A hybrid variational quantum circuit approach for stabilizer states classifiers

TL;DR

The paper addresses the challenge of entanglement classification for pure multipartite states by framing it as orbit learning under groups like LU, LC, and SLOCC. It introduces a hybrid variational quantum circuit with classical neural-network post-processing to capture the non-linear structure of entanglement orbits directly from quantum data, avoiding full state tomography. On four-qubit graph states, the approach achieves near-perfect classification across six graph-state classes, accurately identifies LC-stabilizer and LU orbits, and demonstrates robust performance when evaluating against the full Hilbert space. This NISQ-friendly framework leverages amplitude encoding and polynomial resource scaling, offering a practical method for entanglement classification with potential extension to larger systems and broader orbit types.

Abstract

Entanglement classification of pure multipartite quantum states is a challenging problem in quantum information theory that can be mathematically stated as orbit classification for some given group action on the ambient Hilbert space. The group action depends on the grained classification one expects, the finer-grained one being the classification up to local unitary transformation (LU). In this article, we show how a variational quantum circuit approach can be used to learn entanglement orbits, and we apply our findings to build a classifier for four-qubit states.

Figures (4)

• Figure 1: Schematic diagram of a general $n$-qubit simple Variational Quantum Classifier. The pattern indicates execution on a quantum device and indicates execution on a classical device.
• Figure 2: (a) shows the original classification of the dataset. (b) shows the results when a single-qubit VQC is trained to classify this dataset. It can be observed that this VQC can only draw linear boundaries in its attempt to minimize the cost function. (c) shows the results when a two-qubit VQC is trained to classify the dataset. The two features of the dataset were encoded into the amplitudes rath of the qubits with the other two amplitudes treated as free hyperparameters and set to 0 and 0.25. However, it is obvious that even with these additional degrees freedom, this VQC cannot fully grasp the non-linearity required from the problem. Finally, (d) shows the performance of a two-qubit hybrid VQC that conducts classical post-processing using classical NN layers. The classification accuracy is $100\%$ for this case.
• Figure 3: Schematic $n$-qubit diagram of the specific hybrid VQC structure employed in this work. Here, $\boldsymbol{\theta}=\{\boldsymbol{\alpha,\beta,\gamma}\}$ form the set of trainable parameters of the quantum circuit with their upper and lower indices indicating the qubit number and layer number respectively. $G_{AMP}$ indicates amplitude encoding rath and $R_X,R_Y,R_Z$ are single qubit rotation gates round $X,Y,Z$ axes respectively. The diagram only shows one hidden layer of the classical NN for simplicity. In practice, the number of hidden layers and the number of neurons in the hidden layers were varied for each classification task. The pattern indicates execution on a quantum device, indicates repeated layer of the quantum circuit, and indicates execution on a classical device.
• Figure 4: Classification of four-qubit graph states. Note that the star graph corresponds to the four-qubit GHZ state.

Theorems & Definitions (2)

• Example 1
• Example 2