neural-network-quantum-states-grand-canonical

name	neural-network-quantum-states-grand-canonical
description	Neural network quantum state (NQS) architecture for grand canonical ensemble bosonic systems. Enables variational Monte Carlo with variable particle number in Fock space. Activation: neural quantum states, grand canonical ensemble, bosonic wavefunctions, Fock space, variational Monte Carlo, NQS, quantum many-body ground state.

Neural Network Quantum States in Grand Canonical Ensemble

Variational Monte Carlo with neural quantum states capable of representing symmetric bosonic wavefunctions in Fock space, enabling study of systems with variable particle number.

Metadata

• Source: arXiv:2605.07779
• Authors: Anton Hul, Matija Medvidović, Juan Carrasquilla
• Published: 2026-05-08
• Category: Quantum Physics / Machine Learning

Core Methodology

Key Innovation

Extends neural quantum states (NQS) from fixed-particle-number systems to the grand canonical ensemble, enabling:

• Variable particle number systems under chemical potential control
• Symmetric bosonic wavefunctions in Fock space
• Direct computation of one-body reduced density matrices
• Access to observables like condensate fractions and radial density profiles

Technical Framework

1. Fock Space Architecture: Neural network operates in Fock space rather than fixed-particle Hilbert space
2. Symmetric Bosonic States: Enforces bosonic exchange symmetry in the wavefunction representation
3. Monte Carlo Sampling: Combines variational state with Monte Carlo sampling for efficient evaluation
4. Geometric Optimization: Uses geometric optimization for variational energy minimization
5. Chemical Potential Control: System converges to physical boson number under set chemical potential

Results

• Competitive variational energies across 1D and 2D systems
• Converges to correct physical boson number
• Accurate one-body reduced density matrices
• Access to condensate fractions and radial density profiles

Implementation Guide

Prerequisites

• Neural network framework (PyTorch/JAX)
• Monte Carlo sampling infrastructure
• Variational Monte Carlo implementation

Step-by-Step

1. Fock Space Representation: Encode many-body states as occupation number vectors in Fock space
2. Symmetric Architecture: Design neural network that respects bosonic exchange symmetry
3. Chemical Potential: Include μN term in Hamiltonian for grand canonical ensemble
4. MC Sampling: Sample configurations weighted by |ψ(n)|² where n is occupation number
5. Optimization: Minimize E = ⟨ψ|H - μN|ψ⟩/⟨ψ|ψ⟩ using stochastic reconfiguration
6. Observables: Compute 1-RDM and extract condensate fraction

Code Pattern

# Grand canonical NQS energy (conceptual)
def grand_canonical_energy(nqs, hamiltonian, mu, mc_samples):
    """Compute E = ⟨H - μN⟩ in grand canonical ensemble."""
    # Sample occupation number configurations
    configs = sample_fock_space(nqs, mc_samples)
    # Local energy: (H - μN)ψ / ψ
    local_energies = compute_local_energy(nqs, configs, hamiltonian, mu)
    return torch.mean(local_energies), torch.var(local_energies) / len(configs)

Applications

• Bosonic quantum many-body systems
• Bose-Einstein condensates
• Quantum phase transitions with variable particle number
• Superfluid systems
• Quantum optics with variable photon number

Related Skills

• universal-neural-propagator-quantum-dynamics
• quantum-neural-network-data-loading
• quantum-ml-patterns

Pitfalls

• Fock space dimensionality grows rapidly with system size
• MC sampling efficiency depends on wavefunction quality
• Chemical potential tuning requires careful convergence checks