Sample-efficient learning of quantum many-body systems

Abstract

We study the problem of learning the Hamiltonian of a quantum many-body system given samples from its Gibbs (thermal) state. The classical analog of this problem, known as learning graphical models or Boltzmann machines, is a well-studied question in machine learning and statistics. In this work, we give the first sample-efficient algorithm for the quantum Hamiltonian learning problem. In particular, we prove that polynomially many samples in the number of particles (qudits) are necessary and sufficient for learning the parameters of a spatially local Hamiltonian in l_2-norm. Our main contribution is in establishing the strong convexity of the log-partition function of quantum many-body systems, which along with the maximum entropy estimation yields our sample-efficient algorithm. Classically, the strong convexity for partition functions follows from the Markov property of Gibbs distributions. This is, however, known to be violated in its exact form in the quantum case. We introduce several new ideas to obtain an unconditional result that avoids relying on the Markov property of quantum systems, at the cost of a slightly weaker bound. In particular, we prove a lower bound on the variance of quasi-local operators with respect to the Gibbs state, which might be of independent interest. Our work paves the way toward a more rigorous application of machine learning techniques to quantum many-body problems.

60 pages, 3 figures

References in corpus (24)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Probing many-body dynamics on a 51-atom quantum simulator
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Observation of a Many-Body Dynamical Phase Transition with a 53-Qubit Quantum Simulator
• Quantum Simulation of Antiferromagnetic Spin Chains in an Optical Lattice
• Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution
• Probing entanglement entropy via randomized measurements
• Experimental Quantum Hamiltonian Learning
• Hamiltonian Learning and Certification Using Quantum Resources
• Learning a local Hamiltonian from local measurements
• Quantum Graphical Models and Belief Propagation
• Determining a local Hamiltonian from a single eigenstate
• Quantum Overlapping Tomography
• Coherent Imaging Spectroscopy of a Quantum Many-Body Spin System
• Quantum Hamiltonian Learning Using Imperfect Quantum Resources
• Quantum SDP-Solvers: Better upper and lower bounds
• Efficient estimation of nearly sparse many-body quantum Hamiltonians
• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Markov entropy decomposition: a variational dual for quantum belief propagation
• Learning the dynamics of open quantum systems from their steady states
• Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems
• Reconstructing quantum states from local data
• Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions
• Algorithms for the Markov Entropy Decomposition

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• Deep reinforcement learning for preparation of thermal and prethermal quantum states
• Unified Quantum State Tomography and Hamiltonian Learning Using Transformer Models: A Language-Translation-Like Approach for Quantum Systems
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• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Exploring Many-body Interactions Through Quantum Fisher Information
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• Digital Quantum Simulation, Learning of the Floquet Hamiltonian, and Quantum Chaos of the Kicked Top
• Hamiltonian Learning using Machine Learning Models Trained with Continuous Measurements
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians
• Ansatz-free Hamiltonian learning with Heisenberg-limited scaling
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• Learning interacting fermionic Hamiltonians at the Heisenberg limit
• Relationship between the ground-state wave function of a magnet and its static structure factor
• Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States
• Finite temperature tensor network algorithm for frustrated two-dimensional quantum materials
• Hamiltonian-learning quantum magnets with non-local impurity tomography
• Efficient Quantum State Tracking in Noisy Environments
• Time-Efficient Quantum Entropy Estimator via Samplizer
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• Information geometry of bosonic Gaussian thermal states
• Characterizing errors in parameter estimation by local measurements