Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entanglement-Assisted Coding for Arbitrary Linear Computations Over a Quantum MAC

Lei Hu, Mohamed Nomeir, Alptug Aytekin, Yu Shi, Sennur Ulukus, Saikat Guha

TL;DR

This work develops entanglement-assisted coding for arbitrary linear computations over a quantum multi-access channel (LC-QMAC). By leveraging stabilizer formalism and entanglement-assisted quantum error-correcting code ideas, it shows how to transform any linear computation matrix $\mathbf{V}$ into a self-orthogonal form using auxiliary qudits and precoding, enabling joint retrieval of multiple computations with improved rates. The main contribution is a concrete, general achievable region $\mathcal{D}_{\mathsf{LC}}$ and the corresponding rate $R_{\mathsf{LC}} = \dfrac{2K}{\sum_s m_s + c}$, with $c$ minimized via optimal precoding, and special cases where the scheme attains known capacity bounds (e.g., $\Sigma$-QMAC) or capacity in certain configurations. The approach provides a practical pathway to implement linear computations over quantum MACs with entanglement, advancing computation efficiency in quantum networks.

Abstract

We study a linear computation problem over a quantum multiple access channel (LC-QMAC), where $S$ servers share an entangled state and separately store classical data streams $W_1,\cdots, W_S$ over a finite field $\mathbb{F}_d$. A user aims to compute $K$ linear combinations of these data streams, represented as $Y = \mathbf{V}_1 W_1 + \mathbf{V}_2 W_2 + \cdots + \mathbf{V}_S W_S \in \mathbb{F}_d^{K \times 1}$. To this end, each server encodes its classical information into its local quantum subsystem and transmits it to the user, who retrieves the desired computations via quantum measurements. In this work, we propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error-correcting codes (EAQECC). Specifically, given any linear computation matrix, we construct a self-orthogonal matrix that can be implemented using the stabilizer formalism. Also, we apply precoding matrices to minimize the number of auxiliary qudits required. Our scheme achieves more computations per qudit, i.e., a higher computation rate, compared to the best-known methods in the literature, and attains the capacity in certain cases.

Entanglement-Assisted Coding for Arbitrary Linear Computations Over a Quantum MAC

TL;DR

This work develops entanglement-assisted coding for arbitrary linear computations over a quantum multi-access channel (LC-QMAC). By leveraging stabilizer formalism and entanglement-assisted quantum error-correcting code ideas, it shows how to transform any linear computation matrix into a self-orthogonal form using auxiliary qudits and precoding, enabling joint retrieval of multiple computations with improved rates. The main contribution is a concrete, general achievable region and the corresponding rate , with minimized via optimal precoding, and special cases where the scheme attains known capacity bounds (e.g., -QMAC) or capacity in certain configurations. The approach provides a practical pathway to implement linear computations over quantum MACs with entanglement, advancing computation efficiency in quantum networks.

Abstract

We study a linear computation problem over a quantum multiple access channel (LC-QMAC), where servers share an entangled state and separately store classical data streams over a finite field . A user aims to compute linear combinations of these data streams, represented as . To this end, each server encodes its classical information into its local quantum subsystem and transmits it to the user, who retrieves the desired computations via quantum measurements. In this work, we propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error-correcting codes (EAQECC). Specifically, given any linear computation matrix, we construct a self-orthogonal matrix that can be implemented using the stabilizer formalism. Also, we apply precoding matrices to minimize the number of auxiliary qudits required. Our scheme achieves more computations per qudit, i.e., a higher computation rate, compared to the best-known methods in the literature, and attains the capacity in certain cases.

Paper Structure

This paper contains 11 sections, 4 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Model
  4. Communication Scheme
  5. Achievable Region
  6. An Implementation Based on the Stabilizer Formalism
  7. Main Results
  8. Illustrative Examples
  9. The General Achievable Scheme for LC-QMAC
  10. Proof of Lemma \ref{['lemma:symplectic']}
  11. Proof of Lemma \ref{['lemma:c']}

Key Result

Lemma 1

(Stabilizer-Based Implementation N_sum_boxketkar2006nonbinaryashikhmin2001nonbinarycalderbank1998quantum) Consider a distributed system with $N$ parties and a receiver, where each party $n \in [N]$ has a qudit and classical symbols $(x_n, x_{n+N}) \in \mathbb{F}_d^2$. The parties share a quantum sys where $c_\mathbf{v} \mathbf{I}_{d^N} \in \mathcal{L}(\mathcal{V})$ if and only if $c_\mathbf{v} = 1

Figures (1)

  • Figure 1: An example of LC-QMAC with $S=4$ servers and data streams $A, B, C, D$. The user wants $K=2$ linear combinations of the data streams.

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Remark 4
  • Example 3
  • Lemma 2
  • ...and 2 more