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Solving the Quantum Many-Body Problem with Artificial Neural Networks
Giuseppe Carleo, Matthias Troyer
TL;DR
The methods address computational costs in variational ground-state optimization and time-dependent evolution for neural-network quantum states. Stochastic reconfiguration, Monte Carlo sampling, and iterative solvers systematically approach exact results while maintaining favorable scaling with system size and variational parameters.
Problem
Large variational-parameter sets make explicitly forming the covariance matrix computationally expensive, with quadratic cost in the number of variational parameters.
Method
The approach combines stochastic reconfiguration, Metropolis-Hastings sampling, time-dependent variational Monte Carlo, and MINRES-QLP iterative solution of the covariance-system equations.
Results
Increasing hidden-unit density α systematically approaches the exact ground-state energy and exact time-evolved physical properties.
Takeaways & Limitations
Iterative solvers avoid explicitly forming the covariance matrix and reduce the dominant solver complexity to O(Nvar × NMC), matching the complexity of Monte Carlo sampling in the challenging case described.
Abstract
from arXiv · showhide
The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the non-trivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form, for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with variable number of hidden neurons. A reinforcement-learning scheme is then demonstrated, capable of either finding the ground-state or describing the unitary time evolution of complex interacting quantum systems. We show that this approach achieves very high accuracy in the description of equilibrium and dynamical properties of prototypical interacting spins models in both one and two dimensions, thus offering a new powerful tool to solve the quantum many-body problem.
Appendix A: Stochastic Optimization For The Ground State
The ground-state network parameters are optimized by minimizing variational energy with stochastic reconfiguration, using a covariance matrix and regularization when needed.
- Optimization method: Stochastic Reconfiguration optimizes the network by minimizing the variational energy with respect to the network weights.The target stationary solution satisfies ∇E(W⋆) = 0.
- Numerical stabilization: The covariance matrix can be non-invertible, so its Moore-Penrose pseudo-inverse or explicit regularization is used.The implementation uses a decaying regularization parameter λ(p) = max(λ0b^p, λmin).
- Convergence: The optimization systematically approaches the exact energy as the hidden-units density α increases.Weights are initialized with small random values before iterative optimization.
Appendix B: Time-Dependent Variational Monte Carlo
Time-dependent variational Monte Carlo solves the many-body Schrödinger dynamics within the neural-network ansatz by minimizing a Fubini–Study distance, with stable evolution reported as α increases.
- Method: Time-dependent variational Monte Carlo solves the many-body Schrödinger equation using a neural-network quantum-state variational ansatz.The method is used when dynamical properties are of interest.
- Residual minimization: The residuals depend on variational-parameter derivatives and measure the distance between exactly and variationally evolved states.For unnormalized states, the relevant Hilbert-space distance is the Fubini–Study metric.
- Equations of motion: Minimizing the distance between short-time exact and variational evolutions yields the equations of motion for the network parameters.The construction uses Φ = Ψ + δ∂tΨ and Φ′ = Ψ − iδHΨ at lowest order in δ.
- Numerical treatment: The correlation matrix is inverted with its Moore-Penrose pseudo-inverse because diagonal regularization generally cannot be applied.The correlation matrix and forces are defined analogously to the ground-state procedure.
- Convergence: The time-evolved physical properties systematically approach exact results as the hidden-units density α increases.The procedure is reported to be globally stable.
Appendix C: Efficient Stochastic Sampling
The stochastic workflow samples neural-network quantum states with a Metropolis-Hastings Markov chain and uses incremental updates to keep sampling and derivative calculations efficient.
- Convergence diagnostics: Figure 5 tracks variational energy against Stochastic Reconfiguration iterates for different α values in a periodic N = 40 Heisenberg chain.The left panel shows the full optimization and the right panel zooms near the exact energy.
- Convergence diagnostics: Figure 6 tracks transverse polarization after a TFI quench and the relative residual error for different hidden-unit densities.The variational curves approach the exact solution while residual error decreases with increasing α.
- Stochastic sampling: A Markov chain samples many-body configurations according to the squared wave-function modulus |ΨM(S)|^2.Each Metropolis-Hastings step proposes flipping a random spin and accepts the new configuration probabilistically.
- Incremental updates: Look-up tables for effective angles are updated after accepted spin flips to compute acceptances and variational derivatives efficiently.The tables are updated when a new configuration is accepted.
- Computational cost: The computational cost of a Monte Carlo sweep is O(N × M) = O(αN^2), matching the cost of derivative and local-energy calculations.A sweep consists of O(N) single-spin-flip moves.
Appendix D: Iterative Solver
The dominant computational bottleneck is solving covariance-matrix linear systems, which iterative solvers reduce by avoiding explicit matrix formation.
- Bottleneck: Solving the linear systems in SR and t-VMC is the most time-consuming operation when the number of variational parameters Nvar is large.Explicitly forming the covariance matrix has quadratic cost in Nvar and the number of Monte Carlo sweeps.
- Iterative solver: MINRES-QLP reduces this cost by iteratively computing the pseudo-inverse without explicitly forming the covariance matrix.The method uses modified conjugate-gradient iterations based on Lanczos tridiagonalization.
- Complexity: The sparse solver has dominant complexity O(Nvar × NMC), enabled by efficient covariance-matrix–vector products.Without translational symmetry, Nvar = αN^2, making solver cost comparable to Monte Carlo sampling.
Appendix E: Implementing Symmetries
The paper exploits physical symmetries to reduce the number of variational parameters in neural-network quantum states. Translation invariance makes each feature act as a shared filter over translated spin configurations.
- Physical Hamiltonian symmetries can be imposed on neural-network quantum states to reduce their variational parameters.
- The symmetry construction uses linear transformations that map spin configurations to symmetry-transformed configurations while enforcing invariance of the neural-network representation.
- For each feature, the connectivity matrix contains αs × N elements, and the construction is equivalent to a standard neural-network quantum state with M = S × αs hidden variables.
- With site translation invariance, the symmetry-group orbit contains S = N elements, and each feature acts as a filter on N translated copies of a spin configuration.
- Each translation-invariant feature has a pool of N hidden variables that apply the same filter to symmetry-transformed spin images.