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Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
Yusuke Nomura, Andrew S. Darmawan, Youhei Yamaji, Masatoshi Imada
TL;DR
Accurate ground-state wave functions for strongly interacting quantum spin and fermionic systems remain difficult to construct. The paper combines an RBM correlation factor with pair-product wave functions and quantum-number projections, achieving substantially improved accuracy in square-lattice Heisenberg and Hubbard models.
Problem
Accurate ground-state wave functions for strongly interacting and entangled quantum spin and fermionic systems are difficult to construct.
Method
RBM+PP combines an RBM neural-network correlation factor with an entangled pair-product reference wave function and quantum-number projections.
Results
RBM+PP substantially improves accuracy beyond the separate methods in both the Heisenberg and Hubbard models on square lattices.
Takeaways & Limitations
RBM+PP offers a high-accuracy variational solver applicable to both quantum-spin and interacting fermionic problems.
Abstract
from arXiv · showhide
We develop a machine learning method to construct accurate ground-state wave functions of strongly interacting and entangled quantum spin as well as fermionic models on lattices. A restricted Boltzmann machine algorithm in the form of an artificial neural network is combined with a conventional variational Monte Carlo method with pair product (geminal) wave functions and quantum number projections. The combination allows an application of the machine learning scheme to interacting fermionic systems. The combined method substantially improves the accuracy beyond that ever achieved by each method separately, in the Heisenberg as well as Hubbard models on square lattices, thus proving its power as a highly accurate quantum many-body solver.
I. INTRODUCTION
The paper addresses accurate ground-state wave functions for strongly interacting quantum systems by combining RBM learning with entangled reference states, extending the approach to fermions.
- Accurate ground-state wave functions for many-body interacting systems remain a central challenge in condensed matter physics.
- VMC provides accurate ground-state wave functions for quantum spins and fermions on various lattices.
- RBM variational wave functions use hidden neurons to mediate entanglement among physical quantum spins and are self-optimized through machine learning.
- The original RBM uses a product reference state, leaving nonlocal entanglement to the RBM correlation factor alone.
- RBM+PP combines a flexible RBM correlation factor with an entangled pair-product reference state and significantly outperforms the original RBM on the 2D Heisenberg model.
- The pair-product reference incorporates nonlocal fermionic correlations and the fermionic sign, enabling application to interacting fermion systems.
- For fermionic RBM states, the Fermi-sea reference supplies primitive fermionic entanglement and signs while fermionic modes are mapped to visible-layer spins.
B. PP wave function
The pair-product wave function supplies an entangled reference state that captures nonlocal correlations and optimizes fermionic nodal structure within the Pfaffian framework.
- The pair-product state is used as the reference state in RBM+PP to incorporate more involved entanglement directly into the wave function.
- For a real-space configuration, the pair-product amplitude is expressed as a Pfaffian and calculated efficiently.
- The Pfaffian wave function can optimize nodal structure, which is crucial for accurately describing fermionic wave functions.
- In spin models, the pair-product wave function is supplemented by the Gutzwiller factor to prohibit double occupation.
- The pair-product wave function can represent resonating valence bond wave functions.
C. RBM+PP wave function
RBM+PP combines the neural-network correlation factor with a pair-product reference, using direct entanglement, symmetry projections, and flexible many-body correlations.
- RBM+PP uses N(x) as the correlation factor and a pair-product reference for itinerant fermions or a Gutzwiller-projected pair-product reference for spins.
- Compared with mVMC, RBM+PP replaces empirical Gutzwiller and Jastrow factors with the more flexible neural-network factor N(x).
- Unlike P-RBM, RBM+PP provides direct entanglement among physical variables through pair-product parameters in addition to hidden-layer connections.
- The neural-network factor can represent both Gutzwiller and Jastrow factors as well as correlations involving more than two bodies.
- The method imposes translationally invariant RBM parameters and projects the reference state onto total-spin-zero and zero-momentum subspaces.
D. Machine learning of variational parameters
The variational parameters are optimized by Monte Carlo-based stochastic reconfiguration or natural-gradient methods for square-lattice Heisenberg and Hubbard models.
- Physical quantities and parameter derivatives are estimated efficiently by Markov-chain Monte Carlo sampling over the wave-function probability distribution.
- Stochastic reconfiguration, also called natural gradient, optimizes the variational parameters with respect to energy.
- RBM+PP improves accuracy over P-RBM at additional computational cost.
- The calculations target 2D S = 1/2 antiferromagnetic Heisenberg and Hubbard models on square lattices.
- The Hubbard model uses onsite repulsion U to control correlation strength, with periodic and antiperiodic boundary conditions in the two lattice directions.
IV. RESULTS
RBM+PP substantially improves ground-state accuracy for the 8 × 8 Heisenberg model by combining an RBM correlation factor with entangled reference states and quantum projections. The method also accurately captures spin correlations, while variance extrapolation further refines its energy estimate.
- Heisenberg model: RBM+PP substantially improves the accuracy of independent mVMC and P-RBM schemes for the 2D Heisenberg model.
- Reference states: Using the same RBM correlation factor, the projected pair-product reference state gives the lowest energy, whereas the product reference state gives the highest.
- Variance extrapolation: Variance extrapolation to zero variance provides a more accurate ground-state energy estimate and works better for RBM+PP because its variance is already small.
- Energy accuracy: The RBM+PP energy reaches a relative error of order 10^-5, or 0.001 percent, relative to SSE-QMC.
- Correlation functions: RBM+PP also achieves high accuracy for the spin structure factor S(π, π) in the Heisenberg model.
B. Hubbard model
For the half-filled 8 × 8 Hubbard model, RBM+PP improves ground-state energy accuracy over F-RBM and mVMC, with accuracy improving as the reference wave function is enhanced. It also approaches exact AF-QMC results for spin correlations as hidden-variable density increases.
- Energy accuracy: RBM+PP achieves significantly higher energy accuracy than F-RBM for both U/t = 4 and U/t = 8, where F-RBM errors are several percent.Both RBM and pair-product parameters are optimized in the RBM+PP calculation.
- Energy accuracy: RBM+PP surpasses mVMC accuracy, indicating an advantage for the neural-network correlation factor over empirical Gutzwiller-Jastrow factors.The comparison uses mVMC wave functions with Gutzwiller-Jastrow correlation factors.
- Energy accuracy: At U/t = 4, the best RBM+PP relative error is comparable to TNVMC’s ∼0.25 percent result at tensor bond dimension D = 16.RBM+PP additionally avoids the involved tensor-network contraction procedure described for TNVMC.
- Energy accuracy: The figure plots RBM+PP relative energy error to AF-QMC versus 1/α for U/t = 4 and 8, comparing RBM+PP, F-RBM, mVMC, and TNVMC.AF-QMC reference energies are E/t = −0.8642(2) and −0.5259(3), respectively; error bars are Monte Carlo standard errors.
- Spin correlations: For the spin structure factor S(π, π), increasing α brings RBM+PP values closer to the exact AF-QMC value at both interaction strengths.Table II compares RBM+PP with mVMC and AF-QMC results.
V. DISCUSSION
The discussion positions RBM+PP as a flexible variational ansatz for spin and fermionic systems, while identifying representability and optimization issues for fermionic nodes and large hidden-variable densities. A deep Boltzmann machine is proposed as a future extension.
- Representability: For fermionic problems, the pair-product state is expected to reproduce nodal structure, but its optimized nodes may still differ from the exact ones.Complex RBM parameters are suggested as a possible way to adjust nodes beyond the Pfaffian wave-function framework.
- Representability: The Hubbard energy curve’s convergence as 1/α approaches zero remains an open question because optimization becomes increasingly difficult at larger α.This computational difficulty may hamper the expected convergence.
- Summary: RBM+PP combines an RBM correlation factor with a pair-product reference state and is reported to improve accuracy for both Heisenberg and Hubbard models.The summary states that it outperforms simple neural-network wave functions and mVMC.
- Summary: RBM+PP is presented as applicable to both spin and fermionic problems with high accuracy and reasonable computational cost.The stated scope includes bosonic or spin systems and interacting fermionic systems.
- Future perspectives: A deep Boltzmann machine is proposed as a future extension because it may represent certain many-body wave functions more efficiently than an RBM.Using multiple hidden layers would require additional Monte Carlo sampling for hidden spins.
Appendix A: Ability of RBM to represent Gutzwiller and Jastrow factors
The appendix explains how RBM correlation factors can reproduce Gutzwiller and Jastrow factors by mediating interactions among visible physical variables through hidden neurons.
- Gutzwiller factor: The Gutzwiller factor PG = exp(−g n_i↑ n_i↓) can be recast in a form consistent with the neural-network factor N.The recasting is stated up to a trivial constant factor and one-body potential.
- Gutzwiller factor: A hidden neuron can mediate the interaction between the two physical variables representing spin-up and spin-down occupations at site i.The visible variables are defined as σ_2i = 2n_i↑−1 and σ_2i−1 = 2n_i↓−1.
- Jastrow factor: The appendix states that the neural-network correlation factor N can likewise represent the Jastrow factor.The displayed passage introduces this correspondence after establishing the Gutzwiller-factor construction.
- Wave-function forms: Tables III and IV summarize the wave-function forms used for the Heisenberg and Hubbard model calculations.
1. Stabilization factor
The stochastic reconfiguration optimization uses a positive definite S matrix and diagonal stabilization factors to control parameter updates and convergence. The implementation distinguishes stabilization settings for neural-network and pair-product parameters.
- Optimization role: Stochastic reconfiguration updates variational parameters through an optimization equivalent to imaginary-time evolution in the variational subspace.The method uses a small imaginary-time step and an S matrix to determine updates.
- Stabilization mechanism: The stabilization factor modifies the diagonal elements of the S matrix to improve optimization stability.The diagonal modification includes a scaling factor and a constant shift.
- Stabilization mechanism: The constant shift ǫ2 strongly stabilizes optimization but can slow convergence, so it is typically kept small during early iterations.The reported early-iteration range is ǫ2 ∼10^-7-10^-6.
- Parameter-dependent settings: Smaller ǫ1 values can lower the energy for neural-network parameters, whereas too-small ǫ1 can destabilize pair-product parameter optimization.The neural-network-related range is ǫ1 ∼10^-5-10^-4, while ǫ1 ≲10^-3 can destabilize f ↑↓ij optimization.
- Initialization: Initial RBM parameters are small random numbers, and the lowest-energy wave function from several random seeds is selected.This initialization and selection procedure is applied before optimization proceeds.
2. Particle-hole transformation of Hubbard model
The paper motivates transforming Hamiltonians into solver-suitable representations, contrasting conventional real-space VMC with machine-learning approaches. It also reports using a staggered particle-hole transformation and examines Hubbard-model energy errors versus interaction strength.
- Representation choice: Hamiltonian transformations can provide representations better suited to a chosen theoretical or numerical solver.The paper frames representation choice as part of analyzing many-body Hamiltonians.
- Conventional VMC: Conventional VMC analyzes real-space interacting models directly because its empirical correlation factors are defined in the real-space basis.The comparison is made with machine-learning approaches.
- Hubbard transformation: The authors use a staggered particle-hole transformation for the Hubbard model, while noting that it may not be optimal.The transformation maps c_i↓ to (−1)^i c†... in the reported formulation.
- Energy comparison: For the 8 × 8 Hubbard model, the figure reports RBM+PP relative energy error against AF-QMC and compares the t/U = 0 point with an 8 × 8 Heisenberg result.The Hubbard calculation uses periodic–antiperiodic boundary conditions.
Appendix D: U dependence of energy in RBM+PP
Appendix D examines how RBM+PP energy accuracy varies with interaction strength in the 2D 8 × 8 Hubbard model. The reported error decreases as U/t increases.
- System and ansatz: The analysis studies RBM+PP energy in the 2D 8 × 8 Hubbard model.The reported wave function uses NLK=0 and α = 32.
- Interaction dependence: The RBM+PP energy error decreases with increasing U/t.This is the central interaction-strength trend reported for the calculation.
- Accuracy trend: The appendix therefore reports improved RBM+PP accuracy toward larger interaction strength in this Hubbard-model calculation.This statement follows directly from the reported decrease in error with U/t.