Impure codes exceeding the pure bounds for quantum local recovery

Abstract: Literature provides several bounds for quantum local recovery, which essentially consider the number of message qudits, the distance, the length, and the locality of the involved codes. We give a family of $J$-affine variety codes that result in impure CSS codes. These quantum codes exceed several of the above mentioned bounds that apply to pure quantum locally recoverable codes. We also discuss a connection between bounds on quantum local recovery and on weight-constrained stabilizer codes.

• The paper introduces a novel construction of impure quantum LRCs that violate fundamental pure code bounds via CSS-derived J-affine variety codes.
• It leverages the disparity between classical locality and stabilizer weight, enabling the codes to surpass Singleton-, Griesmer-, and Plotkin-like bounds.
• Explicit numeric examples demonstrate significant parameter improvements, suggesting practical benefits for enhanced fault-tolerant quantum error correction.

Impure Quantum Locally Recoverable Codes Violating Pure Code Bounds

Introduction and Context

This paper introduces a family of impure quantum locally recoverable codes (QLRCs) derived from $J$-affine variety codes via the Calderbank-Shor-Steane (CSS) construction. These codes are shown to systematically violate several established upper bounds—such as Singleton-like, Griesmer-like, and Plotkin-like bounds—on the code parameters for pure quantum locally recoverable codes. The existence of such constructions demonstrates a categorical separation between the achievable parameter spaces of pure and impure QLRCs, fundamentally questioning the universality of widely used bounds when impurity is allowed. The paper also analyzes the relation between quantum locality and stabilizer generator weight constraints and identifies intrinsic limits in translating locality constraints into weight-constrained stabilizer bounds.

Bounds for Quantum Local Recovery

The study begins by summarizing key quantum bounds relevant for locally recoverable codes:

• Singleton-like bounds for classical and quantum LRCs, most notably the quantum Singleton-like bound for pure CSS codes,
• Tighter bounds for the Hermitian case in quantum codes ([qlrc24], [li2025improvedboundsoptimalconstructions]),
• Bounds for multiple erasure correction, where only pure code bounds exist for $\delta > 2$,
• The connection between generator weight in stabilizer codes and locality: small generator weights imply low locality for erasure recovery, but the converse does not hold.

All known bounds for multiple erasures either inherently assume code purity or restrict to pure code constructions.

Construction of Impure QLRCs from $J$-Affine Variety Codes

The central technical contribution is the explicit family of codes constructed as follows:

1. Affine Variety Code Template: Consider $C(\Delta_{H,V,a,b})$, a code over $\mathbf{F}_q$ based on evaluations of bivariate polynomials at a structured grid in $\mathbf{F}_q^2$, determined by $(H,V,a,b)$ parameters and certain divisibility conditions.
2. CSS Construction: The code $C(\Delta_{H,V,a,b})$ is chosen such that it contains its Euclidean dual, facilitating the use of the CSS construction. The quantum code $Q(C(\Delta_{H,V,a,b}))$ has dimension depending combinatorially on $a, b$ and is typically highly impure.
3. Relative Parameters: The construction exploits the gap between $\delta > 2$0 and $\delta > 2$1 that is characteristic of impure CSS codes.
4. Locality Analysis: The classical code's locality, characterized combinatorially via the grid structure, transfers to the quantum code through the CSS method.

Through analytical (and example-driven) evaluation, the paper establishes conditions on the parameters such that the resulting quantum code violates the bounds that are provably unbreakable for pure codes.

Explicit Violations of Purity-Bounded Results

Three main families of violations are proven:

• Singleton-like Bound Violation: For a broad range of parameters (especially $\delta > 2$2 odd, $\delta > 2$3), the constructed codes have $\delta > 2$4, exceeding the Singleton-like bound by an unbounded margin as $\delta > 2$5 increases. This shows the impossibility of "patching" pure bounds for impurity by the addition of a constant correction.
• Griesmer-like & Plotkin-like Bounds Violation: For $\delta > 2$6, specific choices of $\delta > 2$7, and $\delta > 2$8, the results show $\delta > 2$9 and $J$0 combinations outside feasible regions of pure quantum codes as prescribed by Griesmer and Plotkin-inspired bounds. The violations increase with growing block length.
• Gap Between Locality and Stabilizer Weight: A family of examples is constructed demonstrating that low locality does not imply low stabilizer weight—specifically, codes with locality $J$1 that require some parity checks of weight $J$2.

Representative Numeric Examples

Notable concrete examples are provided:

• $J$3 QLRC with locality $J$4, violating Singleton, Griesmer, and Plotkin bounds for pure codes.
• $J$5 QLRC with locality $J$6 and message distance gap indicative of large violation of pure bounds.

These codes are explicitly constructed and analyzed for impurity and their parameter gaps with pure quantum codes.

Implications and Theoretical Significance

The results:

• Demonstrate the inadequacy of pure code bounds for impure QLRCs, particularly for high-performance scenarios such as multiple erasure correction or high-rate settings.
• Highlight the non-equivalence between locality and stabilizer generator weight, implying caution when interpreting physical device constraints in QECC design via locality notions alone.
• Suggest that new upper bounds for impure QLRCs—especially for correcting multiple erasures—require fundamentally different techniques from those effective for pure codes.

Directions for Future Research

Future work should:

• Develop upper bounds for impure QLRCs in the multiple-erasure setting, likely requiring new invariants that account for the subtleties of undetectable low-weight errors present in impure codes.
• Investigate other code families (e.g., algebraic geometry codes) that, via generalized CSS or Hermitian constructions, may yield further examples of impure codes exceeding known bounds.
• Explore the practical impact of these impure code structures for quantum devices, particularly regarding syndrome measurement cost and fault-tolerance under realistic physical assumptions.

Conclusion

This paper establishes that impure quantum locally recoverable codes constructed from $J$7-affine variety codes via the CSS method can systematically exceed the fundamental bounds for pure quantum codes, including Singleton-like, Griesmer-like, and Plotkin-like bounds. This demonstrates that the space of achievable parameters for impure QLRCs is significantly larger than that for pure QLRCs. The work motivates a directed search for new upper bounds for impure codes and further investigation into their role in quantum information theory and fault-tolerant quantum computation.

Reference: "Impure codes exceeding the pure bounds for quantum local recovery" (2604.03569)