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Simulating disordered quantum Ising chains via dense and sparse restricted Boltzmann machines.

S Pilati1, P Pieri1,2

  • 1School of Science and Technology, Physics Division, Università di Camerino, 62032 Camerino (MC), Italy.

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Summary
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Sparse restricted Boltzmann machines (RBMs) improve quantum many-body simulations of disordered Ising chains by reducing computational cost and enhancing accuracy. These networks outperform dense RBMs, offering unbiased predictions for ground-state energies and magnetization profiles.

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Area of Science:

  • Quantum physics
  • Computational condensed matter physics
  • Machine learning applications in physics

Background:

  • Generative artificial neural networks, specifically restricted Boltzmann machines (RBMs), are effective variational wave functions for quantum many-body systems.
  • Simulating disordered quantum systems presents challenges due to the lack of translational invariance, complicating parameter optimization in standard dense RBMs.

Purpose of the Study:

  • To investigate the efficacy of sparse RBMs for simulating disordered quantum Ising chains.
  • To compare the performance of sparse RBMs against dense RBMs in terms of parameter reduction and simulation accuracy.
  • To utilize sparse RBMs as guiding functions in projective quantum Monte Carlo (PQMC) simulations for improved efficiency and bias reduction.

Main Methods:

  • Implementation of sparse RBMs with localized connectivity between visible and hidden neurons to reduce model parameters.
  • Assessment of sparse RBM performance based on connection range and comparison with dense RBMs.
  • Training RBM wave functions using an unsupervised learning scheme with projective quantum Monte Carlo (PQMC) algorithms.
  • Performing PQMC simulations guided by sparse RBMs to obtain ground-state energies and magnetization profiles.

Main Results:

  • Sparse RBMs demonstrate superior performance compared to dense RBMs for disordered quantum Ising chains.
  • The sparse connectivity in RBMs facilitates the training process and reduces computational cost.
  • PQMC simulations guided by sparse RBMs yield unbiased ground-state energies and magnetization profiles at the ferromagnetic quantum critical point.
  • Obtained magnetization profiles align with the Fisher-de Gennes scaling relation for conformally invariant systems.

Conclusions:

  • Sparse RBMs offer a more efficient and accurate approach for simulating disordered quantum many-body systems than dense RBMs.
  • The use of sparse RBMs in PQMC simulations significantly reduces computational cost and mitigates biases.
  • The study provides unbiased predictions for critical properties of quantum Ising chains, validating theoretical scaling relations.