Quantum Matter Classification Via Network Theory Links Wave Function Snapshots to Collective Properties

Understanding the phases of quantum matter represents a major challenge in modern physics, and researchers are increasingly turning to powerful computers and simulators to probe these complex states. Riccardo Andreoni, Vittorio Vitale, and Cristiano Muzzi, alongside colleagues from institutions including the International School for Advanced Studies (SISSA) and The Abdus Salam International Centre for Theoretical Physics (ICTP), have developed a new theoretical framework that links the snapshots obtained from these simulations to the underlying phases of matter. Their approach combines principles of data complexity with network analysis, allowing them to classify quantum states without relying on assumptions about the system’s dynamics. This method identifies a minimal set of measurements needed to describe a quantum state and then builds a ‘wave-function network’ to reveal its key characteristics, offering a fully interpretable and experimentally relevant way to analyse the output of quantum computations and simulations.

Wave Function Networks Characterise Quantum Systems

This work introduces a new approach to characterise quantum many-body systems by applying concepts from network theory. The method constructs networks directly from snapshots of the quantum wave function, using the single-particle density matrix to define connections between nodes. This allows scientists to identify emergent network structures that reflect the underlying quantum correlations and entanglement within the system. By analysing key network properties, such as clustering and path length, the team demonstrates the ability to classify different quantum phases of matter, including those with topological order and symmetry-breaking transitions. The research establishes a direct link between the microscopic quantum state and macroscopic network characteristics, offering a new perspective on understanding complex quantum systems. Furthermore, the approach proves robust against disorder and provides a valuable tool for analysing quantum systems beyond the reach of traditional methods.

Entanglement, Many-Body Systems, and DMRG Techniques

This body of work encompasses a broad range of research in quantum physics, condensed matter physics, network science, data analysis, and computational methods. Core to much of the research is the study of many-body physics, focusing on entanglement entropy and utilising techniques like density matrix renormalization group (DMRG) and matrix product states (MPS) to investigate strongly correlated quantum systems. Scientists explore quantum phase transitions, examining transitions between different states of matter driven by changes in parameters like temperature or magnetic field. Research also focuses on integrable systems and their solutions, providing benchmarks for more complex systems, and a strong emphasis on entanglement as a measure of quantum correlations and its connection to topological phases of matter.

Spin chains serve as fundamental models for studying quantum magnetism and many-body effects. Beyond purely quantum physics, this research incorporates network analysis, particularly the study of higher-order networks that move beyond simple pairwise connections to consider more complex interactions. Scientists investigate the structure and dynamics of complex networks, including scale-free networks and their properties, and apply concepts from statistical mechanics to understand network behaviour. A rapidly growing area involves applying data science techniques to physics, with a major focus on topological data analysis (TDA).

This involves using techniques like persistent homology and Mapper to extract meaningful features from complex data, identify patterns, and understand underlying structure. Researchers also employ methods for estimating the intrinsic dimensionality of data, and apply machine learning algorithms to analyse physical data and make predictions. Computational methods play a crucial role, with a strong emphasis on tensor network methods like DMRG, MPS, and the ITensor library as efficient ways to simulate quantum many-body systems. Cutting-edge research combines these areas, applying TDA to analyse quantum states, identify topological phases, and understand entanglement, and extending TDA to study lattice gauge theory. Scientists are also exploring data-driven discovery in quantum simulators and combining persistent homology with Kolmogorov complexity to quantify the complexity of quantum systems. Overall, this research represents a highly interdisciplinary program that combines advanced techniques from multiple fields to push the boundaries of our understanding of complex systems.

Wave Function Networks Reveal Intrinsic Complexity

This work presents a new theoretical framework for characterising stochastic sampling of many-body wave functions, achieved through collective projective measurements, and connects these snapshots to collective properties of quantum matter. The research leverages manifold learning and network theory to manage the vast amount of correlations present in many-body snapshots, employing a principle similar to Occam’s razor to identify a minimal-complexity measurement basis. Scientists embed data from snapshots in distinct metric spaces, utilising Parisi’s two-replica overlap functions to define distances between samples, and then construct wave function networks where nodes represent snapshots and links are determined by these distances. The team estimates the intrinsic dimension of the data manifolds in each basis, identifying the minimal-complexity basis as the one with the smallest intrinsic dimension, effectively quantifying the minimal degrees of freedom required to capture the samples.

This approach allows for data compression and provides a proxy for Kolmogorov complexity, enabling efficient representation of complex quantum states. Once the minimal-complexity basis is identified, scientists characterise the corresponding network via its degree distribution, diagnosing the presence or absence of correlations in data space. Applying this framework to one-dimensional translational invariant Hamiltonians, representing paramagnetic, critical, symmetry-broken, and symmetry-protected topological states, the research demonstrates its versatility. The team investigates how the minimal intrinsic dimension varies as a function of real-space scale using a Kadanoff-type decimation scheme directly on the samples, successfully distinguishing between local and topological real-space correlations. This framework is highly interpretable, state agnostic, and scalable, typically requiring only a few thousand samples for experimentally relevant system sizes, offering a powerful new tool for analysing complex quantum systems and their properties.

Wave Function Complexity Reveals Material Phases

This research presents a new framework for characterising phases of matter by analysing snapshots of many-body wave functions obtained from computer simulations. The team developed a method that links the complexity of these snapshots to the underlying physical properties of the system, moving beyond traditional order parameters. By identifying a minimal-complexity measurement basis and constructing wave-function networks, scientists can classify phases without assuming knowledge of the system’s dynamics. The findings demonstrate that ordered phases exhibit structured networks with low intrinsic dimension and clustering, while symmetry-protected topological phases are characterised by high complexity and random networks, indicating the absence of a local order parameter.

This approach allows for the identification of phases based solely on the statistical properties of wave function snapshots, offering a powerful tool for analysing complex quantum systems. The authors acknowledge that their method relies on the availability of wave function snapshots, which are currently limited by computational resources. Future research will focus on incorporating more advanced network mathematics, such as discrete homology, and applying this method to time-dependent dynamics and gauge theories, potentially broadening its applicability to a wider range of physical phenomena.