Explainable Representation Learning of Small Quantum States

TL;DR

The paper investigates how unsupervised representation learning can yield interpretable encodings of quantum states. By applying a $eta$-VAE to two-qubit density matrices generated from a parameterized circuit, and introducing information scrambling to suppress local features, the authors show that the latent representation tracks entanglement via a quantity closely related to concurrence $C[ ho]$. Optimizing the regularization strength $eta$ yields disentangled latent factors, with a single latent variable capturing the entanglement information, and the approach extends to random pure states, depolarized states, and three-qubit state subpartitions. This work provides a proof-of-concept that machine-learned representations of quantum states can be made interpretable and physically meaningful, potentially enabling scalable insights into quantum systems from unsupervised learning.

Abstract

Unsupervised machine learning models build an internal representation of their training data without the need for explicit human guidance or feature engineering. This learned representation provides insights into which features of the data are relevant for the task at hand. In the context of quantum physics, training models to describe quantum states without human intervention offers a promising approach to gaining insight into how machines represent complex quantum states. The ability to interpret the learned representation may offer a new perspective on non-trivial features of quantum systems and their efficient representation. We train a generative model on two-qubit density matrices generated by a parameterized quantum circuit. In a series of computational experiments, we investigate the learned representation of the model and its internal understanding of the data. We observe that the model learns an interpretable representation which relates the quantum states to their underlying entanglement characteristics. In particular, our results demonstrate that the latent representation of the model is directly correlated with the entanglement measure concurrence. The insights from this study represent proof of concept towards interpretable machine learning of quantum states. Our approach offers insight into how machines learn to represent small-scale quantum systems autonomously.

Figures (12)

• Figure 1: Conceptual overview. a) Quantum states $\rho(\alpha)$ are generated by a two-qubit quantum circuit consisting of a Hadamard, a Controlled-$R_y$ gate parameterized by the angle $\alpha$, and two single-qubit rotations. b) Data are encoded from a density matrix into a stochastic latent representation $z$ using the trained encoder network. c) Latent variables $z=(z_0,z_1)$ are visualized to analyze the relation of structure of the learned representation and encoded properties. In this figure, the two-dimensional latent space is color-coded by an entanglement measure of underlying states (the concurrence). Here, the low entanglement region is colored purple and the high entanglement region is colored yellow.
• Figure 2: Schematic overview of VAE architecture. The input $\textbf{x}$ is compressed by the neural network-based encoder into the latent space, represented as $\textbf{z}$, serving as an information bottleneck. The decoder network then uses the information from the latent space to reconstruct $\textbf{x}^*$.
• Figure 3: The $\rho$-VAE learns to extract the parameter $\alpha$ from quantum states to structure its latent space. The correlation between the one-dimensional latent space $z$ of $\rho$-VAE and parameter $\alpha$ of encoded density matrices (blue, mean and standard deviation of $10$ samples). The error bars are contained within the markers. The regression of encoded quantum states (black) shows that the correlation has a small sinusoidal feature but is sufficiently characterized by a linear function with $r^2>0.99$. Inset: The final loss of $\rho$-VAE trained on quantum states $\rho(\alpha)$ at $\beta=0$ with latent space dimensions $N\in [1,8]$ (mean and standard deviation of $9$ experiments) indicates that a one-dimensional latent space has sufficient information capacity.
• Figure 4: The $\rho_s$-VAE learns an efficient but uninterpretable representation of quantum states with information scrambling $\rho_s(\alpha)$. Three-dimensional latent space $z=(z_0,z_1,z_2)$ of $\rho_s$-VAE trained with $\beta=0$. Each encoded density matrix is color-coded by its corresponding concurrence value. Inset: The final loss of $\rho_s$-VAE trained on quantum states $\rho_s(\alpha)$ at $\beta=0$ with latent space dimensions $N\in [1,8]$ (mean and standard deviation of $9$ experiments) indicates that a three-dimensional latent space has sufficient information capacity.
• Figure 5: Tuning the $\beta$ parameter of $\rho_s$-VAE leads to a compressed representation of quantum states. a) Regularization loss $\mathcal{L}_{KL}^{(i)}$ contributed by each latent variable $z_i$ of $\rho_s$-VAE at different $\beta$ values. The $N=8$ latent variables are normalized and presented in descending order of loss values. b-e) Two-dimensional latent space $(z_0,z_1)$ of two largest $\mathcal{L}_{KL}^{(i)}$ at $\beta \in (0.01,0.4,0.75,1.0)$ values. The color-coding is identical to Fig. \ref{['fig:exp_2']} and indicates the concurrence value of the encoded quantum states.