Quantum Entanglement Halves the Oblivious Update Bandwidth

Abstract: We consider $(n,k)$ MDS-coded distributed storage over $\mathbb{F}_q$ with per-node storage $α$ symbols. For the oblivious update problem, where a single message symbol changes and neither helpers nor the stale node know which, the classical lower bound is $αk \log_2 q$ bits. We prove that when the $k$ contacted helpers share prior quantum entanglement, the update bandwidth is $\lceil α/2 \rceil \cdot k \log_2 q$ bits-equivalent, a factor approaching 2 reduction. For $α= 2$, a $[[k, k-2]]_q$ CSS code achieves bandwidth $k \log_2 q$ with one qudit per helper. For general $α$, a $[[\lceil α/2 \rceil k, \lceil α/2 \rceil k - α]]_q$ CSS code achieves the bound with $\lceil α/2 \rceil$ qudits per helper. The matching converse uses the superdense coding bound: the stale node holds all transmitted qudits and hence the entangled partners, so each helper's channel supports at most $D^2$ distinguishable signals for dimension $D$. The result holds for all $(n,k)$ pairs with sufficiently large prime $q$.

• The paper demonstrates that quantum entanglement reduces oblivious update bandwidth by half compared to classical models in MDS-coded storage systems.
• It employs a CSS stabilizer code framework and superdense coding to encode data into qudits, effectively halving the required communication resources.
• Numerical verifications confirm consistent efficiency gains across various parameter configurations, suggesting scalability for quantum-assisted protocols.

Quantum Entanglement Halves the Oblivious Update Bandwidth

Introduction

The paper investigates the utilization of quantum entanglement in enhancing update bandwidth efficiency within distributed storage systems utilizing MDS codes. Specifically, it reveals that sharing quantum entanglement among helper nodes can reduce the oblivious update bandwidth by a factor of two compared to classical limits. This finding exploits prior entanglement to substantially decrease the quantum resources required for data updates in distributed networks.

Problem Formulation and Classical Baseline

Using (n, k) MDS-coded storage systems over finite fields $F_q$, the study addresses the oblivious update problem wherein a single message symbol changes without helpers or stale nodes knowing the symbol that was altered. Classical models established by Nakkiran et al. required a bandwidth of at least $2k \log_2 q$ bits for facilitating oblivious updates. This paper contrasts these limitations by integrating quantum computations to achieve efficient data updates.

Entanglement-Assisted Model and Results

The entanglement-assisted model extends the prior work by Hu et al., which demonstrated bandwidth reduction for repairs through quantum entanglement. It utilizes CSS stabilizer codes where helper nodes encode classical data into qudits. The stale node then extracts information through stabilizer measurements. In this quantumized setting, the total bandwidth becomes $[\alpha/2] k \log_2 q$ bits-equivalent, where each helper transmits $[\alpha/2]$ qudits, effectively halving the channel resources compared to classical methods.

Through the proposed methodology, once the qudits are transmitted, the stale node possesses both the transmitted qudits and the entangled partners, aligning with the superdense coding scenario. Consequently, this configuration translates to each qudit delivering $2 \log_2 q$ classical bits, aligning the requirement with half the channel resources.

Achievability and Numerical Verification

The paper delineates the construction of a CSS code to achieve half the classical update bandwidth with examples provided for the specific parameter set $(n, k) = (3, 2)$. Lemmas are presented to establish the existence of CSS parity-check matrices, imperative for fulfilling the superdense coding framework. The proposed protocol has been numerically confirmed over various parameter configurations and satisfies the fidelity requirements for entanglement sharing.

Discussion and Theoretical Implications

The results exemplify a systematic advancement in utilizing quantum resources for distributed storage updates, yielding a universal factor-of-two improvement. The theoretical implications extend to proving that entanglement effectively halves the communication needed in these systems, offering insights into the capabilities of shared quantum resources across communication networks.

The work foresees speculative paths for future investigations: examining the trade-offs between shared entanglement and bandwidth, determining optimization in helper numbers beyond $k$, and identifying additional benefits with non-MDS code implementations. Such endeavors could further bridge the gap in resource-efficient communication practices leveraging quantum entanglement.

Conclusion

This study provides a comprehensive resolution to the quantum-assisted oblivious update bandwidth problem in MDS-coded storage systems, establishing a bandwidth requirement that universally achieves a factor-of-two reduction in channel resources. The innovative application of shared quantum entanglement underscores a promising avenue that holds potential across varied distributed storage contexts. The continued exploration of quantum-enhanced protocols could further revolutionize bandwidth efficiency, reinforcing the advantageous role of quantum paradigms in contemporary storage technologies.