Distributed Quantum Storage Code

• Distributed quantum storage codes are advanced protocols that spread quantum information across multiple nodes using CSS code constructions to ensure resilience against errors and erasures.
• They integrate classical secure storage techniques with quantum CSS codes to achieve stringent capacity bounds and maintain secrecy under quantum constraints.
• Practical implementations optimize performance in heterogeneous networks, enhancing quantum cloud storage, network coding, and secure quantum data recovery.

Distributed quantum storage codes are robust protocols designed to encode quantum information across multiple physically separated nodes, providing resilience against a range of erasure or error patterns. These codes address the practical challenges of heterogeneous node capacities, complex network topologies, and stringent requirements of quantum error correction, all while maintaining the fundamental constraints imposed by quantum mechanics, such as the no-cloning theorem and entropy inequalities.

1. Capacity of Distributed Quantum Storage

Capacity $C(\mathcal{G})$ is formally defined as the maximum quantum information that can be reliably stored and recovered from $N$ distributed nodes (with arbitrary sizes), subject to specified erasure patterns and allowed scaling. This is modeled via a storage graph $\mathcal{G}$, where nodes $Q_n$ have sizes $\lambda_n$, and decoding sets—subsets of nodes sufficient to reconstruct the quantum message—are represented as hyperedges.

Specific results derived for notable graph families include:

Graph Type	Capacity Formula	Uniform Node Example
MDS $(N,K)$	$C(\mathcal{M}_{N,K}) = \sum_{j=1}^{2K-N}\lambda_j$	$C_u(\mathcal{M}_{N,K}) = 2K-N$
Wheel $\mathcal{W}_n$	$$C(\mathcal{W}_n) \leq \min\{\lambda_1,(\lambda_1+\lambda_2)/3,...\}$$	$$C(\mathcal{W}_4) = \min(\lambda_1, \lambda_2, (\lambda_1+\lambda_2)/3)$$
Fano $\mathcal{F}_7$	$C(\mathcal{F}_7)=1$	All sizes
Intersection $1_{(\Delta,m)}$	$C_u(1_{(\Delta,m)}) = \binom{\Delta-2}{m-2}$

The achievability of these bounds is demonstrated using quantum Calderbank-Shor-Steane (CSS) codes (Sun et al., 12 Oct 2025).

2. CSS Code Construction and Mapping to Classical Secure Storage

Achievability employs quantum CSS codes whose construction leverages classical secure storage codes. Each storage process is modeled as a classical linear code: a secret vector $\mathbf{a}$ is encoded into shares $Y_{Q_n}$ via matrices $A$ and $B$ over $\mathbb{F}_q$,

$$\left(Y_{Q_1}, ..., Y_{Q_n}\right) = \mathbf{a} \cdot A + \mathbf{b} \cdot B$$

where $\mathbf{b}$ is securely random noise that protects secrecy. Decoding sets must allow recovery of $\mathbf{a}$, while complementary sets must guarantee information-theoretic secrecy (zero mutual information). These properties are checked by rank and mutual information conditions on restricted submatrices.

Quantum capacity is achieved by lifting these classical encodings to the quantum setting via CSS codes, preserving both recoverability and secrecy in the presence of the quantum no-cloning restriction. Solutions often require interference alignment techniques and algebraic constructions, such as Vandermonde matrices with aligned terms across multiple shares.

3. Nontrivial Alignment in Code Design

Optimal code designs in distributed quantum storage require nontrivial alignment structures. In contrast to classical codes—which often have independently random assignments—quantum codes must align noise (redundant information) across shares intricately. Examples include:

• Wheel graphs: Shares may include common alignment terms (e.g., $a+b_1$ shared among several nodes) carefully arranged to ensure decodability from any decoding set and zero leakage elsewhere.
• Intersection graphs: Shares are constructed via subset sums and alternate signs, engineering dependence among overlapping shares for the right secrecy/recovery properties.

This suggests that generic random linear codes are generally sub-optimal in such distributed quantum settings.

4. Quantum Information Inequalities: Converse Bounds

Converse bounds are proven using quantum information inequalities adapted to the topology of the given storage graph:

• Strong subadditivity (SSA): Submodular partitioning of node entropy leads to wheel-type bounds. For partitions reflecting decoding sets, entropy of the reference $R$ is bounded by

$$H(R) \leq \frac{\Lambda(\mathcal{Q}_1)+\cdots+\Lambda(\mathcal{Q}_k)}{2k-1}$$

when $\mathcal{Q}_i$ are subsets of nodes with paired relations.

• Weak monotonicity: Used to establish monogamy bounds, limits mutual information between disjoint sets to $2H(R)$, which directly connects to security constraints.
• Intersection bounds: For decoding sets $\mathcal{D}(e_i)$ and $\mathcal{D}(e_j)$,

$$C(\mathcal{G}) \leq \Lambda(\mathcal{D}(e_i) \cap \mathcal{D}(e_j))$$

These inequalities are not generic but are precisely tailored to the connectivity and erasure pattern structure imposed by the storage graph.

5. Practical Impact and Optimization for Heterogeneous Systems

The theoretical framework provides immediate relevance for quantum cloud systems, quantum networks, and any future distributed quantum computing infrastructure. Practical implications include:

• Ability to determine the optimal message size to store across heterogeneously sized nodes and arbitrary erasure patterns.
• Direct code constructions for irregular topologies, promoting tailored quantum CSS codes over generic random codes.
• Insights into node selection, scaling, and recovery policies shaped by graph-theoretic and entropic constraints.

Challenges remain in achieving codes over small alphabet fields, understanding capacity additivity under system concatenation, and addressing worst-case graphs with minimal capacity.

6. Connections to Quantum Network Coding and Secret Sharing

The formalism generalizes and unifies classical secret sharing, quantum secret sharing, and network coding models. The need for alignment and quantum information inequalities reflects deep underlying connections to interference alignment in wireless networks and private information retrieval. Given the unique no-cloning and secrecy constraints of quantum information, distributed quantum storage code design is necessarily more restrictive and nuanced than its classical counterparts.

7. Future Directions

Potential research advancements include:

• Exploring capacity super-additivity and potential quantum advantages in concatenated systems.
• Extending constructions for practical finite fields and low-resource implementations.
• Systematic investigation of extremal graph topologies that minimize quantum storage capacity.
• Development of scalable protocols for initialization and decoding aligned with theoretical capacity and code structure.
• Cross-disciplinary transfer of interference alignment and entropic inequality techniques from network information theory to quantum distributed storage.

In sum, distributed quantum storage codes embody advanced alignment-based coding schemes, leveraging CSS code constructions and quantum entropy inequalities to provide rigorous bounds and implementations for robust, heterogeneous quantum memories in networked quantum systems (Sun et al., 12 Oct 2025).