Neural network for excess noise estimation in continuous-variable quantum key distribution under composable finite-size security

TL;DR

The paper tackles finite-size CV-QKD security by integrating neural-network parameter estimation for the critical quantities $t$ and $\xi$, while maintaining composable security through the failure probability $\epsilon_{PE}$. By deriving worst-case confidence intervals for NN predictions via a delta-method–inspired framework and training on synthetic Gaussian-channel data, the authors show that neural estimators can yield tighter bounds on excess noise than standard MLE, leading to measurable increases in the secret-key rate under collective Gaussian attacks. The key contribution is demonstrating operationally equivalent security to traditional PE methods while enabling larger secure distances and higher key rates in realistic resource-limited regimes, with practical implications for real-time CV-QKD systems. The approach leverages offline training and offline Jacobian precomputation to keep online inference lightweight, suggesting a viable path for integrating data-driven estimators into composable quantum cryptographic protocols without compromising security guarantees.

Abstract

Parameter estimation is a critical step in continuous-variable quantum key distribution (CV-QKD), especially in the finite-size regime where worst-case confidence intervals can significantly reduce the achievable secret-key rate. We provide a finite-size security analysis demonstrating that neural networks can be reliably employed for parameter estimation in CV-QKD with quantifiable failure probabilities $ε_{PE}$, endowed with an operational interpretation and composable security guarantees. Using a protocol that is operationally equivalent to standard approaches, our method produces significantly tighter confidence intervals, unlocking higher key rates even under collective Gaussian attacks. The proposed approach yields tighter confidence intervals, leading to a quantifiable increase in the secret-key rate under collective Gaussian attacks. These results open up new perspectives for integrating modern machine learning techniques into quantum cryptographic protocols, particularly in practical resource-constrained scenarios.

Figures (5)

• Figure 1: Schematic representation of the parameter estimation step in a CV-QKD protocol. The two parallel branches illustrate the operational equivalence between the conventional MLE and the proposed neural-network estimator. In both branches, Alice and Bob use the same set of correlated quadrature measurements $\{x_i, y_i\}_m$ to compute a worst-case upper bound $\sigma^2_{\max}$ on the noise variance and a lower bound $t_{\min\text{-MLE}}$ on the channel transmittance. These bounds are then used to construct the same worst-case covariance matrix $\Gamma_{\epsilon_{PE}}$, from which the smooth Holevo information $\chi_{\epsilon_{PE}}(y:E)$ entering the secret key rate is evaluated.
• Figure 2: Neural network architecture for estimating the scaled excess noise $t^2 \xi$.
• Figure 3: Standard deviation between the estimated channel parameters $\sigma^2_{\text{max-NN}}$ and $\sigma^2_{\text{max-MLE}}$ and the real values $\sigma^2$, using $m = 10^4$, $10^5$, $10^6$, $10^7$ and $10^8$ signals. The curves show that the average distance between the neural network estimation and the real values is smaller, which implies more precise estimations if compared to standard MLE method.
• Figure 4: Comparison between the estimated and real values of $\sigma^2$. Dot-dashed line, dashed line and dotted line corresponds, respectively, to $m =10^6$, $10^7$ and $10^8$ signals. In all cases, the estimated values was never inferior to the real values.
• Figure 5: Secret-key rate using the discussed protocol with parameters described in Tab. \ref{['tab:parameters']}. Dot-dashed line, dashed line and dotted line corresponds, respectively, to $N = 2 \cdot 10^6$, $2 \cdot 10^7$ and $2 \cdot 10^8$ signals. The use of neural networks allowed a gain of $6.1$ km, $1.3$ km and $3.0$ km, respectively. In all cases, the estimated values was never superior to the real values.