Despite their fundamental importance in dictating the quantum-mechanical properties of a system, ground states of many-body local quantum Hamiltonians form a set of measure zero in the many-body Hilbert space. Hence determining whether a given many-body quantum state is ground-stateable is a challenging task. Here we propose an unsupervised machine learning approach, dubbed Entanglement Clustering ("EntanCl"), to separate out ground-stateable wave functions from those that must be excited-state wave functions using entanglement structure information. EntanCl uses snapshots of an ensemble of swap operators as input and projects these high-dimensional data to two dimensions, preserving important topological features of the data associated with distinct entanglement structure using the uniform manifold approximation and projection. The projected data are then clustered using K-means clustering with k = 2. By applying EntanCl to two examples, a one-dimensional free fermion model and the two-dimensional toric code, we demonstrate that EntanCl can successfully separate ground states from excited states with high computational efficiency. Being independent of a Hamiltonian and associated energy estimates, EntanCl offers a new paradigm for addressing quantum many-body wave functions in a computationally efficient manner.
Despite their fundamental importance in dictating the quantum-mechanical properties of a system, ground states of many-body local quantum Hamiltonians form a set of measure zero in the many-body Hilbert space. Hence determining whether a given many-body quantum state is ground-stateable is a challenging task. Here we propose an unsupervised machine learning approach, dubbed Entanglement Clustering (“EntanCl”), to separate out ground-stateable wave functions from those that must be excited-state wave functions using entanglement structure information. EntanCl uses snapshots of an ensemble of swap operators as input and projects these high-dimensional data to two dimensions, preserving important topological features of the data associated with distinct entanglement structure using the uniform manifold approximation and projection. The projected data are then clustered using K-means clustering with k=2. By applying EntanCl to two examples, a one-dimensional free fermion model and the two-dimensional toric code, we demonstrate that EntanCl can successfully separate ground states from excited states with high computational efficiency. Being independent of a Hamiltonian and associated energy estimates, EntanCl offers a new paradigm for addressing quantum many-body wave functions in a computationally efficient manner.Received 15 May 2020Accepted 29 April 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023212Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement entropyEntanglement measuresMachine learningQuantum entanglementTechniquesVariational wave functional methodsCondensed Matter, Materials & Applied PhysicsQuantum Information
We discuss and demonstrate an unsupervised machine-learning procedure to detect topological order in quantum many-body systems. Using a restricted Boltzmann machine to define a variational Ansatz for the low-energy spectrum, we sample wave functions with probability decaying exponentially with their variational energy; this defines our training dataset that we use as input to a diffusion map scheme. The diffusion map provides a low-dimensional embedding of the wave functions, revealing the presence or absence of superselection sectors and, thus, topological order. We show that for the diffusion map, the required similarity measure of quantum states can be defined in terms of the network parameters, allowing for an efficient evaluation within polynomial time. However, possible ``gauge redundancies'' have to be carefully taken into account. As an explicit example, we apply the method to the toric code.
Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with nonconvex or nested structures [A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Advances in Neural Information Processing Systems 14: Proceedings of the 2001 Conference (MIT Press, Cambridge, MA, 2002), pp. 849--856]. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a low-dimensional space where clustering is more efficient. Despite its success in clustering tasks, spectral clustering suffers in practice from a fast-growing running time of
O(n3), where
n is the number of points in the data set. In this work we propose an end-to-end quantum algorithm performing spectral clustering, extending a number of works in quantum machine learning. The quantum algorithm is composed of two parts: the first is the efficient creation of the quantum state corresponding to the projected Laplacian matrix, and the second consists of applying the existing quantum analog of the
k-means algorithm [I. Kerenidis, J. Landman, A. Luongo, and A. Prakash,
q-means: A quantum algorithm for unsupervised machine learning, in Advances in Neural Information Processing Systems 32: Proceedings of the 2019 Conference (Curran Associates, Red Hook, NY, 2020), pp. 4136--4146]. Both steps depend polynomially on the number of clusters, as well as precision and data parameters arising from quantum procedures, and polylogarithmically on the dimension of the input vectors. Our numerical simulations show an asymptotic linear growth with
n when all terms are taken into account, significantly better than the classical cubic growth. This work opens the path to other graph-based quantum machine learning algorithms, as it provides routines for efficient computation and quantum access to the incidence, adjacency, and projected Laplacian matrices of a graph.
We propose a framework to design concurrently a frustration-free quantum many-body Hamiltonian and its numerically exact ground states on a sufficiently large finite-size cluster in one and two dimensions using an elementary matrix product state (MPS) representation. Our approach strategically chooses a local cluster Hamiltonian, which is arranged to overlap with neighboring clusters on a designed lattice. The frustration-free Hamiltonian is given as the sum of the cluster Hamiltonians by ensuring that there exists a state that has its local submanifolds as the lowest-energy eigenstate of every cluster. The key to find such a solution is a systematic protocol, which projects out excited states on every cluster using MPS and effectively entangles the cluster states. This algorithm not only offers an exact solution to general frustration-free Hamiltonian, but even provide the excited eigenstates by setting a proper target. It has several advantages, including the ability to achieve the calculation of states at nearly equal cost in one and two dimensions including those belonging to gapless or long-range entangled classes, flexibility in designing Hamiltonians unbiasedly across various forms of models, and numerically feasible validation of exactness through energy calculations. It enables the exploration of exact phase boundaries and the analysis of even a spatially nonuniform random system, providing platforms for quantum simulations and benchmarks.
The quantum many-body problem is ultimately a curse of dimensionality: the state of a system with many particles is determined by a function with many dimensions, which rapidly becomes difficult to efficiently store, evaluate and manipulate numerically. On the other hand, modern machine learning models like deep neural networks can express highly correlated functions in extremely large-dimensional spaces, including those describing quantum mechanical problems. We show that if one represents wavefunctions as a stochastically generated set of sample points, the problem of finding ground states can be reduced to one where the most technically challenging step is that of performing regression-a standard supervised learning task. In the stochastic representation the (anti)symmetric property of fermionic/bosonic wavefunction can be used for data augmentation and learned rather than explicitly enforced. We further demonstrate that propagation of an ansatz towards the ground state can then be performed in a more robust and computationally scalable fashion than traditional variational approaches allow.
The ability of modern quantum simulators--both digital and analogue--to generate large ensembles of single-shot projective "snapshots" has opened a data-rich avenue for the study of quantum many-body systems. Unsupervised machine learning analysis of such snapshots has gained traction, with numerous works reconstructing phase diagrams by learning and clustering low-dimensional representations of quantum states. Here, we forgo such representation learning in favour of distance learning: we infer the pairwise distances between quantum states--already sufficient for clustering--directly from snapshots. Specifically, we use a single neural discriminator to estimate Csiszar f-divergences--statistical distances between distributions--in an unsupervised manner. The resulting clusters reveal regimes with different dominant correlations, often coinciding with, but not limited to, conventionally defined phases of matter. Beyond phase-diagram exploration, we connect the infinitesimal limit of the inferred divergences to the Fisher information metric and analyse its finite-size scaling. This yields critical exponents of the discovered transitions and enables snapshot-based analysis of universality classes. We apply distance learning to a diverse set of systems characterised by conventional local order parameters (1D transverse-field and 2D classical Ising models), non-local topological order (extended toric code), and higher-order correlations (fermionic t-J model on a triangular lattice). In all cases, we correctly recover boundaries between distinct correlation regimes and, where applicable, quantitatively match established critical behaviour. Finally, we show that distances to suitably chosen reference snapshot distributions help identify the dominant correlations within the discovered clusters, positioning distance learning as a versatile information-geometric probe of quantum many-body physics.