The experimentally reconstructe wavefunctions from the QCQMC paper are available for download from zenodo.
Setup
First install recirq:
try:
import recirq
except ImportError:
%pip install git+https://github.com/quantumlib/ReCirq
import urllib.request
import tarfile
stream = urllib.request.urlopen("https://zenodo.org/records/10141262/files/qcqmc_data.tar.gz")
with tarfile.open(fileobj=stream, mode='r|gz') as tf:
tf.extractall()
# Look at the README from the downloaded data
from IPython.display import Markdown, display
display(Markdown('qcqmc_data/README.md'))
Wavefunctions and Hamiltonians from "Unbiasing fermionic quantum Monte Carlo with a quantum computer"
This repository contains 4 directories which contain the wavefunctions and hamiltonians necessary to reproduce the results from the paper using ipie. Each system directory follows the layout:
- fqe_hams: The Hamiltonian matrix elements in hdf5 format. Note that both the active space hamiltonian (datasets: h1, h2, and e0) and the full space Hamiltonian (h1_full, LpqX_full and e0_full) are provided. LpqX_full is the Cholesky factor for the full space eris. (pq|rs) = np.einsum('pqX,rsX', LpqX, LrsX).
- fqe_wfns: The (experimentally reconstructed shadow) wavefunction in an FQE format.
- ipie_ham: The Hamiltonian in ipie format.
- ipie_wfns: The (experimentally reconstructed shadow) wavefunction in an ipie format.
We also include a script variational_energy.py which can be used to read the FQE wavefunction and compute the variational energy of the shadow wavefunction:
python variational_energy.py --ham_file diamond_dzvp/fqe_hams/chem_ham_0.800.h5 --wfn_file diamond_dzvp/fqe_wfns/wfn_0.800_50000
Variational energy (-9.121080838729604+2.7755575615628914e-17j)
which can be compared to value of -9.121081 from the first row of Table S10 in the SI. Note that for N2 the FQE and ipie variational energies differ slightly due to cholesky thresholds in the latter.
The AFQMC simulations can be run using ipie using the ipie code: https://github.com/JoonhoLee-Group/ipie, e.g.
from ipie.qmc.afqmc import AFQMC
afqmc = AFQMC.build_from_hdf5(
(nalpha, nbeta), ham_file, wfn_file, num_walkers=num_walkers, num_blocks=num_blocks
)
afqmc.run()
where nalpha, and nbeta are the number of alpha and beta electrons, ham_file and wfn_file are paths to the ipie_ham/ and ipie_wfn files respectively, and num_walkers and num_blocks are the number of walkers and number of Monte Carlo blocks.
Note the filenames typically follow a pattern of wfn_param_num_cliffords, where param may be the intermolecular distance and numcliffords is the N{Cliffords} in, for example, Table S7.
In the case of diamond note that the distances in the filenames should be multiplied by 3.6 A to obtain the first column of S10.
Loading the wavefunctions
Here we will reproduce some of the results from Section E of the SI from QCQMC.
The variational energy of the trial wavefunction as a function of the number of cliffords used for shadow tomography is reported in Table S1, which we have reproduced here:
| NCliffords | repeat 1 | repeat 2 | repeat 3 | repeat 4 |
|---|---|---|---|---|
| 10 | -1.800644 | -1.764747 | -1.813274 | -1.658202 |
| 16 | -1.823041 | -1.802192 | -1.840494 | -1.730591 |
| 28 | -1.906644 | -1.839835 | -1.843326 | -1.746749 |
| 47 | -1.925654 | -1.888527 | -1.860863 | -1.809656 |
| 80 | -1.909567 | -1.869456 | -1.887139 | -1.846339 |
| 136 | -1.930880 | -1.902309 | -1.889992 | -1.879164 |
| 229 | -1.944249 | -1.921523 | -1.903710 | -1.890947 |
| 387 | -1.947362 | -1.934682 | -1.910477 | -1.901883 |
| 652 | -1.952416 | -1.939853 | -1.912790 | -1.905250 |
| 1100 | -1.955544 | -1.944651 | -1.915073 | -1.909122 |
| 1856 | -1.955028 | -1.945966 | -1.909558 | -1.908038 |
| 3129 | -1.953877 | -1.947763 | -1.913386 | -1.908835 |
| 5276 | -1.954697 | -1.947323 | -1.912284 | -1.909315 |
| 8896 | -1.954930 | -1.947458 | -1.913889 | -1.913068 |
| 15000 | -1.954356 | -1.948894 | -1.913894 | -1.913082 |
Each column represents an independent partitioned shadow tomography experiment.
We can reproduce this table from the data by using FQE to read the wavefunctions and Hamiltonian and recompute the variational energy of the shadow trial wavefunction.
First we need to parse the wavefunction and Hamiltonian before computing the energy using FQE:
import fqe
import h5py
import numpy as np
def compute_energy(wfn_file: str, ham_file: str) -> float:
"""Compute the variational energy of the experimentally reconstructed wavefunction.
Args:
wfn_file: The path to the FQE wavefunction.
ham_file: The path to the hamiltonian.
Returns:
The variational energy
"""
fqe_wfn = fqe.wavefunction.Wavefunction()
fqe_wfn.read(filename=wfn_file)
with h5py.File(ham_file, 'r') as fh5:
ecore = fh5["e0"][()]
h1e_act = fh5["h1"][:]
eri_act = fh5["h2"][:]
# get integrals into openfermion order
of_eris = np.transpose(eri_act, (0, 2, 3, 1))
# ... and then into FQE format
fqe_ham = fqe.hamiltonians.restricted_hamiltonian.RestrictedHamiltonian((h1e_act, np.einsum('ijlk', -0.5 * of_eris)), e_0=ecore)
return fqe_wfn.expectationValue(fqe_ham)
Now we can glob all of the wavefunctions from the zenodo repo and recompute the variational energy:
import glob
import pandas as pd
for column in [1, 2, 3, 4]:
wavefunctions = glob.glob(f'qcqmc_data/h4_sto3g/fqe_wfns/column{column}/wfn*')
n_cliffords = [int(x.split('/')[-1].split('_')[-1]) for x in wavefunctions]
energies = [compute_energy(wfn_file, f'qcqmc_data/h4_sto3g/fqe_hams/column{column}/chem_ham.h5').real for wfn_file in wavefunctions]
if column == 1:
df = pd.DataFrame({'n_clifford': n_cliffords, f'repeat {column}': energies})
else:
new_df = pd.DataFrame({'n_clifford': n_cliffords, f'repeat {column}': energies})
df = pd.merge(df, new_df, on='n_clifford', how='outer')
reproduced_table = df.sort_values(by='n_clifford')
print(reproduced_table)
n_clifford repeat 1 repeat 2 repeat 3 repeat 4 0 10 -1.800644 -1.764747 -1.813274 -1.658202 1 16 -1.823041 -1.802192 -1.840494 -1.730591 2 28 -1.906644 -1.839835 -1.843326 -1.746749 3 47 -1.925654 -1.888527 -1.860863 -1.809656 4 80 -1.909567 -1.869456 -1.887139 -1.846339 5 136 -1.930880 -1.902309 -1.889992 -1.879164 6 229 -1.944249 -1.921523 -1.903710 -1.890947 7 387 -1.947362 -1.934682 -1.910477 -1.901883 8 652 -1.952416 -1.939853 -1.912790 -1.905250 9 1100 -1.955544 -1.944651 -1.915073 -1.909122 10 1856 -1.955028 -1.945966 -1.909558 -1.908038 11 3129 -1.953877 -1.947763 -1.913386 -1.908835 12 5276 -1.954697 -1.947323 -1.912284 -1.909315 13 8896 -1.954930 -1.947458 -1.913889 -1.913068 14 15000 -1.954356 -1.948894 -1.913894 -1.913082
Given these inputs we can also re-run AFQMC using the shadow wavefunctions as a trial wavefuncktion. To do so, we first need to parse the wavefunction and hamiltonian which is provided in ipie format in the zenodo repo:
from ipie.qmc.afqmc import AFQMC
num_elec = (2, 2) # H4 molecule, 4 electrons, ms = 0
n_clifford = 15_000
ham_file = 'qcqmc_data/h4_sto3g/ipie_ham/column3/ham.h5'
wfn_file = f'qcqmc_data/h4_sto3g/ipie_wfns/column3/wfn_{n_clifford}.h5'
afqmc = AFQMC.build_from_hdf5(num_elec, ham_file, wfn_file, num_blocks=1_000, num_walkers=100)
afqmc.trial.calculate_energy(afqmc.system, afqmc.hamiltonian)
afqmc.run(estimator_filename=f'h4_sto3g_n_clifford_{n_clifford}.h5')
# Have shared memory: False
# Time to read integrals: 0.004739
# Time to pack Cholesky vectors: 0.472707
# Number of orbitals: 4
# Approximate memory required by Cholesky vectors 0.000002 GB
# Approximate memory required by packed Cholesky vectors 0.000001 GB
# Approximate memory required total 0.000003 GB
# Number of Cholesky vectors: 16
# Number of fields: 16
# Finished setting up GenericRealChol object.
# Using generalized Wick's theorem for the PHMSD trial
# Number of determinants in trial: 36
# Setting the first determinant in expansion as the reference wfn for Wick's theorem.
# Number of frozen orbitals: 0
# Number of occupied electrons in active space trial: (2, 2)
# Number of orbitals in active space trial: 4
# Setting additional member variables for Wick's theorem
# Reference alpha determinant: [0 2]
# Reference beta determinant: [0 2]
# Number of alpha determinants at each level: [5, 24, 6]
# Number of beta determinants at each level: [5, 24, 6]
# Computing 1-RDM of the trial wfn for mean-field shift.
# Using first 36 determinants for evaluation.
# Tr(G_T): (4.000000000000001-4.710946609976203e-18j)
# Constructing half rotated Cholesky vectors.
# Shape of alpha half-rotated Cholesky: (1, 16, 8)
# Shape of beta half-rotated Cholesky: (1, 16, 8)
# Half-Rotating Cholesky for determinant.
# Constructing half rotated Cholesky vectors.
# Shape of alpha half-rotated Cholesky: (1, 16, 16)
# Shape of beta half-rotated Cholesky: (1, 16, 16)
# Half-Rotating Cholesky for determinant.
# random seed is 25966688
# Computing trial wavefunction energy.
# Variational energy of trial wavefunction: -1.9138939545263898
# Using pair_branch population control algorithm.
# target weight is 100
# total weight is 100
# ipie version: 0.7.1
# Calculation uuid: bb17cb1e-4e54-11ef-982f-d38e1b77be85.
# Approximate memory available per node: 117.9093 GB.
# Running on 1 MPI rank.
# Root processor name: kokoro-gcp-ubuntu-prod-91717751
# Python interpreter: 3.10.14 (main, Apr 6 2024, 18:45:05) [GCC 9.4.0]
# Using numpy v1.26.4 from: /tmpfs/src/tf_docs_env/lib/python3.10/site-packages/numpy.
# - BLAS lib: openblas64
# - BLAS version: 0.3.23.dev
# - BLAS include directory: /usr/local/lib
# Using scipy v1.14.0 from: /tmpfs/src/tf_docs_env/lib/python3.10/site-packages/scipy.
# Using h5py v3.11.0 from: /tmpfs/src/tf_docs_env/lib/python3.10/site-packages/h5py.
# Package mpi4py not found.
# Package cupy not found.
# MPI communicator : <class 'ipie.qmc.comm.FakeComm'>
# Available memory on the node is 117.909 GB
# Setting up estimator object.
# Writing estimator data to h4_sto3g_n_clifford_15000.h5
# Finished settting up estimator object.
Block Weight WeightFactor HybridEnergy ENumer EDenom ETotal E1Body E2Body
0 1.0000000000000000e+02 1.0000000000000000e+02 0.0000000000000000e+00 -1.9330645767050612e+02 1.0000000000000000e+02 -1.9330645767050612e+00 -4.3439318841445109e+00 2.4108673074394495e+00
1 1.0044840551495349e+02 1.0094905343575985e+02 -4.7906424091774608e-01 -1.9402332567974247e+02 1.0000000000000000e+02 -1.9402332567974248e+00 -4.3373974575984668e+00 2.3971642008010416e+00
2 9.9992381990174110e+01 1.0019449857195362e+02 -4.8811436844486600e-01 -1.9413652784791685e+02 9.9999999999999986e+01 -1.9413652784791688e+00 -4.3308547170547413e+00 2.3894894385755721e+00
3 9.9996608985369136e+01 9.9975091327327036e+01 -4.9675060479817051e-01 -1.9398097232282879e+02 1.0000000000000001e+02 -1.9398097232282876e+00 -4.3273895088187295e+00 2.3875797855904421e+00
4 9.9992003417193317e+01 9.9966483749364414e+01 -4.9474693335935999e-01 -1.9388274328142143e+02 9.9999999999999986e+01 -1.9388274328142145e+00 -4.3246353645822024e+00 2.3858079317679874e+00
5 9.9992689065177970e+01 9.9990150571134507e+01 -4.9965313145427520e-01 -1.9409326635864849e+02 1.0000000000000001e+02 -1.9409326635864845e+00 -4.3198235034172052e+00 2.3788908398307207e+00
6 9.9965475754283872e+01 9.9926216127460847e+01 -4.8298902271334371e-01 -1.9392182109505495e+02 9.9999999999999972e+01 -1.9392182109505502e+00 -4.3134754117010088e+00 2.3742572007504585e+00
7 9.9994360716791931e+01 9.9971696851330350e+01 -4.8881745482240779e-01 -1.9459491783209302e+02 1.0000000000000000e+02 -1.9459491783209302e+00 -4.3117000496842568e+00 2.3657508713633271e+00
8 1.0000203634518667e+02 9.9997031170855124e+01 -4.9870953141613522e-01 -1.9500565685072095e+02 1.0000000000000000e+02 -1.9500565685072095e+00 -4.3024541543641126e+00 2.3523975858569033e+00
9 9.9986560519495470e+01 9.9960218819181733e+01 -4.9616478694366595e-01 -1.9466713592431634e+02 9.9999999999999972e+01 -1.9466713592431641e+00 -4.3044645370414134e+00 2.3577931777982499e+00
10 9.9983365478920533e+01 9.9955547610356604e+01 -4.9263571033786452e-01 -1.9488565108055224e+02 1.0000000000000003e+02 -1.9488565108055218e+00 -4.3001696570236483e+00 2.3513131462181263e+00
11 9.9996208293318077e+01 9.9991577942243552e+01 -4.9965517249672486e-01 -1.9576685950525734e+02 1.0000000000000003e+02 -1.9576685950525727e+00 -4.2957289819841851e+00 2.3380603869316121e+00
12 1.0000038219286020e+02 9.9994315059002517e+01 -5.1604617125208141e-01 -1.9608080343405089e+02 1.0000000000000000e+02 -1.9608080343405090e+00 -4.2913032054983971e+00 2.3304951711578874e+00
13 9.9983894032048596e+01 9.9980506706425558e+01 -5.1586505013621176e-01 -1.9624033415957740e+02 1.0000000000000000e+02 -1.9624033415957740e+00 -4.2886379147826004e+00 2.3262345731868259e+00
14 9.9968998000720703e+01 9.9935482281504491e+01 -5.0103942695576920e-01 -1.9629658611618794e+02 1.0000000000000000e+02 -1.9629658611618794e+00 -4.2848546452452503e+00 2.3218887840833697e+00
15 9.9984651410143357e+01 9.9951966573856666e+01 -5.0024383517119764e-01 -1.9559991874262363e+02 9.9999999999999986e+01 -1.9559991874262366e+00 -4.2759970188138698e+00 2.3199978313876333e+00
16 9.9991678875670758e+01 9.9972757618634603e+01 -5.0926609340452755e-01 -1.9633889804309828e+02 9.9999999999999972e+01 -1.9633889804309836e+00 -4.2805151908788215e+00 2.3171262104478383e+00
17 9.9994103346450359e+01 9.9981766335984233e+01 -5.1389695559512816e-01 -1.9721042793718905e+02 1.0000000000000001e+02 -1.9721042793718901e+00 -4.2747844514640709e+00 2.3026801720921806e+00
18 9.9989135940274764e+01 9.9967807592645258e+01 -5.2101085998521590e-01 -1.9802600957077604e+02 1.0000000000000000e+02 -1.9802600957077605e+00 -4.2780784595307040e+00 2.2978183638229437e+00
19 9.9992709747317917e+01 1.0000233525585328e+02 -5.3607442038761233e-01 -1.9746410757492490e+02 9.9999999999999986e+01 -1.9746410757492494e+00 -4.2772893493054625e+00 2.3026482735562124e+00
20 9.9973881349781678e+01 9.9946956170603102e+01 -5.2967894763484746e-01 -1.9768780982469568e+02 1.0000000000000001e+02 -1.9768780982469565e+00 -4.2760832303392746e+00 2.2992051320923190e+00
21 9.9965124537332144e+01 9.9908904195788196e+01 -5.1346332967970210e-01 -1.9821863005784203e+02 1.0000000000000003e+02 -1.9821863005784197e+00 -4.2764076873739993e+00 2.2942213867955799e+00
22 9.9997840023995849e+01 1.0000796401685443e+02 -5.3400563175117421e-01 -1.9816713845131005e+02 9.9999999999999986e+01 -1.9816713845131009e+00 -4.2787035782154064e+00 2.2970321937023055e+00
23 9.9984252138909255e+01 9.9942232509974858e+01 -5.3066075380987132e-01 -1.9837865770749991e+02 1.0000000000000001e+02 -1.9837865770749987e+00 -4.2843116210561059e+00 2.3005250439811062e+00
24 9.9965639797594179e+01 9.9925999094868658e+01 -5.1842542871841957e-01 -1.9752002166981296e+02 1.0000000000000001e+02 -1.9752002166981293e+00 -4.2874458750981006e+00 2.3122456583999709e+00
25 9.9975072555273130e+01 9.9932440703585200e+01 -5.1155613425320645e-01 -1.9724343734011069e+02 9.9999999999999986e+01 -1.9724343734011072e+00 -4.2906716161303899e+00 2.3182372427292828e+00
26 9.9987330426705753e+01 9.9980008067910717e+01 -5.3023117329641323e-01 -1.9791365670241723e+02 9.9999999999999972e+01 -1.9791365670241730e+00 -4.2876416136854747e+00 2.3085050466613009e+00
27 9.9951240060471918e+01 9.9896314813192362e+01 -5.1790228369025804e-01 -1.9778915299350956e+02 1.0000000000000001e+02 -1.9778915299350952e+00 -4.2889491097761399e+00 2.3110575798410444e+00
28 9.9982349731332647e+01 9.9942425369609339e+01 -5.3543577034068379e-01 -1.9731663629275641e+02 1.0000000000000000e+02 -1.9731663629275642e+00 -4.2911225077060697e+00 2.3179561447785040e+00
29 9.9977238408446354e+01 9.9967222677803036e+01 -5.3194964302100267e-01 -1.9742523259617224e+02 1.0000000000000000e+02 -1.9742523259617224e+00 -4.2938179776005647e+00 2.3195656516388428e+00
30 9.9972166004473507e+01 9.9927682060773392e+01 -5.1961365868612197e-01 -1.9769273839896013e+02 1.0000000000000001e+02 -1.9769273839896011e+00 -4.2908038640498303e+00 2.3138764800602289e+00
31 9.9997691879264735e+01 9.9968858287104013e+01 -5.3471725550160298e-01 -1.9822250819167763e+02 9.9999999999999972e+01 -1.9822250819167770e+00 -4.2868086825730698e+00 2.3045836006562932e+00
32 9.9967891397537969e+01 9.9936743470592305e+01 -5.2553481387816547e-01 -1.9872127280623070e+02 9.9999999999999986e+01 -1.9872127280623073e+00 -4.2880174094246097e+00 2.3008046813623011e+00
33 9.9979533912465328e+01 9.9922539554086654e+01 -5.2154565929158991e-01 -1.9872696391453249e+02 9.9999999999999986e+01 -1.9872696391453253e+00 -4.2868889974458542e+00 2.2996193583005295e+00
34 9.9969520118273834e+01 9.9955017755617774e+01 -5.1904488827544148e-01 -1.9704268694374164e+02 9.9999999999999986e+01 -1.9704268694374167e+00 -4.2872657601001576e+00 2.3168388906627415e+00
35 9.9984679769875456e+01 9.9956745976557031e+01 -5.2150898726237060e-01 -1.9676142172232122e+02 1.0000000000000000e+02 -1.9676142172232123e+00 -4.2867411123955916e+00 2.3191268951723796e+00
36 9.9979386596302263e+01 9.9945060311090586e+01 -5.1762063550107096e-01 -1.9697244687297291e+02 1.0000000000000001e+02 -1.9697244687297288e+00 -4.2915945226319705e+00 2.3218700539022419e+00
37 9.9985890382596878e+01 9.9973584401407336e+01 -5.2270793759946221e-01 -1.9693634101971949e+02 1.0000000000000001e+02 -1.9693634101971946e+00 -4.2905778781541350e+00 2.3212144679569398e+00
38 9.9971958322943919e+01 9.9932015347663310e+01 -5.0993916300696429e-01 -1.9664654928376174e+02 1.0000000000000000e+02 -1.9664654928376175e+00 -4.2875540857935697e+00 2.3210885929559528e+00
39 9.9977406252912488e+01 9.9939114020784700e+01 -5.1110299331631015e-01 -1.9660208607436584e+02 9.9999999999999986e+01 -1.9660208607436589e+00 -4.2901012666226777e+00 2.3240804058790192e+00
40 9.9988107952643134e+01 9.9978942713291730e+01 -5.1790195235886483e-01 -1.9661036678354992e+02 9.9999999999999986e+01 -1.9661036678354995e+00 -4.2836947660784999e+00 2.3175910982430010e+00
41 9.9979788160977932e+01 9.9952535080722953e+01 -5.1763894890909978e-01 -1.9648230697631939e+02 1.0000000000000000e+02 -1.9648230697631939e+00 -4.2869399355251110e+00 2.3221168657619176e+00
42 9.9985712121736142e+01 9.9960481387687040e+01 -5.2932208275349324e-01 -1.9686017515561923e+02 1.0000000000000000e+02 -1.9686017515561924e+00 -4.2900338121293764e+00 2.3214320605731835e+00
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943 9.9960578585272273e+01 9.9898025291292058e+01 -5.2796891615635932e-01 -1.9851644891130491e+02 9.9999999999999986e+01 -1.9851644891130495e+00 -4.2898485532738189e+00 2.3046840641607695e+00
944 9.9920338156874109e+01 1.0029311723049783e+02 -5.0826926144792384e-01 -1.9714839487985492e+02 9.9999999999999972e+01 -1.9714839487985498e+00 -4.2965727120772410e+00 2.3250887632786914e+00
945 9.9980583206785852e+01 9.9943054758733538e+01 -5.1637567682374996e-01 -1.9718528259504166e+02 1.0000000000000003e+02 -1.9718528259504160e+00 -4.2979540641335765e+00 2.3261012381831603e+00
946 9.9964497683018280e+01 9.9926118889061243e+01 -5.1891844632431861e-01 -1.9712440322182496e+02 1.0000000000000000e+02 -1.9712440322182496e+00 -4.2909559675222351e+00 2.3197119353039866e+00
947 9.9969733832050068e+01 9.9908049387960475e+01 -5.2935819747604684e-01 -1.9705680330270968e+02 1.0000000000000003e+02 -1.9705680330270963e+00 -4.3030198305114791e+00 2.3324517974843828e+00
948 9.9961708074391794e+01 9.9907384773178677e+01 -5.2106721671424483e-01 -1.9675997848689443e+02 9.9999999999999986e+01 -1.9675997848689446e+00 -4.3039854309669083e+00 2.3363856460979631e+00
949 9.9968362749030362e+01 9.9917400008293313e+01 -5.1022428608590953e-01 -1.9662799633163633e+02 1.0000000000000000e+02 -1.9662799633163632e+00 -4.3066202761067629e+00 2.3403403127903992e+00
950 9.9986337139394749e+01 9.9962532485934915e+01 -5.3304542663892651e-01 -1.9670962574541343e+02 1.0000000000000001e+02 -1.9670962574541340e+00 -4.2988485671243808e+00 2.3317523096702470e+00
951 9.9938654389754248e+01 9.9885986213932156e+01 -5.0442582028755323e-01 -1.9689398904860397e+02 1.0000000000000000e+02 -1.9689398904860398e+00 -4.2948266409608982e+00 2.3258867504748588e+00
952 9.9985004618489043e+01 9.9928659699410233e+01 -5.2031543117068213e-01 -1.9735139192910873e+02 1.0000000000000001e+02 -1.9735139192910869e+00 -4.2906642223336435e+00 2.3171503030425571e+00
953 9.9976158595240818e+01 9.9957390344670600e+01 -5.3822518924442109e-01 -1.9739150482458635e+02 1.0000000000000003e+02 -1.9739150482458629e+00 -4.2793486148590860e+00 2.3054335666132220e+00
954 9.9940212550627606e+01 9.9879190753953040e+01 -5.2041838588280409e-01 -1.9737445360448413e+02 1.0000000000000000e+02 -1.9737445360448413e+00 -4.2768054194344236e+00 2.3030608833895823e+00
955 9.9973773140504051e+01 9.9925619834435238e+01 -5.3636775708135453e-01 -1.9701763488236287e+02 1.0000000000000000e+02 -1.9701763488236288e+00 -4.2837159007644496e+00 2.3135395519408202e+00
956 9.9943733292371604e+01 9.9845568199364777e+01 -5.1334919785789568e-01 -1.9780270666612043e+02 1.0000000000000001e+02 -1.9780270666612041e+00 -4.2748226135804925e+00 2.2967955469192889e+00
957 9.9969687251498200e+01 9.9882378059736027e+01 -5.4825754226196755e-01 -1.9806842440793289e+02 1.0000000000000000e+02 -1.9806842440793291e+00 -4.2729226018569566e+00 2.2922383577776277e+00
958 9.9947309853503739e+01 9.9900924538948715e+01 -5.3987658986322562e-01 -1.9705139143264628e+02 9.9999999999999986e+01 -1.9705139143264632e+00 -4.2759558630947527e+00 2.3054419487682902e+00
959 9.9946374360371451e+01 9.9829368842961870e+01 -5.1138640191604490e-01 -1.9717856715523794e+02 9.9999999999999986e+01 -1.9717856715523798e+00 -4.2759705136425890e+00 2.3041848420902102e+00
960 9.9964022408972966e+01 9.9885884229718627e+01 -5.1855711622994527e-01 -1.9690519739804949e+02 1.0000000000000001e+02 -1.9690519739804946e+00 -4.2777576943698241e+00 2.3087057203893298e+00
961 9.9938209326101074e+01 9.9882048128372958e+01 -4.9595408066224256e-01 -1.9704823614280500e+02 1.0000000000000000e+02 -1.9704823614280500e+00 -4.2812884741164838e+00 2.3108061126884341e+00
962 1.0000255359362620e+02 9.9976564236211033e+01 -5.2513762111640616e-01 -1.9650284516839898e+02 1.0000000000000001e+02 -1.9650284516839895e+00 -4.2870542894607704e+00 2.3220258377767817e+00
963 9.9964910189429474e+01 9.9913935536473460e+01 -5.1412609709855939e-01 -1.9648441603591453e+02 9.9999999999999986e+01 -1.9648441603591458e+00 -4.2874520455264191e+00 2.3226078851672733e+00
964 9.9974202946545319e+01 9.9941500024492726e+01 -5.2002342628454257e-01 -1.9686854298038412e+02 9.9999999999999986e+01 -1.9686854298038416e+00 -4.2957554885246720e+00 2.3270700587208299e+00
965 9.9959040252420550e+01 9.9910539460192922e+01 -5.0452294275864196e-01 -1.9656258943827820e+02 9.9999999999999986e+01 -1.9656258943827822e+00 -4.2940276161971038e+00 2.3284017218143216e+00
966 9.9995043482811909e+01 9.9968978565837332e+01 -5.3256770586125768e-01 -1.9699860945749498e+02 1.0000000000000000e+02 -1.9699860945749499e+00 -4.2969617423610256e+00 2.3269756477860750e+00
967 9.9955140794147482e+01 9.9911142672681194e+01 -5.2717714490254675e-01 -1.9755734030071346e+02 9.9999999999999986e+01 -1.9755734030071350e+00 -4.3023408825465959e+00 2.3267674795394617e+00
968 9.9977281249295046e+01 9.9934062621393409e+01 -5.2690893207233780e-01 -1.9734278730322794e+02 9.9999999999999986e+01 -1.9734278730322798e+00 -4.3022515022355217e+00 2.3288236292032423e+00
969 9.9974125986092176e+01 9.9938472408728472e+01 -5.2277513188134472e-01 -1.9711455636686608e+02 9.9999999999999986e+01 -1.9711455636686612e+00 -4.3014838266638851e+00 2.3303382629952236e+00
970 9.9958175274440876e+01 9.9909646806068977e+01 -5.1043884920850480e-01 -1.9714206273725634e+02 1.0000000000000001e+02 -1.9714206273725632e+00 -4.3010493028940706e+00 2.3296286755215077e+00
971 9.9972724126921534e+01 9.9934367433147784e+01 -5.1534909392192141e-01 -1.9647356672938156e+02 1.0000000000000001e+02 -1.9647356672938152e+00 -4.2999407028561309e+00 2.3352050355623155e+00
972 9.9967979493479220e+01 9.9934998641796497e+01 -5.1500435523222954e-01 -1.9598298206718826e+02 1.0000000000000003e+02 -1.9598298206718821e+00 -4.2976524897021733e+00 2.3378226690302908e+00
973 9.9958361431987584e+01 9.9920893646861117e+01 -5.0605679502145884e-01 -1.9590036538160388e+02 1.0000000000000000e+02 -1.9590036538160389e+00 -4.2956533701809327e+00 2.3366497163648945e+00
974 9.9971808870256936e+01 1.0047008490710323e+02 -5.1176847877588416e-01 -1.9573085551022169e+02 1.0000000000000001e+02 -1.9573085551022165e+00 -4.2971470701576102e+00 2.3398385150553924e+00
975 9.9966512812092148e+01 9.9925653599757609e+01 -5.0023450030094552e-01 -1.9635284156603640e+02 9.9999999999999972e+01 -1.9635284156603647e+00 -4.2982133325402074e+00 2.3346849168798411e+00
976 9.9967498564823515e+01 9.9926424025970334e+01 -5.0175975349103719e-01 -1.9600203885633712e+02 1.0000000000000001e+02 -1.9600203885633709e+00 -4.3034043658681780e+00 2.3433839773048075e+00
977 9.9985110261805332e+01 9.9960903746050661e+01 -5.1220150174495949e-01 -1.9606656811378153e+02 1.0000000000000001e+02 -1.9606656811378149e+00 -4.3015981436159008e+00 2.3409324624780861e+00
978 9.9967331531048501e+01 9.9915340827582767e+01 -5.1046767979001462e-01 -1.9642455454592124e+02 9.9999999999999986e+01 -1.9642455454592127e+00 -4.3007857489925554e+00 2.3365402035333425e+00
979 9.9961464881491210e+01 9.9928725793080744e+01 -5.0668210640114941e-01 -1.9636653145251688e+02 9.9999999999999986e+01 -1.9636653145251692e+00 -4.3008886400622481e+00 2.3372233255370798e+00
980 9.9965526860254982e+01 9.9915450675908673e+01 -5.0717282760810711e-01 -1.9683853281687669e+02 1.0000000000000003e+02 -1.9683853281687662e+00 -4.3027761799977577e+00 2.3343908518289913e+00
981 9.9951494532001149e+01 9.9899097104728980e+01 -5.0146051178007445e-01 -1.9772393361980215e+02 1.0000000000000001e+02 -1.9772393361980212e+00 -4.3045034085805236e+00 2.3272640723825018e+00
982 9.9967839901429443e+01 9.9912837115124645e+01 -5.0891581266394714e-01 -1.9694068674691070e+02 1.0000000000000001e+02 -1.9694068674691068e+00 -4.3072000054816568e+00 2.3377931380125512e+00
983 9.9992193498914503e+01 9.9983255409884094e+01 -5.3555603422618914e-01 -1.9702163664825730e+02 1.0000000000000000e+02 -1.9702163664825731e+00 -4.2986585986987702e+00 2.3284422322161973e+00
984 9.9944321438622026e+01 9.9878825830368257e+01 -5.1226121175045514e-01 -1.9649070327352135e+02 9.9999999999999986e+01 -1.9649070327352138e+00 -4.2981277721203082e+00 2.3332207393850939e+00
985 9.9976116260992640e+01 9.9943880663076641e+01 -5.2181547491846592e-01 -1.9646727608645307e+02 9.9999999999999986e+01 -1.9646727608645311e+00 -4.2983389447654412e+00 2.3336661839009105e+00
986 9.9951828456033553e+01 9.9894029516740844e+01 -4.9632188362889490e-01 -1.9673524876632413e+02 1.0000000000000001e+02 -1.9673524876632411e+00 -4.2977442060521387e+00 2.3303917183888974e+00
987 9.9962655055043200e+01 9.9915664566296712e+01 -4.9894966613502961e-01 -1.9721537925174596e+02 1.0000000000000001e+02 -1.9721537925174593e+00 -4.2992400480853190e+00 2.3270862555678602e+00
988 9.9995405173817872e+01 9.9983043023571227e+01 -5.3042038950543546e-01 -1.9639016674252704e+02 1.0000000000000000e+02 -1.9639016674252705e+00 -4.2960015095689918e+00 2.3320998421437205e+00
989 9.9912137677484068e+01 9.9826220672401959e+01 -5.1257519915011973e-01 -1.9510053447924940e+02 1.0000000000000001e+02 -1.9510053447924938e+00 -4.2978825884695633e+00 2.3468772436770688e+00
990 9.9943302183469257e+01 9.9824886186697896e+01 -4.8840804874600735e-01 -1.9487145589745617e+02 1.0000000000000000e+02 -1.9487145589745618e+00 -4.3008462229614484e+00 2.3521316639868872e+00
991 9.9981341009453288e+01 9.9943153824432116e+01 -5.0331808269049261e-01 -1.9481874518488399e+02 9.9999999999999986e+01 -1.9481874518488402e+00 -4.3070273455508499e+00 2.3588398937020094e+00
992 9.9980582835063998e+01 9.9959742564952833e+01 -5.1666350604724021e-01 -1.9560885541440368e+02 1.0000000000000001e+02 -1.9560885541440365e+00 -4.3071114926275795e+00 2.3510229384835428e+00
993 9.9965646326152367e+01 9.9921684009934111e+01 -5.0928295737972640e-01 -1.9577088818644577e+02 9.9999999999999986e+01 -1.9577088818644581e+00 -4.3112791541483739e+00 2.3535702722839154e+00
994 9.9976192754234560e+01 9.9924059158642535e+01 -5.0859876654571168e-01 -1.9607809182558108e+02 9.9999999999999986e+01 -1.9607809182558111e+00 -4.3055090805334242e+00 2.3447281622776130e+00
995 9.9965393357737852e+01 9.9916139089242719e+01 -5.0763703190172738e-01 -1.9579258466673727e+02 9.9999999999999972e+01 -1.9579258466673735e+00 -4.3076610240487456e+00 2.3497351773813722e+00
996 9.9966734124668932e+01 1.0058945172687605e+02 -5.1156908798279899e-01 -1.9683912098660616e+02 1.0000000000000003e+02 -1.9683912098660610e+00 -4.3093369174833134e+00 2.3409457076172537e+00
997 9.9982433838614057e+01 9.9941979279643419e+01 -5.2119282790149069e-01 -1.9766239457377606e+02 9.9999999999999972e+01 -1.9766239457377612e+00 -4.3126230670019536e+00 2.3359991212641917e+00
998 9.9985089066416847e+01 9.9955118148884367e+01 -5.3125739667077987e-01 -1.9782890158890939e+02 9.9999999999999986e+01 -1.9782890158890942e+00 -4.3042658887031280e+00 2.3259768728140324e+00
999 9.9969632083037013e+01 9.9932675304680686e+01 -5.3884006550808972e-01 -1.9910269032634218e+02 1.0000000000000000e+02 -1.9910269032634218e+00 -4.3024156370146596e+00 2.3113887337512371e+00
1000 9.9955602135887588e+01 9.9895023264070716e+01 -5.3546116505182706e-01 -1.9854319744883011e+02 9.9999999999999957e+01 -1.9854319744883018e+00 -4.3079953441296812e+00 2.3225633696413803e+00
Finally we can analyze the result and compare it to the literature value which we agree with within error bars.
from ipie.analysis.extraction import extract_observable
import matplotlib.pyplot as plt
from ipie.analysis.blocking import reblock_minimal
paper_result = -1.96922
analyzed_result = reblock_minimal(afqmc.estimators.filename, start_block=100)
notebook_energy = analyzed_result.ETotal_ac.values[0]
notebook_error = analyzed_result.ETotal_error_ac.values[0]
data = extract_observable(afqmc.estimators.filename)
plt.plot(data.ETotal, marker='o', label="AFQMC", color="C0")
plt.axhline(paper_result, label="Exact Result", color="C1")
plt.axhline(notebook_energy, label="Notebook result", color="C2")
plt.legend()
plt.xlabel("Block number")
plt.ylabel("Total Energy (Ha)")
print(f"paper result = {paper_result}")
print(f"notebook result = {notebook_energy} +/ {notebook_error}")
paper result = -1.96922 notebook result = -1.9678660760529634 +/ 0.0008669053611567947
