Unsupervised Learning to Recognize Quantum Phases of Matter

TL;DR

This work addresses the challenge of mapping quantum phase diagrams without labeled data by introducing an unsupervised kernel-based clustering framework that uses ground-state fidelities as the kernel: $K_{ij}=|ra{GS(oldsymbol{x}_i)}GS(oldsymbol{x}_j) angle|^2$. Ground states are represented as Matrix Product Operators and the method leverages spectral clustering, with the number of phases $c$ determined via Elbow and Silhouette criteria. The authors demonstrate the approach on two one-dimensional spin-$ frac{1}{2}$ models, the ANNNI and Cluster-Ising models, showing that the resulting phase diagrams align with known results and even identify a symmetry-protected topological (SPT) phase, albeit with finite-size and bond-dimension limitations. This label-free technique enables autonomous phase discrimination and offers a pathway toward applying quantum-generated kernels to experimental data, potentially extending to complex systems like spin liquids or spin glasses.

Abstract

Drawing the quantum phase diagram of a many-body system in the parameter space of its Hamiltonian can be seen as a learning problem, which implies labelling the corresponding ground states according to some classification criterium that defines the phases. In this work we adopt unsupervised learning, where the algorithm has no access to any priorly labeled states, as a tool for determining quantum phase diagrams of many-body systems. The algorithm directly works with quantum states: given the ground-state configurations for different values of the Hamiltonian parameters, the process uncovers the most significant way of grouping them based on a similarity criterion that refers to the fidelity between quantum states, that can be easily estimated, even experimentally. We benchmark our method with two specific spin-$\frac{1}{2}$ chains, with states determined via tensor network techniques. We find that unsupervised learning algorithms based on spectral clustering, combined with ``silhouette'' and ``elbow'' methods for determining the optimal number of phases, can accurately reproduce the phase diagrams. Our results show how unsupervised learning can autonomously recognize and possibly unveil novel phases of quantum matter.

Figures (6)

• Figure 1: Outline of the clustering procedure for quantum ground states, as described in the introduction. The process begins with randomly producing ground states of a target Hamiltonian, either using classical approximations (e.g. tensor networks), a quantum hardware with physical spins, or a quantum computer. The kernel is then estimated either in hardware or via tensor network techniques. Next, a kernel-driven clustering model groups the ground states based on their correlation structure, capturing quantum similarities, including entanglement properties. Finally, we present the resulting clusters.
• Figure 2: Graphical representation of a typical local Hamiltonian, illustrating how the automaton structure translates into the explicit MPO representation.
• Figure 3: Local interaction in the MPO formalism for ANNNI model using finite-state automata graphichal representation.
• Figure 4: Results for the ANNNI Hamiltonian \ref{['eq:annni']} with $N = 51$ spins, where the ground states are simulated with a bond dimension $\chi = 20$. (a) Average silhouette score as a function of the number of clusters $c$; the inset shows the elbow point, identified by a significant change in the WCSS dependence on $c$. (b) Silhouette plot for $c=4$, as determined by Fig. \ref{['fig:annni']}a. (c) The predicted phase diagram.
• Figure 5: Local interaction in the MPO formalism for Cluster-Ising model using finite-state automata graphichal representation.