Data supporting "Simple Fermionic backflow states via a systematically improvable tensor decomposition"

This repository contains numerical data used to generate the plots in the paper Simple Fermionic Backflow States via a Systematically Improvable Tensor Decomposition, which introduces the CPD backflow ansatz.

The data corresponds to variational quantum Monte Carlo (VMC) calculations for different ansatze or quantum chemistry methods on various fermionic systems, including the Fermi-Hubbard model, water molecule, and hydrogen lattice systems. The methods used to obtain ground state energies are specified as follows:

• variational Monte Carlo (VMC) with the CPD backflow ansatz using support dimension M=1 and no exchange cutoff
• VMC with the CPD backflow ansatz using support dimension M=1 and an exchange cutoff K=5
• VMC with the Gaussian Process state (GPS) multiplied by a Slater determinant using support dimension M=72
• density matrix renormalization group (DMRG) with the matrix product state tensor network using bond dimension M=1024
• unrestricted CCSD (UCCSD)
• restricted Hartree-Fock (RHF)

Each dataset is stored in CSV format, with columns containing numerical results for specific quantities.

Data Description

1. hubbard2d_U8.csv

• Description: VMC ground state energies obtained by the CPD backflow ansatz on the 4x4 Fermi-Hubbard model at U/t = 8 (Figure 2).
• Columns:
• index: Row index.
• filling: Electron filling fraction (e.g., 1.0 for half-filled, <1 for doped).
• M: Support dimension of the CPD ansatz.
• n_samples: Number of samples used in optimization.
• energy: Computed ground-state energy (Hartree).
• rel_error: Relative energy error compared to exact diagonalization.
• rel_errorbar: Error bar for the relative energy error.
• n_params: Number of variational parameters in the model.

2. h2o.csv

• Description: VMC ground state energies obtained by the CPD backflow ansatz on the water molecule in the 6-31G basis set (Figure 3 and 4)
• Columns:
• index: Row index.
• M: Support dimension of the CPD backflow ansatz.
• n_samples: Number of samples used in the optimization.
• energy: Computed ground-state energy (in Hartree).
• rel_corr_error: Relative correlation energy error compared to exact diagonalization.
• rel_corr_errorbar: Error bar for the relative correlation energy error.
• n_params: Number of variational parameters in the model.

3. hsheet.csv

• Description: Ground state energies for the 6x6 hydrogen lattice in the STO-6G basis set at different interatomic distances obtained by the CPD backflow and GPS times SD ansatze, DMRG, UCCSD and RHF methods (Figure 5).
• Columns:
• index: Row index.
• distance: Interatomic distance (Å).
• M: Support dimension of the CPD backflow ansatz, GPS times SD or DMRG.
• exchange_cutoff: Exchange cutoff applied to the CPD backflow ansatz.
• n_samples: Number of samples used in the optimization.
• energy: Computed ground-state energy (Hartree).
• errorbar: Standard error of the computed energy.
• n_params: Number of variational parameters.
• uuid: Unique identifier for the CPD backflow model.

4. hsheet_corr.csv

• Description: Correlation function data for the CPD backflow ansatz on the 6x6 hydrogen lattice at two different interatomic distances (Figure 6).
• Columns:
• index: Row index.
• distance: Interatomic distance (Å).
• r: Radial distance between hydrogen atoms in the lattice.
• C: Computed spin-spin correlation function.

5. runtime.csv

• Description: Runtime data of the VMC iteration for the scaling analysis of the CPD backflow ansatz performed on progressively longer hydrogen chains (Figure 7)
• Columns:
• index: Row index.
• n_atoms: Number of hydrogen atoms in the system.
• pruning_threshold: Energy threshold used to prune Hamiltonian elements.
• runtime: Mean runtime per VMC optimization step (seconds).
• error: Standard error of the runtime measurement.

6. Reduced density matrices

• Description: Reduced density matrices for the CPD backflow wavefunctions on the 6x6 hydrogen lattice at interatomic distance 1.2Å and 3.0Å used to compute the correlation function data in Figure 6

Usage Notes

• The files are formatted as comma-separated values (CSV) and can be loaded using Python (pandas), MATLAB, or Excel.
• Energy values are in Hartree (Eh) unless specified otherwise.
• The data can be used to regenerate plots and validate the numerical results presented in the paper using the Python scripts in the plots folder of the main GitHub repository.

For any questions or further clarification, please contact the corresponding authors.