We learned in the high-level notebook how to setup the quantum side of the calculation, which involves specifying a Hamiltonian, defining a trial wavefunction ansatz before performing shadow tomography to extract the AFQMC trial wavefunction. In this example, we will repeat these steps again for the case of H4 which was studied in the paper. Then we will outline how to interface the output of the quantum half of the calculation with ipie to perform AFQMC with the quantum trial wavefunction.
Setup
First install recirq:
try:
import recirq
except ImportError:
%pip install git+https://github.com/quantumlib/ReCirq
Define an SCF Job
We first setup a PyscfHamiltonianParams object which defines the SCF job we will run using pyscf. Here we are simulation H4 in the (minimal) sto-3g basis.
from recirq.qcqmc.hamiltonian import PyscfHamiltonianParams, HamiltonianData
pyscf_params = PyscfHamiltonianParams(
name="TEST_square_H4",
n_orb=4,
n_elec=4,
geometry=(("H", (0, 0, 0)), ("H", (0, 0, 1.23)), ("H", (1.23, 0, 0)), ("H", (1.23, 0, 1.23))),
basis="sto3g",
multiplicity=1,
charge=0,
save_chkfile=True,
overwrite_chk_file=True,
)
pyscf_hamiltonian = HamiltonianData.build_hamiltonian_from_pyscf(pyscf_params)
chk_path = pyscf_params.base_path.with_suffix(".chk")
Build Perfect Pairing Wavefunction
Next we build a perfect pairing wavefunction based upon our pyscf_params
import numpy as np
from recirq.qcqmc.trial_wf import (
PerfectPairingPlusTrialWavefunctionParams,
TrialWavefunctionData
)
pp_params = PerfectPairingPlusTrialWavefunctionParams(
'pp_plus_test',
hamiltonian_params=pyscf_params,
heuristic_layers=tuple(),
do_pp=True,
restricted=False,
initial_orbital_rotation=None,
initial_two_body_qchem_amplitudes=np.asarray([0.3, 0.4]),
do_optimization=True,
use_fast_gradients=True
)
trial_wf = TrialWavefunctionData.build_pp_plus_trial_from_dependencies(
pp_params, dependencies={pyscf_params: pyscf_hamiltonian}, do_print=True
)
Building Trial Wavefunction
Optimization restart 0
energy (-0.9036540062626459+0j), max|grad| 0.4055778794357425
energy (-1.2562368588436836+0j), max|grad| 0.43313791305139393
.energy (-1.5679374247963844+7.888609052210118e-31j), max|grad| 0.18111782362662004
.energy (-1.6837246665467358+0j), max|grad| 0.17092036783044715
.energy (-1.6110669589234181+0j), max|grad| 0.24059875134725753
energy (-1.7390726256610076-7.888609052210118e-31j), max|grad| 0.18367926366417667
.energy (-1.7998087490345682-1.5777218104420236e-30j), max|grad| 0.1517612491290922
.energy (-1.85760482706117+0j), max|grad| 0.11159886713967626
.energy (-1.8729342965890314+7.888609052210118e-31j), max|grad| 0.1216392843358692
.energy (-1.8833608675750924-3.944304526105059e-31j), max|grad| 0.04744855668934871
.energy (-1.8915925472404727+0j), max|grad| 0.043464113681721546
.energy (-1.896150318701043+0j), max|grad| 0.06268423948369739
.energy (-1.8984417758128584+0j), max|grad| 0.029698836620385877
.energy (-1.8994746204928887+1.5777218104420236e-30j), max|grad| 0.016201854102027454
.energy (-1.899957201496921+0j), max|grad| 0.013304133970140785
.energy (-1.9008511510081867+2.341930812374879e-31j), max|grad| 0.013590539661908211
.energy (-1.9024862069591846-1.9721522630525295e-31j), max|grad| 0.01776285826411418
.energy (-1.9052419928245046+7.888609052210118e-31j), max|grad| 0.03273195822714455
.energy (-1.9077304802944566-1.5777218104420236e-30j), max|grad| 0.06430668054864518
.energy (-1.9109588448091852-1.5777218104420236e-30j), max|grad| 0.0456020728724604
.energy (-1.9167439088358718+1.5777218104420236e-30j), max|grad| 0.030847960479975674
.energy (-1.9245237883922313+9.860761315262648e-32j), max|grad| 0.05417187731080087
.energy (-1.9296292702179798+5.9164567891575885e-31j), max|grad| 0.025592029356486093
.energy (-1.9311761633082778-3.944304526105059e-31j), max|grad| 0.029638055385661645
.energy (-1.933215914812636+3.1554436208840472e-30j), max|grad| 0.01621076088582836
.energy (-1.9334865022840528+0j), max|grad| 0.05061349409744201
energy (-1.9343719716192522+0j), max|grad| 0.01912404951615153
.energy (-1.9355257556500183-1.5777218104420236e-30j), max|grad| 0.015830794257891478
.energy (-1.935838772234188+0j), max|grad| 0.0037725673337152825
.energy (-1.9359314607080256+1.1832913578315177e-30j), max|grad| 0.003200160407726419
.energy (-1.935949268389063+1.5777218104420236e-30j), max|grad| 0.0018919063116268726
.energy (-1.9359611849206226-1.5777218104420236e-30j), max|grad| 0.0015724061558558594
.energy (-1.9359718407742434-1.5777218104420236e-30j), max|grad| 0.002200219431710977
.energy (-1.9359908818544875+0j), max|grad| 0.0027990086027876444
.energy (-1.9360254775825618-3.944304526105059e-31j), max|grad| 0.0034162005029662727
.energy (-1.9360872529493829+7.888609052210118e-31j), max|grad| 0.004092624834665993
.energy (-1.9361891445655823-7.888609052210118e-31j), max|grad| 0.004789327611432818
.energy (-1.9363129739874951+7.888609052210118e-31j), max|grad| 0.006438470215631584
.energy (-1.936409828806081-6.902532920683853e-31j), max|grad| 0.004456081295501152
.energy (-1.9365429132704115+4.930380657631324e-32j), max|grad| 0.0021233065187949984
.energy (-1.9365688598431472+1.5777218104420236e-30j), max|grad| 0.0005844751564333227
.energy (-1.936569194425596+0j), max|grad| 0.0006212049182390798
.energy (-1.9365697065818228-1.9721522630525295e-31j), max|grad| 0.000396322023845012
.energy (-1.9365703915500676-1.5777218104420236e-30j), max|grad| 0.00035847268845497156
.energy (-1.9365715608439755+7.888609052210118e-31j), max|grad| 0.0005875338345701087
.energy (-1.9365736450508706-7.888609052210118e-31j), max|grad| 0.0009275529494447987
.energy (-1.936577364856836+0j), max|grad| 0.001290087498159842
.energy (-1.9365837600802127+1.5777218104420236e-30j), max|grad| 0.0016835767447708205
.energy (-1.9365936870844989+7.888609052210118e-31j), max|grad| 0.002066433013532154
.energy (-1.9366057394306582+3.944304526105059e-31j), max|grad| 0.002171053716593041
.energy (-1.9366157496909846+0j), max|grad| 0.0015707204870063496
.energy (-1.9366194954891824+1.5777218104420236e-30j), max|grad| 0.00036139858612830895
.energy (-1.9366196680654788+1.5777218104420236e-30j), max|grad| 0.00011315080525932123
.energy (-1.9366197270827676-1.5777218104420236e-30j), max|grad| 6.430515389602392e-05
.energy (-1.9366197458722914-3.944304526105059e-31j), max|grad| 7.331252496905818e-05
.energy (-1.9366197673277403+7.888609052210118e-31j), max|grad| 4.9780835403821e-05
.energy (-1.9366197715277558-7.888609052210118e-31j), max|grad| 2.5640979216920167e-05
.energy (-1.9366197732374273+4.733165431326071e-30j), max|grad| 2.658368168230504e-05
.energy (-1.9366197751441474+0j), max|grad| 2.8157268404369024e-05
.energy (-1.9366197785809476+1.5777218104420236e-30j), max|grad| 4.345122893493406e-05
.energy (-1.93661978495203+7.888609052210118e-31j), max|grad| 6.284515144451495e-05
.energy (-1.936619796829076+3.944304526105059e-31j), max|grad| 8.776692882146378e-05
.energy (-1.9366198187972676-3.1554436208840472e-30j), max|grad| 0.00012194120079247437
.energy (-1.9366198585693368-3.1554436208840472e-30j), max|grad| 0.0001701334299161085
.energy (-1.9366199276036058+1.5777218104420236e-30j), max|grad| 0.0002373872242281216
.energy (-1.9366200380462466+0j), max|grad| 0.00032367056761343577
.energy (-1.9366201880972573+0j), max|grad| 0.0003985907193036062
.energy (-1.9366203391133157+0j), max|grad| 0.00038025048350678084
.energy (-1.9366205555842493-3.1554436208840472e-30j), max|grad| 0.00023727562931326692
.energy (-1.93662064426736+3.944304526105059e-31j), max|grad| 6.282714761231305e-05
.energy (-1.9366206498974865-3.944304526105059e-31j), max|grad| 1.2028184630336895e-05
.energy (-1.936620650696911-4.733165431326071e-30j), max|grad| 3.6026151661679147e-06
. message: Optimization terminated successfully.
success: True
status: 0
fun: -1.936620650696911
x: [ 1.529e+00 1.601e+00 ... 1.504e+00 4.454e-01]
nit: 69
jac: [ 4.615e-07 -1.193e-06 ... 1.574e-07 -8.279e-08]
hess_inv: [[ 3.209e-01 -1.605e-02 ... -1.289e-01 3.218e-01]
[-1.605e-02 2.131e+00 ... -1.185e+00 2.527e+00]
...
[-1.289e-01 -1.185e+00 ... 6.735e+01 -1.023e+02]
[ 3.218e-01 2.527e+00 ... -1.023e+02 1.811e+02]]
nfev: 72
njev: 72
Hartree-Fock Energy:
(-1.779243269890757+0j)
Sector N = 4 : S_z = 0
a'0011'b'0011' (1+0j)
--------------------------------------------------------------------------------
FCI Energy:
-1.9695121652162788
--------------------------------------------------------------------------------
message: Optimization terminated successfully.
success: True
status: 0
fun: -1.936620650696911
x: [ 1.529e+00 1.601e+00 ... 1.504e+00 4.454e-01]
nit: 69
jac: [ 4.615e-07 -1.193e-06 ... 1.574e-07 -8.279e-08]
hess_inv: [[ 3.209e-01 -1.605e-02 ... -1.289e-01 3.218e-01]
[-1.605e-02 2.131e+00 ... -1.185e+00 2.527e+00]
...
[-1.289e-01 -1.185e+00 ... 6.735e+01 -1.023e+02]
[ 3.218e-01 2.527e+00 ... -1.023e+02 1.811e+02]]
nfev: 72
njev: 72
--------------------------------------------------------------------------------
Ansatz Energy:
-1.936620650696911
Sector N = 4 : S_z = 0
a'0011'b'0011' (-0.46387401399583944-1.6169791200448813e-15j)
a'0011'b'0101' (-0.49319604867937417-1.4418533641341602e-15j)
a'0011'b'1010' (0.09453242557421507+2.408351838269207e-16j)
a'0011'b'1100' (0.10050769884960999+3.006492624302304e-16j)
a'0101'b'0011' (0.4932121133952673+1.240300180269105e-15j)
a'0101'b'0101' (0.4638724215893809+9.917773254129042e-16j)
a'0101'b'1010' (-0.10051252390245798-1.836909530733566e-16j)
a'0101'b'1100' (-0.09453345932512375-2.362883109660596e-16j)
a'1010'b'0011' (-0.09453244426471545-3.956524777942727e-16j)
a'1010'b'0101' (-0.10050753163472044-4.301436594064678e-16j)
a'1010'b'1010' (0.040495383211038925+1.1647041837925531e-16j)
a'1010'b'1100' (0.043055478757829016+1.3775127414607236e-16j)
a'1100'b'0011' (0.10051091749766967+4.610840789037729e-16j)
a'1100'b'0101' (0.0945318069870978+4.036484688163533e-16j)
a'1100'b'1010' (-0.04305727968792324-1.1641620827063104e-16j)
a'1100'b'1100' (-0.04049580296637009-1.3711430536973712e-16j)
Basis Rotation Matrix:
[[-0.0019289 +0.j 0. +0.j 0.20844642+0.j 0. +0.j
0.97799642+0.j 0. +0.j -0.00832942+0.j 0. +0.j]
[ 0. +0.j -0.0017375 +0.j 0. +0.j -0.20844971+0.j
0. +0.j -0.97801093+0.j 0. +0.j -0.00635044+0.j]
[-0.70709082+0.j 0. +0.j -0.0055334 +0.j 0. +0.j
-0.00623724+0.j 0. +0.j -0.70707358+0.j 0. +0.j]
[ 0. +0.j -0.70708361+0.j 0. +0.j 0.00480579+0.j
0. +0.j 0.00482323+0.j 0. +0.j -0.70709717+0.j]
[-0.70707664+0.j 0. +0.j -0.00587992+0.j 0. +0.j
0.0058808 +0.j 0. +0.j 0.70708802+0.j 0. +0.j]
[ 0. +0.j 0.70709486+0.j 0. +0.j -0.00514814+0.j
0. +0.j 0.00443232+0.j 0. +0.j -0.70708607+0.j]
[-0.00784059+0.j 0. +0.j 0.97800046+0.j 0. +0.j
-0.20844549+0.j 0. +0.j 0.0020259 +0.j 0. +0.j]
[ 0. +0.j 0.00682626+0.j 0. +0.j 0.97800773+0.j
0. +0.j -0.20845076+0.j 0. +0.j -0.00160098+0.j]]
Two Body Rotation Parameters:
[1.52877842 1.60143649]
We can comare the wavefunction of the optimized ansatz to the exact ground state energy:
print(f"Trial wavefunction energy: {trial_wf.ansatz_energy}")
print(f"Exact ground state energy: {trial_wf.fci_energy}")
Trial wavefunction energy: -1.9366206506969077 Exact ground state energy: -1.9695121652162788
Next, we build an experiment which will combine the trial wavefunction circuit with that required for shadow tomography. In this case the experiment will be simulated using a statevector simulator.
from recirq.qcqmc.blueprint import BlueprintParamsTrialWf, BlueprintData
from recirq.qcqmc.experiment import SimulatedExperimentParams, ExperimentData
blueprint_params = BlueprintParamsTrialWf(
name="blueprint_test_medium",
trial_wf_params=pp_params,
n_cliffords=100,
qubit_partition=(tuple(qubit for qubit in pp_params.qubits_jordan_wigner_ordered),),
seed=1,
)
blueprint = BlueprintData.build_blueprint_from_dependencies(blueprint_params, dependencies={pp_params: trial_wf})
simulated_experiment_params = SimulatedExperimentParams(
name="test_1",
blueprint_params=blueprint.params,
noise_model_name="None",
noise_model_params=(0,),
n_samples_per_clifford=31,
seed=1,
)
experiment = ExperimentData.build_experiment_from_dependencies(
params=simulated_experiment_params, dependencies={blueprint.params: blueprint}
)
The experimental output is post-processed to extract the reconstructed trial wavefunction:
from typing import Dict
from recirq.qcqmc.data import Params, Data
from recirq.qcqmc.analysis import OverlapAnalysisData, OverlapAnalysisParams
analysis_params = OverlapAnalysisParams(
"TEST_analysis", experiment_params=experiment.params, k_to_calculate=(1,)
)
all_dependencies: Dict[Params, Data] = {
pyscf_params: pyscf_hamiltonian,
pp_params: trial_wf,
blueprint_params: blueprint,
simulated_experiment_params: experiment,
}
analysis = OverlapAnalysisData.build_analysis_from_dependencies(
analysis_params, dependencies=all_dependencies
)
Finally, we save the wavefunction in a format that is acceptable for ipie.
from recirq.qcqmc.convert_to_ipie import save_wavefunction_for_ipie
ipie_data = save_wavefunction_for_ipie(
hamiltonian_data=pyscf_hamiltonian, trial_wf_data=trial_wf, overlap_analysis_data=analysis, do_print=False
)
print(f"Reconstructed shadow wavefunction energy: {ipie_data.variational_energy}")
print(f"Ideal trial wavefunction energy: {trial_wf.ansatz_energy}")
print(f"FCI energy: {trial_wf.fci_energy}")
Reconstructed shadow wavefunction energy: -1.8771197609628356 Ideal trial wavefunction energy: -1.9366206506969077 FCI energy: -1.9695121652162788
Run AFQMC
Now that we have built our quantum wavefunction, we can use it as a trial wavefunction in an AFQMC simulation. First, we need to build a factorized Hamiltonian which is required for AFQMC.
In particular, we require the three-index Cholesky tensor (ham.chol below) which satisfies
\[ (pq|rs) = \sum_X L_{pq}^X L_{rs}^X. \]
from ipie.systems.generic import Generic
from ipie.utils.from_pyscf import generate_hamiltonian_from_chk
num_elec = (pyscf_params.n_elec // 2,) * 2
system = Generic(num_elec)
chk_path = pyscf_params.base_path.with_suffix(".chk")
ham = generate_hamiltonian_from_chk(str(chk_path))
assert ham.H1.shape == (2, pyscf_params.n_orb, pyscf_params.n_orb)
assert ham.chol.shape[0] == pyscf_params.n_orb * pyscf_params.n_orb
Next, we need to build a trial wavefunction from quantum wavefunction. In practice, the quantum trial is expanded as a linear combination of (orthogonal) Slater Determinants. Within ipie, this style of trial wavefunction is defined as a ParticleHole trial wavefunction.
import h5py
from ipie.trial_wavefunction.particle_hole import ParticleHole
# Read quantum wavefunction from file.
with h5py.File(ipie_data.path, 'r') as fh5:
coeffs = fh5["coeffs_rotated"][:]
occa = fh5["occa_rotated"][:]
occb = fh5["occb_rotated"][:]
wfn = ParticleHole((coeffs, occa, occb), num_elec, pyscf_params.n_orb)
wfn.half_rotate(ham)
wfn.calculate_energy(system, ham)
print(f"Trial wavefunction variational energy in ipie: {wfn.energy.real}")
assert np.isclose(wfn.energy.real, ipie_data.variational_energy)
Trial wavefunction variational energy in ipie: -1.8771197609628336
Note that the variational energy might differ slightly from the result compute from the previous section. This is because the cholesky factorization uses a threshold of \(1\times10^{-5}\) a stopping criteria for convergence. Reducing the parameter chol_cut in generate_hamiltonian_from_chk will yield better agreement.
At this point, we are ready to run AFQMC. We need to build an AFQMC driver class which takes as input the factorized Hamiltonian and our ParticleHole trial wavefunction.
from ipie.qmc.afqmc import AFQMC
qmc_driver = AFQMC.build(num_elec, ham, wfn, num_blocks=300, num_walkers=50, seed=7)
# random seed is 7
from recirq.qcqmc.config import OUTDIRS
import pathlib
ipie_path = pathlib.Path(pyscf_hamiltonian.params.path_prefix + OUTDIRS.DEFAULT_QMC_DIRECTORY)
if not ipie_path.is_dir():
ipie_path.mkdir(parents=True)
qmc_driver.run(estimator_filename=f'{ipie_path}/estimates.h5')
# Using pair_branch population control algorithm.
# target weight is 50
# total weight is 50
# ipie version: 0.7.1
# Calculation uuid: 24501ff2-ec50-11ef-b8b6-4735a95cd5b3.
# Approximate memory available per node: 117.9093 GB.
# Running on 1 MPI rank.
# Root processor name: kokoro-gcp-ubuntu-prod-2033379579
# Python interpreter: 3.10.16 (main, Dec 4 2024, 08:53:37) [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 v1.15.1 from: /tmpfs/src/tf_docs_env/lib/python3.10/site-packages/scipy.
# Using h5py v3.12.1 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 data/afqmc/estimates.h5
# Finished settting up estimator object.
Block Weight WeightFactor HybridEnergy ENumer EDenom ETotal E1Body E2Body
0 5.0000000000000000e+01 5.0000000000000000e+01 0.0000000000000000e+00 -9.6486304211488687e+01 5.0000000000000000e+01 -1.9297260842297739e+00 -4.3439319178107789e+00 2.4142058335810050e+00
1 5.0244433021531144e+01 5.0518752257436617e+01 -5.1480637820850661e-01 -9.6652905726715318e+01 4.9999999999999993e+01 -1.9330581145343066e+00 -4.3381857300470426e+00 2.4051276155127361e+00
2 4.9972807280256106e+01 5.0051684340529746e+01 -4.7661057456165529e-01 -9.7306249370556429e+01 5.0000000000000000e+01 -1.9461249874111286e+00 -4.3279047904644310e+00 2.3817798030533019e+00
3 5.0011569797916898e+01 5.0004270514043455e+01 -5.1187080315422628e-01 -9.7466683634549312e+01 5.0000000000000007e+01 -1.9493336726909860e+00 -4.3253085724128253e+00 2.3759748997218391e+00
4 4.9980812287753643e+01 4.9970432248304306e+01 -4.9087031780061963e-01 -9.7093058372178888e+01 4.9999999999999993e+01 -1.9418611674435782e+00 -4.3194201294604841e+00 2.3775589620169058e+00
5 4.9994801257977485e+01 4.9974671017998880e+01 -4.8858192683037621e-01 -9.7291505142812056e+01 4.9999999999999993e+01 -1.9458301028562415e+00 -4.3163812727238051e+00 2.3705511698675630e+00
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Next, we can visualize the data which is by default saved to estimates.0.h5.
from ipie.analysis.extraction import extract_observable
import matplotlib.pyplot as plt
data = extract_observable(qmc_driver.estimators.filename)
plt.plot(data.ETotal, marker='o', label="AFQMC", color="C0")
plt.axhline(-1.969512, label="Exact Result", color="C1")
plt.legend()
plt.xlabel("Block number")
plt.ylabel("Total Energy (Ha)")
Text(0, 0.5, 'Total Energy (Ha)')
Through visual inspection, we can determine when the calculation has equilibrated, after which we should be sampling the (approximate) ground state. We can use samples from this point on to estimate the AFQMC energy and error bar. The function reblock_minimal will perform the necessary error analysis taking into account the serial temporal correlation in the AFQMC data.
from ipie.analysis.blocking import reblock_minimal
reblock_minimal(qmc_driver.estimators.filename, start_block=20)
Note that the number of walkers and the number of blocks is probably too low to obtain statistically significant results and we would advise increasing both of these. For larger scale simulations, we can use MPI to distribute the work among many MPI processes. See ipie for further details.
Next steps
The results presented above are worse than reported in the paper. In practice one needs to:
- Carefully converge the trial wavefunction parameters by scanning over multiple random restarts.
- Converge the reconstructed shadow wavefunction with respect to:
n_cliffords - The AFQMC part of the simulations can be sped up using MPI parallelism. See ipie for more information. The number of walkers should also be increased to obtain better statistics and reduce any population control biases. The number of samples (
num_blocks) should also be increased (ETotal_nsamp_acshould be about 100 for statistically significant results.)