The learnability scaling of quantum states: restricted Boltzmann machines

Abstract

Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements. As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial. We empirically study the scaling of restricted Boltzmann machines (RBMs) applied to reconstruct ground-state wavefunctions of the one-dimensional transverse-field Ising model from projective measurement data. We define a learning criterion via a threshold on the relative error in the energy estimator of the machine. With this criterion, we observe that the number of RBM weight parameters required for accurate representation of the ground state in the worst case - near criticality - scales quadratically with the number of qubits. By pruning small parameters of the trained model, we find that the number of weights can be significantly reduced while still retaining an accurate reconstruction. This provides evidence that over-parametrization of the RBM is required to facilitate the learning process.

8 pages, 5 figures

References in corpus (32)

• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Entanglement renormalization
• Many-body quantum state tomography with neural networks
• Experimental realization of a long-range antiferromagnet in the Hubbard model with ultracold atoms
• A Convergence Theory for Deep Learning via Over-Parameterization
• Tensor networks for complex quantum systems
• SNIP: Single-shot Network Pruning based on Connection Sensitivity
• Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator
• Efficient Representation of Quantum Many-body States with Deep Neural Networks
• Reconstructing quantum states with generative models
• Learning Thermodynamics with Boltzmann Machines
• Efficient tomography of a quantum many-body system
• Equivalence of restricted Boltzmann machines and tensor network states
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• Stochastic series expansion method for quantum Ising models with arbitrary interactions
• Constructing exact representations of quantum many-body systems with deep neural networks
• Integrating Neural Networks with a Quantum Simulator for State Reconstruction
• Perfect Sampling with Unitary Tensor Networks
• Efficient variational simulation of non-trivial quantum states
• On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization
• Machine learning quantum states in the NISQ era
• Deconstructing Lottery Tickets: Zeros, Signs, and the Supermask
• Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
• A topological insight into restricted Boltzmann machines
• Making Trotters Sprint: A Variational Imaginary Time Ansatz for Quantum Many-body Systems
• Ultrafast Variational Simulation of Non-trivial Quantum States with Long Range Interactions
• The Impact of Neural Network Overparameterization on Gradient Confusion and Stochastic Gradient Descent
• QuCumber: wavefunction reconstruction with neural networks
• Understanding Quantum Tunneling using Diffusion Monte Carlo Simulations
• The Learnability of Quantum States

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• Convolutional restricted Boltzmann machine aided Monte Carlo: An application to Ising and Kitaev models
• Three Learning Stages and Accuracy-Efficiency Tradeoff of Restricted Boltzmann Machines
• Neural networks in quantum many-body physics: a hands-on tutorial
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• Quantum error reduction with deep neural network applied at the post-processing stage
• Empirical Sample Complexity of Neural Network Mixed State Reconstruction
• Quantum error mitigation in the regime of high noise using deep neural network: Trotterized dynamics
• Improving neural network performance for solving quantum sign structure
• Efficiency of neural-network state representations of one-dimensional quantum spin systems
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• Seeding neural network quantum states with tensor network states
• Efficient bit encoding of neural networks for Fock states