⚛️ quantum physics

Entanglement-assisted quantum locally recoverable codes: bounds and constructions with availability

This paper introduces entanglement-assisted quantum locally recoverable codes with availability that enable multiple local recovery sets for erasure correction, establishes a Singleton-like bound for them, and provides both random and explicit constructions derived from various classical code families.

Original authors: Gretchen L. Matthews, Julia Shapiro

Published 2026-08-11

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CC BY 4.0

Original authors: Gretchen L. Matthews, Julia Shapiro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to store a massive library of digital memories in a futuristic cloud. In the real world, things break. Hard drives crash, servers go offline, and cables get cut. In the world of quantum computing, where information is stored in fragile particles called qudits, things break even faster. The big challenge for scientists is: how do we fix a broken piece of information without having to look at the entire library to find the missing page?

This is where "Locally Recoverable Codes" (LRCs) come in. Think of them as a smart filing system. Instead of needing the whole book to fix a torn page, you only need a small, specific group of nearby pages to reconstruct the damage. It's like having a backup copy of your favorite recipe hidden in your kitchen drawer, your car glovebox, and your friend's house. If the kitchen copy gets spilled on, you don't need to call the library; you just grab the one from the car.

But there's a catch in the quantum world. For a long time, scientists thought you could only have one of these backup groups for any single piece of data. If that one group was also damaged, you were stuck. This was because of strict mathematical rules (called "dual containment") that made it impossible to have multiple, separate backup groups that didn't overlap in a way that caused confusion.

Enter the concept of "entanglement." In quantum physics, entanglement is like a magical, invisible thread that ties two particles together across any distance. If you change one, the other changes instantly. Scientists have discovered that if you share these "entangled threads" between the sender and the receiver, you can break the old rules. It's like having a secret handshake that lets you coordinate repairs even when the usual rules say you can't.

This paper, written by Gretchen L. Matthews and Julia Shapiro, explores a new kind of quantum code that uses these magical threads to create "availability." They ask: Can we design a system where a single broken piece of data can be fixed by any one of several different, separate groups of neighbors? The answer is yes, but only if we use entanglement.

The authors define these new codes, which they call "Entanglement-Assisted Quantum Locally Recoverable Codes with Availability" (EAQLRCs). They prove that by sharing entangled pairs (which they call "ebits"), we can have multiple, disjoint recovery sets. This means if one backup group is damaged, you can instantly switch to a completely different group without any conflict.

The paper doesn't just say "it's possible"; it builds the blueprints. The authors establish a new mathematical limit (a "Singleton-like bound") that tells us the absolute best performance we can hope for with these codes. They then show how to build these codes using two methods:

Random Constructions: They show that if you randomly pick certain mathematical structures (using something called Vandermonde matrices), you will almost certainly get a working code.
Explicit Constructions: They build specific, concrete examples using advanced mathematical shapes called algebraic-geometry curves (like Tamo–Barg codes, Hermitian curves, and Suzuki curves). They even show how to "fold" these codes to make them work with larger alphabets.

The paper explicitly rules out the idea that you can have this kind of "multiple disjoint backup" availability in standard quantum codes without entanglement. They confirm that without those shared entangled threads, the strict rules of quantum mechanics prevent having more than one independent recovery set for the same data.

In short, this work proves that by borrowing a little bit of "magic" from quantum entanglement, we can build quantum storage systems that are much more robust and flexible. We can recover lost data from multiple different angles, making the future of quantum data storage significantly safer and more reliable. The authors provide the math to prove it works and the specific recipes to build it, paving the way for quantum computers that can survive the inevitable glitches of the real world.

Technical Summary: Entanglement-assisted quantum locally recoverable codes: bounds and constructions with availability

Problem Statement
Locally recoverable codes (LRCs) are essential for distributed storage systems, allowing the recovery of erased symbols using a small subset of other symbols (locality). In the classical setting, "availability" refers to the existence of multiple disjoint recovery sets for any given symbol, enhancing reliability. While quantum locally recoverable codes (QLRCs) have been developed to address similar needs in quantum data storage, a significant limitation exists in the standard Calderbank–Shor–Steane (CSS) framework: disjoint recovery sets for the same coordinate are generally impossible unless the underlying classical codes satisfy a strict dual-containment condition ( $C^\perp \subseteq C'$ ). Previous work [18] demonstrated that in the CSS setting, availability in the sense of disjoint recovery sets cannot be achieved. This paper addresses the gap by introducing entanglement-assisted quantum locally recoverable codes (EAQLRCs) with availability, leveraging pre-shared entanglement to relax the dual-containment constraint and enable multiple disjoint recovery sets.

Methodology
The authors extend the framework of entanglement-assisted quantum error-correcting codes (EAQECCs) to the local recovery setting. The core methodology involves:

Definition of EAQLRCs with Availability: The paper defines an $(r, \delta)$ EAQLRC with availability $t$ as a code where any set of up to $\delta-1$ erased qudits can be recovered from any one of $t$ local recovery sets. Crucially, these $t$ sets intersect exactly in the erased coordinates (or a subset thereof), allowing them to be disjoint outside the erased set. This is achieved by constructing the quantum code $Q = \text{EACSS}(C_1, C_2)$ from two classical linear codes $C_1$ and $C_2$ that do not necessarily satisfy $C_1^\perp \subseteq C_2$ .
Theoretical Bounds: The authors derive a Singleton-like bound for the dimension $k$ of these codes. The bound incorporates the block length $n$ , minimum distance $d$ , locality parameters $(r, \delta)$ , availability $t$ , and the entanglement parameter $c$ (the number of shared maximally entangled pairs). The bound is shown to generalize existing Singleton bounds for QLRCs and standard EAQECCs.
Constructions:
- Random Constructions: Using Vandermonde parity-check matrices, the authors provide a random construction of EAQLRCs based on classical linear codes.
- Explicit Constructions: The paper constructs explicit EAQLRCs by applying the entanglement-assisted framework to several families of classical algebraic-geometry (AG) codes and their folded versions. These include:
  - Tamo–Barg codes and their folded variants.
  - Fiber-product codes (derived from morphisms between algebraic curves).
  - One-point Hermitian codes.
  - Suzuki codes.
- Parameter Calculation: For each construction, the authors calculate the resulting quantum dimension and distance. A critical component of these calculations is determining the dimension of the hull (or relative hull) of the underlying classical codes, which dictates the required amount of shared entanglement $c$ .

Key Contributions and Results

Definition and Feasibility: The paper formally defines EAQLRCs with availability $t$ . It proves that shared entanglement permits $t > 1$ (multiple disjoint recovery sets) for the same set of erasures, a property previously unattainable in the CSS framework without dual containment.
Singleton-like Bound: A new upper bound on the dimension $k$ is established:
$k \le n - 2(d-1) - s_1 - s_2 + c$
where $s_1$ and $s_2$ are functions of $n, d, r, \delta,$ and $t$ . This bound quantifies the trade-off between code rate, locality, availability, and entanglement consumption.
Explicit Code Families: The authors provide explicit parameters for EAQLRCs constructed from Tamo–Barg, fiber-product, Hermitian, and Suzuki codes. For instance, they demonstrate how to construct a $(2, 2)$ EAQLRC with availability 2 from Tamo–Barg codes, and $(r, 2)$ EAQLRCs with availability 2 from fiber-product and Suzuki curves.
Flexibility: The work highlights that the entanglement-assisted approach allows the use of any pair of classical LRCs (even non-dual-containing ones) to construct a quantum code, significantly expanding the pool of available classical codes for quantum applications.

Significance
The paper claims that its primary significance lies in overcoming the structural rigidity of CSS-based quantum codes. By utilizing pre-shared entanglement, the authors demonstrate that it is possible to achieve availability (multiple disjoint recovery sets) in quantum local recovery, a feature previously thought impossible in the disjoint-recovery-set sense for quantum codes. This flexibility is particularly valuable for LRCs with strict locality and availability constraints, where enforcing dual containment would severely restrict achievable parameters.

The work establishes a theoretical foundation (via the Singleton-like bound) and provides concrete construction methods (via AG codes) for these new codes. The authors note that while the constructions are explicit, determining whether they are optimal with respect to the derived Singleton bound requires further computation of hull dimensions and minimum weights, which remains an open direction. Additionally, the paper suggests that these codes are relevant for future quantum memory architectures where erasure models are applicable, such as in neutral-atom and superconducting systems where detectable faults can be mapped to erasures.

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