In the most basic formulation, a qLRC of locality $r$ permits recovery of any single erased coordinate using only $r$ other coordinates; more generally, a quantum $(r,\delta)$-locally recoverable code allows local correction of any $\delta-1$ erasures inside a set of size at most $r+\delta-1$ (Golowich et al., 2023, Galindo et al., 2024). The subject sits at the intersection of stabilizer-code theory, CSS and Hermitian constructions, algebraic coding theory, and locality-aware storage design. The literature has developed from initial definitions and existential or explicit constructions to sharp algebraic criteria, Singleton-like and stronger bounds, optimal families, intersecting-recovery-set variants, hierarchical locality, and explicit distinctions between pure and impure quantum codes (Golowich et al., 2023, Galindo et al., 2024, Li et al., 8 Dec 2025, Galindo et al., 4 Apr 2026).

1. Foundational definitions and locality models

The initial formalization of qLRCs defines a quantum code $C \subseteq (\mathbb{C}^a)^{\otimes n}$ to have local recoverability with locality $r$ if, for each code qudit $i\in[n]$, there exists a subset $I_i$ with $i\in I_i$, $r$0, and a local recovery channel

$r$1

such that for any code state $r$2,

$r$3

Equivalently, any erased qudit can be recovered by only accessing at most $r$4 other code qudits (Golowich et al., 2023).

The $r$5 generalization adopts the classical LRC paradigm of local groups that tolerate multiple erasures. A quantum code $r$6 is a quantum $r$7-LRC if for each $r$8, there exists a set $r$9 containing $(r,\delta)$0 with $(r,\delta)$1 such that, for every $(r,\delta)$2 with $(r,\delta)$3, $(r,\delta)$4 allows the recovery of erasures at $(r,\delta)$5 from nodes in $(r,\delta)$6 (Galindo et al., 2024). The same work introduces $(r,\delta)$7-local recoverability as a more granular formulation: erasures on $(r,\delta)$8 can be corrected using only the information in $(r,\delta)$9 (Galindo et al., 2024).

An earlier algebraic precursor appears in direct-product constructions from monomial-Cartesian codes. There, locality is formulated through $\delta-1$0-availability: if each component code $\delta-1$1 is locally recoverable of locality $\delta-1$2, then the direct product $\delta-1$3 has $\delta-1$4-availability with locality $\delta-1$5, meaning each symbol can be recovered from $\delta-1$6 pairwise disjoint recovery sets of sizes $\delta-1$7 (López et al., 2019).

This basic vocabulary already indicates a central structural tension. Classical LRCs often exploit multiple disjoint recovery sets, but in the quantum setting even the weakest form of local correctability with two disjoint recovery sets is impossible: if a qudit has two disjoint local recovery sets, then that qudit is unentangled and contains no information about the encoded state (Golowich et al., 2023). This no-cloning obstruction is one of the defining differences between classical and quantum locality.

2. Stabilizer, CSS, Hermitian, and dual-containing frameworks

Most explicit qLRC constructions proceed through stabilizer or CSS machinery. If $\delta-1$8 are classical $\delta-1$9 codes with $r+\delta-1$0, the CSS construction yields a quantum code with parameters

$r+\delta-1$1

where $r+\delta-1$2 is determined by the classical distances of $r+\delta-1$3 and $r+\delta-1$4 (Luo et al., 2023, Zhou et al., 24 Jul 2025). A commonly used specialization is the dual-containing case $r+\delta-1$5, which gives

$r+\delta-1$6

or $r+\delta-1$7, depending on the precise construction hypothesis (Luo et al., 2023, Zhou et al., 24 Jul 2025, Li et al., 8 Dec 2025, Li et al., 19 Aug 2025).

For stabilizer codes defined from a symplectic self-orthogonal code $r+\delta-1$8, local recoverability admits a necessary and sufficient criterion in terms of puncturing and shortening. If $r+\delta-1$9 is a stabilizer code, then $C \subseteq (\mathbb{C}^a)^{\otimes n}$0 is quantum $C \subseteq (\mathbb{C}^a)^{\otimes n}$1-locally recoverable if and only if, for every $C \subseteq (\mathbb{C}^a)^{\otimes n}$2, there exists a set $C \subseteq (\mathbb{C}^a)^{\otimes n}$3 with $C \subseteq (\mathbb{C}^a)^{\otimes n}$4 such that for any subset $C \subseteq (\mathbb{C}^a)^{\otimes n}$5 with $C \subseteq (\mathbb{C}^a)^{\otimes n}$6,

$C \subseteq (\mathbb{C}^a)^{\otimes n}$7

This equality is both necessary and sufficient for $C \subseteq (\mathbb{C}^a)^{\otimes n}$8-local recoverability (Galindo et al., 2024).

A major simplification occurs for Euclidean or Hermitian dual-containing classical codes. If $C \subseteq (\mathbb{C}^a)^{\otimes n}$9 or $r$0, and $r$1 or $r$2, then the associated quantum code is a quantum $r$3-LRC if and only if $r$4 is a classical $r$5-LRC (Galindo et al., 2024). This equivalence underlies a large fraction of the later literature, including constructions from affine variety codes, matrix-product codes, BCH and homothetic-BCH codes, cyclic codes, and Hermitian constructions (Galindo et al., 2024, Cao et al., 5 Aug 2025, Galindo et al., 30 Jan 2026, Rajpurohit et al., 8 Jun 2026).

The Hermitian route has become especially prominent. Several works construct qLRCs by first building Hermitian dual-containing classical LRCs over $r$6, then applying the Hermitian quantum construction to obtain

$r$7

or

$r$8

codes with inherited locality (Li et al., 8 Dec 2025, Li et al., 19 Aug 2025). This suggests that the dual-containing condition is not merely a technical requirement for quantization; it is the principal conduit by which classical locality is transferred into the quantum setting.

3. Bounds, optimality, and asymptotic tradeoffs

The earliest qLRC literature established Singleton-like constraints analogous to classical locality bounds. For a qLRC $r$9 with parameters $i\in[n]$0 and locality $i\in[n]$1, one bound is

$i\in[n]$2

(Luo et al., 2023, Sharma et al., 2024, Li et al., 8 Dec 2025). A second Singleton-like inequality derived from classical LRC bounds is

$i\in[n]$3

(Luo et al., 2023). For qLRCs built from dual-containing $i\in[n]$4-LRCs, the literature uses the quantum Singleton-like bound

$i\in[n]$5

as the main optimality criterion (Galindo et al., 2024, Zhou et al., 24 Jul 2025, Cao et al., 5 Aug 2025, Galindo et al., 30 Jan 2026, Rajpurohit et al., 8 Jun 2026, Guruswami et al., 4 Jun 2026).

The 2023 CSS-based study also derived a quantum CM bound from the Cadambe-Mazumdar classical bound. As $i\in[n]$6 with fixed $i\in[n]$7, the $i\in[n]$8-CM bound is tighter than the earlier Singleton-like inequalities. In asymptotic rate-relative-distance notation $i\in[n]$9, $I_i$0, that work states

$I_i$1

$I_i$2

and

$I_i$3

with the third being tightest in some settings (Luo et al., 2023).

For pure qLRCs from the Hermitian construction, several stronger bounds were introduced in 2025. The paper on improved bounds for pure qLRCs gives a pure Singleton-like bound,

$I_i$4

a pure Griesmer-like bound,

$I_i$5

a pure Plotkin-like bound,

$I_i$6

and a pure sphere-packing-like bound

$I_i$7

(Li et al., 8 Dec 2025). That paper states the hierarchy

$I_i$8

and reports that all these new bounds for pure qLRCs are strictly tighter than the GG Singleton-like bound (Li et al., 8 Dec 2025).

Optimality is usually defined as equality in the relevant Singleton-like bound. In the CSS-based framework, a pure qLRC constructed from $I_i$9 is optimal if and only if $i\in I_i$0 and $i\in I_i$1 have the same minimum distance $i\in I_i$2 and dimension $i\in I_i$3, both attain equality in the classical LRC Singleton-like bound, and

$i\in I_i$4

(Luo et al., 2023). In the $i\in I_i$5 setting, optimal quantum codes are defined by equality in

$i\in I_i$6

(Zhou et al., 24 Jul 2025, Cao et al., 5 Aug 2025).

A major qualification emerged in 2026: bounds proved for pure qLRCs do not extend automatically to impure codes. A family of impure CSS codes from $i\in I_i$7-affine variety codes exceeds several pure-code bounds, including Singleton-like, Griesmer-like, and Plotkin-like inequalities (Galindo et al., 4 Apr 2026). This is one of the central controversies in the area: “optimality” depends essentially on whether one is in the pure or impure regime.

4. Explicit constructions and code families

The initial broad constructions came from three sources: quantum Tamo-Barg codes, random qLRCs, and qLRCs from AEL distance amplification (Golowich et al., 2023). The CSS quantum Tamo-Barg construction produces explicit qLRCs based on algebraic evaluation codes. Folded qTB codes improve the distance-rate tradeoff, and the paper states that they have a close-to-optimal rate-distance tradeoff, an efficient decoder, and permit good spatial locality in a physical implementation (Golowich et al., 2023). Random qLRCs nearly meet the Singleton-like bound with alphabet $i\in I_i$8, while the AEL-amplified family offers efficient construction and efficient decoding up to half the code distance (Golowich et al., 2023).

A parallel algebraic line starts from monomial-Cartesian codes. For $i\in I_i$9, $r$00, and

$r$01

the monomial-Cartesian code $r$02 satisfies $r$03, yielding a quantum code

$r$04

pure to distance $r$05 (López et al., 2019). When $r$06, the resulting codes are quantum MDS with parameters $r$07 (López et al., 2019). That same paper proves that direct products of monomial-Cartesian codes yield $r$08-availability if at least $r$09 components are locally recoverable (López et al., 2019).

The “good polynomial” approach generalizes the qTB viewpoint. A construction based on any good polynomial defines a classical dual-containing LRC and then a CSS qLRC, with locality $r$10, length $r$11, dimension $r$12, and a minimum-distance lower bound derived from expander mixing on Schreier graphs: $r$13 where $r$14 is the smallest prime divisor of $r$15 (Sharma et al., 2024). This removes the earlier restriction that $r$16 be prime (Sharma et al., 2024).

The literature after 2024 broadened substantially. The paper on quantum $r$17-LRCs gives optimal stabilizer examples from dual-containing MDS and affine variety codes, including $r$18 as an optimal quantum $r$19-LRC and $r$20 as an optimal quantum $r$21-LRC (Galindo et al., 2024). The decomposition-theoretic paper of 2025 constructs three infinite families of optimal quantum $r$22-LRCs, including

$r$23

with $r$24 for suitable $r$25 (Zhou et al., 24 Jul 2025).

Other 2025 works develop different algebraic sources. Matrix-product codes yield five infinite families of optimal quantum $r$26-LRCs with flexible parameters (Cao et al., 5 Aug 2025). Hermitian constructions from NMDS codes supporting $r$27-designs yield three explicit families of optimal qLRCs and solve an open problem asking whether methods other than CSS, especially Hermitian, can construct qLRCs with better or more flexible parameters (Li et al., 19 Aug 2025). A pure-code Hermitian study derives qLRC families from quantum Hamming, GRM, and Solomon-Stiffler codes; for example,

$r$28

has locality $r$29 (Li et al., 8 Dec 2025).

In 2026, BCH and homothetic-BCH methods produced pure quantum $r$30-LRCs that are optimal for the Singleton-like bound (Galindo et al., 30 Jan 2026), while cyclic-code methods gave three explicit families of $r$31-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound whenever the codes are pure (Rajpurohit et al., 8 Jun 2026). Construction 2 and Construction 3 in that cyclic-code paper have no bound on their lengths with respect to the field size required to obtain these codes (Rajpurohit et al., 8 Jun 2026).

5. Structural refinements: availability, intersecting recovery sets, and hierarchy

The first systematic refinement beyond basic locality is availability. In the monomial-Cartesian framework, direct products produce codes with $r$32-availability and locality $r$33, where each coordinate has $r$34 pairwise disjoint recovery sets (López et al., 2019). However, quantum no-cloning makes disjoint quantum recovery sets problematic in full generality (Golowich et al., 2023).

This led to the study of intersecting recovery sets. An $r$35-qLRC assigns to each qudit $r$36 $r$37 recovery sets $r$38, each of size at most $r$39 and containing $r$40, with pairwise intersections bounded by $r$41, and each equipped with a local recovery channel $r$42 satisfying

$r$43

for all code states $r$44 (Bu et al., 17 Jan 2025). For these codes, a Singleton-like bound is derived using inclusion-exclusion through the quantity $r$45: $r$46 (Bu et al., 17 Jan 2025). The same paper emphasizes that in the quantum case codes with disjoint multiple recovery sets are trivial, so nontrivial improvement happens only for intersecting sets (Bu et al., 17 Jan 2025). Construction is achieved through a variation of the hypergraph product, using exact Tanner graphs and yielding exact $r$47-qLRCs (Bu et al., 17 Jan 2025).

A further refinement is hierarchical locality. Quantum hierarchical locally recoverable codes (QHLRCs) introduce multiple nested local-recovery levels with parameter sequence $r$48 (Guruswami et al., 4 Jun 2026). Random and explicit $r$49-level QHLRCs are constructed, the explicit families being $r$50-level quantum Tamo-Barg codes (Guruswami et al., 4 Jun 2026). For a CSS code $r$51 built from a dual-containing $r$52-level hierarchical LRC, the Singleton-like bound becomes

$r$53

(Guruswami et al., 4 Jun 2026). An efficient decoding algorithm is also given for the one-level quantum Tamo-Barg $r$54-codes, with runtime $r$55 (Guruswami et al., 4 Jun 2026).

These refinements indicate that locality in quantum coding has become a family of related notions rather than a single parameter. This suggests that the appropriate notion of “local repair” depends strongly on the intended failure model: single erasures, multiple erasures inside a group, overlapping repair neighborhoods, or multi-scale storage architectures.

6. Pure versus impure qLRCs, relation to qLDPC, and current directions

Purity has become a decisive dividing line. The paper on improved bounds for pure qLRCs explicitly states that the existing bounds were not sufficiently tight for pure quantum codes and then provides stronger alternatives tailored to Hermitian constructions (Li et al., 8 Dec 2025). The later work on impure codes shows that a family of impure CSS codes from $r$56-affine variety codes exceeds several bounds that apply to pure qLRCs (Galindo et al., 4 Apr 2026). For example, the paper gives a $r$57 code with locality $r$58 and a $r$59 code with locality $r$60, both violating pure-code bounds (Galindo et al., 4 Apr 2026). The authors emphasize that these bounds do not apply to impure codes (Galindo et al., 4 Apr 2026).

The connection to qLDPC codes is recurrent. The foundational qLRC paper states that every qLRC with locality $r$61 is also a qLDPC code with check weight $r$62, and derives a bound showing that qLDPC codes of constant locality $r$63 must satisfy

$r$64

for arbitrarily large alphabets (Golowich et al., 2023). A later pure-qLRC paper explicitly motivates qLRCs by their relevance to quantum LDPC codes (Li et al., 8 Dec 2025). Another 2026 paper studies a bridge with weight-constrained stabilizer codes, noting that stabilizer generators of weight at most $r$65 imply local recovery with $r$66 for single erasure, but not conversely (Galindo et al., 4 Apr 2026). In particular, there exist qLRCs with small locality but every parity-check matrix has a high-weight row (Galindo et al., 4 Apr 2026).

Several current directions are clear from the literature. One is the systematic lifting of optimal classical $r$67-LRCs to optimal quantum ones, now supported by decomposition theorems, matrix-product criteria, cyclic defining-set conditions, and dual-containing BCH or Hermitian frameworks (Zhou et al., 24 Jul 2025, Cao et al., 5 Aug 2025, Galindo et al., 30 Jan 2026, Rajpurohit et al., 8 Jun 2026). Another is the search for longer codes over small fields: cyclic constructions with $r$68 and no length bound with respect to field size are especially notable here (Rajpurohit et al., 8 Jun 2026). A third is the expansion of algebraic sources, including affine general linear groups, $r$69-design-supporting NMDS codes, and $r$70-affine variety codes (Sharma et al., 2024, Li et al., 19 Aug 2025, Galindo et al., 4 Apr 2026).

A common misconception is that quantum locality is a straightforward transcription of classical locality. The literature shows the opposite. Disjoint local correction is obstructed by no-cloning (Golowich et al., 2023); intersecting recovery sets are essential for nontrivial multiple-recovery-set behavior (Bu et al., 17 Jan 2025); pure-code bounds can fail dramatically for impure constructions (Galindo et al., 4 Apr 2026); and hierarchical locality requires its own separate formalism and bounds (Guruswami et al., 4 Jun 2026). qLRCs therefore form a distinct quantum coding paradigm rather than a direct quantum analogue of classical LRCs.

7. Representative parameter regimes and constructions

The following table summarizes representative families and parameter forms that recur in the literature.

Family or framework	Representative parameters	Notable feature
Monomial-Cartesian CSS codes	$r$71	Dual inclusion and $r$72-availability via direct products (López et al., 2019)
Quantum MDS from monomial-Cartesian, $r$73	$r$74	Quantum MDS special case (López et al., 2019)
Dual-containing MDS-induced q$r$75-LRC	$r$76 with $r$77	Optimal quantum $r$78-LRC (Galindo et al., 2024)
Quantum Hamming LRCs	$r$79	Pure qLRC family from Hermitian construction (Li et al., 8 Dec 2025)
Decomposition-based optimal q$r$80-LRCs	$r$81	One of three infinite optimal families (Zhou et al., 24 Jul 2025)
Cyclic q$r$82-LRCs of unbounded length	$r$83 and $r$84	No bound on lengths with respect to field size in Constructions 2 and 3 (Rajpurohit et al., 8 Jun 2026)
Example of impure bound violation	$r$85, locality $r$86	Exceeds pure Singleton-like, Griesmer-like, and Plotkin-like bounds (Galindo et al., 4 Apr 2026)

Taken together, these results depict a field that has moved rapidly from definition and first constructions to a detailed algebraic and combinatorial theory. The central technical themes are dual containment, locality inherited from classical ingredients, increasingly sharp parameter bounds, and the recognition that quantum locality is constrained by mechanisms absent classically. The resulting body of work now includes explicit optimal pure qLRCs in several regimes, unbounded-length cyclic families, intersecting and hierarchical locality models, and impure constructions that lie beyond the reach of pure-code theory (Golowich et al., 2023, Galindo et al., 2024, Zhou et al., 24 Jul 2025, Li et al., 8 Dec 2025, Galindo et al., 4 Apr 2026, Guruswami et al., 4 Jun 2026).