Unsupervised Detection of Topological Phase Transitions with a Quantum Reservoir

TL;DR

The paper tackles the challenge of identifying topological phase transitions in strongly correlated systems by introducing an unsupervised quantum reservoir computing framework that leverages many-body localized dynamics within a discrete-time crystal circuit. By evolving input states under $U_F(g, \boldsymbol{\phi}, \boldsymbol{h})$ and measuring only local observables $\langle Z_i \rangle$ and $\langle Z_i Z_{i+1} \rangle$, followed by t-SNE embedding and Gaussian Mixture Model clustering, the method uncovers phase structure in the extended SSH model without full density-matrix tomography; it shows that the DTC-MBL processing amplifies distinctions between phases and enables practical, scalable phase diagram reconstruction on NISQ devices. The approach yields MBTI-consistent phase boundaries and demonstrates robustness to noise, highlighting a practical pathway for probing quantum phase transitions in 1D strongly correlated systems. It further suggests potential extensions to higher dimensions through prethermal dynamics or DTC modes, offering a versatile framework for topology-driven quantum many-body studies.

Abstract

In quantum many-body systems, characterizing topological phase transitions typically requires complex many-body topological invariants, which are costly to compute and measure. Inspired by quantum reservoir computing, we propose an unsupervised quantum phase detection method based on a many-body localized evolution, enabling efficient identification of phase transitions in the extended SSH model. The evolved quantum states produce feature distributions under local measurements, which, after simple post-processing and dimensionality reduction, naturally cluster according to different Hamiltonian parameters. Numerical simulations show that the evolution combined with local measurements can significantly amplify distinctions between quantum states, providing an efficient means to detect topological phase transitions. Our approach requires neither complex measurements nor full density matrix reconstruction, making it practical and feasible for noisy intermediate-scale quantum devices.

Figures (5)

• Figure 1: Schematic illustration of the proposed QRC framework. We prepare a set of quantum states under different parameters, which can be obtained on NISQ devices via VQE or in analog quantum simulators by tuning system parameters. In our numerical experiments, these states are generated using DMRG and then evolved under a quantum circuit in the DTC regime. We measure only $\langle Z_i \rangle$ and $\langle Z_i Z_{i+1} \rangle$ to form feature vectors, which are subsequently visualized using t-SNE. The results show that feature vectors from different phases cluster effectively in the feature space, enabling unsupervised learning of phase transitions in strongly correlated systems.
• Figure 2: (a) Schematic representation of the extended SSH model Hamiltonian, illustrating the intra-cell hopping $J$, inter-cell hopping $J'$. (b) Phase diagram where the theoretical phase boundaries are computed based on MBTI. Black diamond markers denote the phase transition points identified by our QRC-based unsupervised method. (c) Visualization of clustering in the t-SNE feature space, showing how the states naturally group according to their underlying phases. (d) Probabilities of phase assignment obtained from a Gaussian Mixture Model (GMM). To enhance clarity and readability, a sparse subset of the 1501 data points is plotted by selecting every 50th point.
• Figure 3: Illustration of t-SNE in the non-MBL DTC region for $\delta = 3.0$. (a) "Identical data" case, where t-SNE is applied directly to the ground-state measurements of $\langle Z_i \rangle$ and $\langle Z_i Z_{i+1} \rangle$. (b) t-SNE after evolution through the thermal region, highlighting how the feature space reorganizes under thermal dynamics. circuit parameters $g=0.5$, $D=5$.
• Figure 4: PCA of ground-state measurements of $\langle Z_i \rangle$ and $\langle Z_i Z_{i+1} \rangle$, showing the first principal component. (a) $\delta = 0.5$. (b) $\delta = 3.0$.
• Figure 5: Illustration of long-time evolution ($0$-$50$ cycles) for three representative states under different circuits. The upper panels show DTC evolution ($g=0.96$) and the lower panels show thermal evolution ($g=0.5$). From left to right, the states are: trivial ($J^\prime/J = 0.9$, $\delta = 1.0$), SB ($\delta = 2.0$, $J^\prime/J = 1.0$), and topological ($\delta = 1.0$, $J^\prime/J = 1.1$). The vertical axis indices 1–128 correspond to $\langle Z_i \rangle$ ($i=1,\dots,128$), and 129–255 correspond to $\langle Z_i Z_{i+1} \rangle$ ($i=1,\dots,127$).