Entanglement estimation of Werner states with a quantum extreme learning machine

TL;DR

The paper tackles estimating entanglement in Werner states using a Quantum Extreme Learning Machine (QELM) framework. It couples random Werner states $\varrho_W(p)=\frac{1-p}{4}\mathbb{I}+p|\psi_{-}\rangle\langle\psi_{-}|$ to a quantum reservoir evolving under a Transverse Ising Hamiltonian and uses local observables $\langle \sigma_i^z\rangle$ fed into a linear readout with weights $\mathbf{w}^{\mathrm{LR}}=X^{+}\mathbf{y}$ to recover the Werner parameter $p$, with robustness to input noise and domain generalization analyzed. Key findings show that noiseless data yield accurate $p$-estimation, noise degrades performance but phase-transition regions can be exploited, and optimal performance is not in the ergodic phase; extending the output layer with two-point correlators further improves robustness. The work demonstrates a practical, scalable route for entanglement estimation on NISQ devices and highlights the potential for domain-generalization to higher-dimensional Werner-like states derived from GHZ correlations.

Abstract

Quantum Extreme Learning Machines (QELMs) have emerged as a potent tool for various quantum information processing tasks. We present a QELM protocol for estimating the amount of entanglement in Werner states. The protocol requires the generation of a sequence of random Werner states, which are then combined with a reservoir state and evolved using an Ising Hamiltonian. A set of observables based on the Bloch basis is constructed and employed to train the system to recognize unseen features. To assess the protocol's robustness, noise is introduced into the input states, and the system's performance under these noisy conditions is analyzed. Additionally, the influence of the magnetic field parameter within the Ising Hamiltonian on the estimation accuracy is investigated.

Figures (6)

• Figure 1: Comparison of Actual Output and Predicted Output in Linear Regression, (a) without noise, (b) with Low Noise $\varepsilon=0.2$, (c) with Moderate Noise $\varepsilon=0.5$, (d) with High Noise $\varepsilon=0.9$. The figure shows the comparison between the actual output and the predicted output obtained from a linear regression model operating in the presence of varying noise levels. The blue data points represent the actual output values, while the red line represents the ideal predictions. As noise increases $(\varepsilon=0.2, \varepsilon=0.5$, and $\varepsilon=0.9$ ), the model's ability to accurately capture the underlying trend faces challenges.
• Figure 2: MSE as a function of the magnetic field $h$ with varying levels of noise (Average over $20$ realizations).
• Figure 3: MSE as a function of $\Delta t$. The system's parameter is $h=0.1$ (average of 20 realizations).
• Figure 4: MSE as a function of $\Delta t$ with different number of measurements and infinite number of measurements $N_m$ and $\varepsilon = 0.2$. The system's parameter is $h=0.1$ (Average of 20 realizations)
• Figure 5: Ability of the QELM model to generalize to unseen data from a new domain. The blue line represents the raw data obtained from the generalized domain (different from the training data). The red line represents the "dressed data". which is constructed by adding the input-output relationship of a single element from the target domain (the domain the model was trained on) to the raw data points in the generalized domain.