Quantum Cross Subspace Alignment Codes via the $N$-sum Box Abstraction

TL;DR

This paper introduces quantum cross subspace alignment codes using the N-sum box abstraction, enabling efficient quantum schemes for private information retrieval and distributed matrix multiplication by translating classical CSA schemes into quantum counterparts.

Contribution

It develops a novel quantum CSA scheme framework leveraging the N-sum box abstraction, bridging classical and quantum channel coding for secure distributed computations.

Findings

Achieves maximal superdense coding gain in quantum CSA schemes.

Translates classical CSA schemes into quantum schemes over a QMAC.

Enables quantum PIR and secure distributed batch matrix multiplication.

Abstract

Cross-subspace alignment (CSA) codes are used in various private information retrieval (PIR) schemes (e.g., with secure storage) and in secure distributed batch matrix multiplication (SDBMM). Using a recently developed $N$-sum box abstraction of a quantum multiple-access channel (QMAC), we translate CSA schemes over classical multiple-access channels into efficient quantum CSA schemes over a QMAC, achieving maximal superdense coding gain. Because of the $N$-sum box abstraction, the underlying problem of coding to exploit quantum entanglements for CSA schemes, becomes conceptually equivalent to that of designing a channel matrix for a MIMO MAC subject to given structural constraints imposed by the $N$-sum box abstraction, such that the resulting MIMO MAC is able to implement the functionality of a CSA scheme (encoding/decoding) \emph{over-the-air}. Applications include Quantum PIR with secure and MDS-coded storage, as well as Quantum SDBMM.