Quantum Private Distributed Matrix Multiplication With Degree Tables

TL;DR

The paper tackles private distributed matrix multiplication (PDMM) in a quantum setting with entangled servers, addressing two privacy regimes. It first extends the state-of-the-art GASP codes to quantum PDMM under a feasibility constraint, achieving super-dense coding (doubling the rate) when satisfied. When the feasibility condition does not hold, it introduces quantum schemes and analyzes CAT/DOG-based approaches, demonstrating that rate gains persist in several cases and can reach up to 1.5× in the low-privacy regime. Overall, the work unifies classical PDMM codes with quantum counterparts across high- and low-privacy regimes, delivering explicit rate expressions and highlighting practical entanglement-assisted strategies for quantum PDMM.

Abstract

In this paper, we explore how quantum resources can be used to increase the rate of private distributed matrix multiplication (PDMM). In PDMM, a user who has two high-dimensional matrices, $A$ and $B$, and lacks the computational capabilities to apply matrix multiplication locally, divides the matrices $A$ and $B$ into $K$ and $L$ sub-blocks, respectively. Then, the user sends them to $N$ servers to apply the required multiplication privately from any $T$ servers. The goal is to reduce the number of servers needed to perform the required matrix multiplication. In the quantum setting, we allow the servers to share an entangled state and respond over quantum channels. Upon receiving the qudits, the user applies measurements to obtain the required multiplication. There are two main regimes in the PDMM literature: The high-privacy regime and the low-privacy regime where $T$ is less than $K$ and $L$. First, in the high-privacy regime, the state-of-the-art classical code is called the gap additive secure polynomial (GASP) code. We define a feasibility requirement in the quantum setting for the GASP code such that the highest performance is achieved when it is satisfied. When it is not satisfied, we address two main concerns. The first is to find a relation between the minimum privacy requirement and the dimensions of the two matrices needed for the feasibility condition to be satisfied. Second, we develop a new family of codes that can work in the quantum setting. Second, since GASP does not work efficiently in the low-privacy regimes compared to cyclic-addition degree tables (CAT) and discretely optimized GASP (DOG), we show that the feasibility condition developed for GASP can be adopted for both CAT and DOG codes as well. In addition, we propose another set of codes that can be used in the low privacy regime in the quantum setting when the feasibility requirement is not satisfied.

Figures (13)

• Figure 1: Quantum circuit for two entangled transmitters with the entangled state $\ket{\beta_{00}} = \frac{1}{\sqrt{2}} \left( \ket{00}+ \ket{11} \right)$ and the Bell measurement circuit on the left, and its $2$-sum box abstraction on the right using the transfer matrix $\mathbf{M}_{2 \times 4}$ to represent the input-output relationship, i.e., the entangled state and the Bell measurement, for the circuit on the left.
• Figure 2: The general setting: Quantum circuit for $N$ entangled transmitters with the entangled state $\ket{\psi}$ with the corresponding quantum measurement circuit at the receiver side on the left, and its $N$-sum box abstraction on the right, using the transfer matrix $\mathbf{M}_{N \times 2N}$ to represent the input-output relationship, i.e., the entangled state and the measurement, for the circuit on the left.
• Figure 3: Min $T$ needed for $K=L$ to satisfy \ref{['eq_feas']}.
• Figure 4: Min $T$ needed for $K$ and $L$ to satisfy \ref{['eq_feas']}.
• Figure 5: $K=L=n^2$ and $T=n^m$, for $m=[2:7]$.

Theorems & Definitions (44)

• Definition 1: GRS code
• Theorem 1: Dual GRS
• Definition 2: Shifted GRS
• Theorem 2
• Definition 3: Quantum density matrices
• Definition 4: Quantum operation
• Definition 5: Trace and partial trace
• Theorem 3: Kraus representation
• Theorem 4: Theorem 1 in nsumbox
• Remark 1