Entanglement Witness Derived By Using Kolmogorov-Arnold Networks

TL;DR

The paper addresses efficient entanglement detection in two-qubit systems by learning interpretable entanglement witnesses with Kolmogorov-Arnold Networks. By representing states with nine Pauli-correlation features $t_{ij}$ and training KANs on a large, uniformly sampled dataset, the authors achieve high classification accuracy ($\approx 94\%$ on clean data, $98\%$ on a symmetric subset) and derive witness functions that require fewer measurements. They demonstrate a systematic method to identify and rank informative features via bootstrap aggregation, enabling witnesses based on as few as four observables, thereby reducing the need for full state tomography. The approach balances interpretability and performance, and while demonstrated for two-qubit states, the authors discuss extensions to larger systems as future work.

Abstract

We utilize Kolmogorov-Arnold Networks to design an interpretable model capable of detecting quantum entanglement within a set of nine-parameter two-qubit states. This network serves as an entanglement witness, achieving an accuracy of $94\%$ in distinguishing entangled states. Additionally, by analyzing the output functions of the KAN models, we explore the significance of each parameter (feature) in identifying the presence of entanglement. This analysis enables us to rank the features and eliminate the less significant ones, leading to the development of new entanglement witness functions that rely on fewer number of features, and hence do not require complete state tomography for their evaluation.

Figures (4)

• Figure 1: The schematic structure of a Kolmogorov-Arnold Neural Network (KAN): This sample model consists of an input layer with 5 nodes, a single hidden layer containing 3 nodes, and an output layer with 1 node. As illustrated, each edge in the network is associated with an activation function that must be optimized during the learning process. The outputs of these activation functions are summed before being passed to the nodes of the subsequent layer.
• Figure 2: The coefficients of observables for a sample output function from the model 9--6--3--1 are displayed, with the y-axis representing the coefficient values of the corresponding features on the x-axis.
• Figure 3: Display the importance of observables in each group, where each line represents a group. For observables on the X-axis, the Y-axis indicates their frequency of occurrence within that group. For example, if a witness requires three observables, refer to the 'Top 3' line and select the three observables with the highest frequencies indicated on the Y-axis.
• Figure 4: Accuracy of each models with different number of reduced feature. All potential size of features, ranging from 1 to 8, are inserted.