Identification of quantum entanglement with Siamese convolutional neural networks and semi-supervised learning

TL;DR

This study uses deep convolutional NNs, a type of supervised machine learning, to identify quantum entanglement for any bipartition in a 3-qubit system and improves the model's accuracy and ability to recognize PPTES.

Abstract

Quantum entanglement is a fundamental property commonly used in various quantum information protocols and algorithms. Nonetheless, the problem of identifying entanglement has still not reached a general solution for systems larger than $2\times3$. In this study, we use deep convolutional NNs, a type of supervised machine learning, to identify quantum entanglement for any bipartition in a 3-qubit system. We demonstrate that training the model on synthetically generated datasets of random density matrices excluding challenging positive-under-partial-transposition entangled states (PPTES), which cannot be identified (and correctly labeled) in general, leads to good model accuracy even for PPTES states, that were outside the training data. Our aim is to enhance the model's generalization on PPTES. By applying entanglement-preserving symmetry operations through a triple Siamese network trained in a semi-supervised manner, we improve the model's accuracy and ability to recognize PPTES. Moreover, by constructing an ensemble of Siamese models, even better generalization is observed, in analogy with the idea of finding separate types of entanglement witnesses for different classes of states.

Figures (7)

• Figure 1: Idea of the supervised training with the training set composed of states we know whether they are entangled or not (verified strategy). The goal is to build a model capable of generalizing to PPTES that are hard to recognize by analytical methods.
• Figure 2: Model architectures: (a) NN for 3-qubit input density matrix (red): CNN backbone with three convolutional layers, each of $2\times2$ kernels (dark blue blocks); five fully-connected layers (green blocks); and finally, 3-element output vector identifying entanglement in three possible bipartitions (light blue). (b) Architecture of the triple Siamese network, where each subnetwork, being the CNN from the model (a), is fed by a different input (original or transformed), but shares the same network parameters (weights). (c) Ensemble of Siamese networks with separate $N_e$ Siamese models trained on different domains determined using separability measure $M$ (being a pretrained autoencoder model).
• Figure 3: Idea of the ensemble method: various states can be distinguished on the basis of some external measure (here defined by a pretrained autoencoder loss $M$) and then learned separately by ensemble of models that work independently in their $M$ domains.
• Figure 4: Example of a circuit generating a 3-qubit random state. It consists of $U$ and $CU$ gates (violet blocks) with random parameters written in each block (and defined in the Appendix).
• Figure 5: Accuracy (blue bars) of the Ensemble models plotted separately for the subsequent models' domains, determined using the separability measure $M$. Ratio (red bars) characterizes the population of the subsequent domains by the states included in various datasets: (a) validataion, (b) Acin PPTES, (c) Horodecki PPTES, (d) UPB PPTES. Purple color means that blue and orange bars overlap.