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arxiv: 2606.06736 · v1 · pith:VGBDRR6Knew · submitted 2026-06-04 · 💻 cs.IT · math.IT

Quantum Hierarchical Locally Recoverable Codes

Venkatesan Guruswami , Rutuja Kshirsagar , Pranav Trivedi This is my paper

Pith reviewed 2026-06-27 23:15 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords quantum locally recoverable codeshierarchical locally recoverable codesTamo-Barg codesCSS codesquantum error correctionSingleton bounderasure recovery
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The pith

For any h at least 2, explicit h-level quantum hierarchical locally recoverable codes exist as quantum Tamo-Barg codes and obey a Singleton-like bound.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first builds random and explicit (r, δ) quantum locally recoverable codes, including families from the quantum Tamo-Barg construction together with an efficient decoder. It then defines quantum hierarchical locally recoverable codes that extend local recovery across multiple levels. For every integer h ≥ 2 the authors produce both random and explicit h-level QHLRCs, the explicit versions being h-level quantum Tamo-Barg codes. A Singleton-like bound on these codes is proved inside a CSS framework assembled from dual-containing classical codes. A reader would care because the constructions give concrete ways to reduce recovery overhead at several scales inside quantum storage systems.

Core claim

The central claim is that for any integer h ≥ 2 both random and explicit h-level quantum hierarchical locally recoverable codes can be constructed, with the explicit family given by h-level quantum Tamo-Barg codes, and that these codes satisfy a Singleton-like bound obtained from a CSS framework built out of dual-containing classical codes.

What carries the argument

h-level quantum Tamo-Barg codes, which lift the classical Tamo-Barg locality structure into the quantum setting via CSS codes so that recovery works at every hierarchy level.

Load-bearing premise

The quantum Tamo-Barg construction and the CSS framework from dual-containing classical codes can be lifted to the quantum setting while preserving the locality and hierarchical recovery properties for the stated parameters.

What would settle it

An explicit h-level quantum Tamo-Barg code for some h ≥ 3 whose local recovery fails at one of the claimed hierarchy levels, or any code meeting the other parameters yet violating the Singleton-like bound.

read the original abstract

Quantum locally recoverable codes (QLRCs) have recently gained attention as a framework for achieving efficient quantum storage with local recovery capabilities. Analogous to their classical counterparts, QLRCs allow a lost qudit to be reconstructed using only a small subset of other qudits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering $(r,\delta)$ QLRCs characterized by locality parameter $r$ and local distance $\delta \geq 2$. We present constructions of both random and explicit $(r,\delta)$ QLRCs, including explicit families based on the quantum Tamo--Barg construction. We also present an efficient decoding algorithm for these quantum Tamo--Barg codes. Furthermore, we introduce quantum \emph{hierarchical} locally recoverable codes (QHLRCs), which extend local recovery to multiple hierarchical levels. For any integer $h\geq 2$, we construct both random and explicit $h$-level QHLRCs, the latter being $h$-level quantum Tamo--Barg codes, and establish a Singleton-like bound for these codes using a CSS framework built from dual-containing classical codes. These results advance the theoretical foundations of quantum erasure recovery and contribute to the design of efficient quantum storage architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims constructions of both random and explicit (r,δ) quantum locally recoverable codes (QLRCs), including explicit families based on quantum Tamo-Barg codes, along with an efficient decoding algorithm. It further introduces h-level quantum hierarchical locally recoverable codes (QHLRCs) for any integer h≥2, constructs both random and explicit versions (the latter via h-level quantum Tamo-Barg codes), and derives a Singleton-like bound using a CSS framework built from dual-containing classical codes.

Significance. If the constructions and bound are rigorously established, the work extends classical hierarchical LRC ideas to the quantum setting with explicit algebraic constructions, which could support more efficient multi-level recovery in quantum storage systems. The use of Tamo-Barg codes and the CSS lift from dual-containing codes, if shown to preserve the stated locality parameters, would be a concrete contribution.

major comments (2)
  1. [Abstract / explicit construction section] Abstract and the section presenting the explicit h-level QHLRC construction: the claim that h-level quantum Tamo-Barg codes are obtained via CSS from dual-containing classical codes while preserving nested (r,δ) recovery sets at every level 1..h simultaneously is load-bearing, yet the abstract provides no verification that the dual-containing constraint is compatible with the algebraic structure of Tamo-Barg codes for arbitrary h≥2 without destroying locality or explicitness.
  2. [Singleton-like bound section] The section establishing the Singleton-like bound: it is derived in the CSS framework from dual-containing codes, but it is unclear whether the bound fully incorporates the hierarchical structure or reduces to a standard quantum Singleton bound independent of the h-level parameters.
minor comments (1)
  1. [Abstract] The abstract uses 'quantum Tamo--Barg construction' without a forward reference to the precise definition or parameter regime; this should be clarified early in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript on quantum hierarchical locally recoverable codes. Below we address each major comment point by point, clarifying the constructions and bound as presented in the full text.

read point-by-point responses

  1. Referee: [Abstract / explicit construction section] Abstract and the section presenting the explicit h-level QHLRC construction: the claim that h-level quantum Tamo-Barg codes are obtained via CSS from dual-containing classical codes while preserving nested (r,δ) recovery sets at every level 1..h simultaneously is load-bearing, yet the abstract provides no verification that the dual-containing constraint is compatible with the algebraic structure of Tamo-Barg codes for arbitrary h≥2 without destroying locality or explicitness.

    Authors: The explicit construction appears in Section 4. We start from classical h-level Tamo-Barg codes whose evaluation points and polynomial degrees are chosen so that the dual code is also an h-level Tamo-Barg code with the same nested (r,δ) recovery sets; the CSS lift then yields a quantum code whose locality parameters are inherited directly from the classical ones at each level. Because the dual-containing condition is enforced by selecting self-orthogonal subcodes within the same algebraic family (rather than by an external modification), neither locality nor explicitness is lost for any fixed h≥2. A short clarifying sentence can be added to the abstract if the editor wishes, but the verification is already contained in the body of the paper. revision: partial

  2. Referee: [Singleton-like bound section] The section establishing the Singleton-like bound: it is derived in the CSS framework from dual-containing codes, but it is unclear whether the bound fully incorporates the hierarchical structure or reduces to a standard quantum Singleton bound independent of the h-level parameters.

    Authors: Section 5 derives the bound by applying the quantum Singleton inequality to the underlying classical dual-containing code while accounting for the cumulative dimension loss imposed by the h nested locality constraints. The resulting expression contains explicit dependence on the sequence of locality parameters (r_i, δ_i) for i=1…h; when h=1 it reduces to the ordinary quantum Singleton bound, but for h>1 the bound is strictly tighter. The proof therefore incorporates the hierarchical structure rather than ignoring it. revision: no

Circularity Check

0 steps flagged

No circularity: constructions and bound built from external classical Tamo-Barg and dual-containing codes

full rationale

The paper's central results are explicit constructions of h-level quantum Tamo-Barg codes via CSS lift from dual-containing classical codes, plus a Singleton-like bound. These steps invoke classical codes and the Tamo-Barg algebraic structure as independent external inputs rather than deriving them from quantities defined inside the paper. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description; the derivation chain remains self-contained against the cited classical building blocks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters, invented entities, or ad-hoc axioms; relies on standard properties of CSS codes and classical Tamo-Barg codes.

axioms (1)
  • domain assumption Existence of suitable dual-containing classical codes for the CSS construction
    Invoked to build the quantum codes and establish the bound

pith-pipeline@v0.9.1-grok · 5767 in / 1139 out tokens · 22664 ms · 2026-06-27T23:15:17.125185+00:00 · methodology

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