How to simulate the local unitary cluster Jastrow (LUCJ) ansatz

In this guide, we show how to use ffsim to simulate the local unitary cluster Jastrow (LUCJ) ansatz. We’ll use it to calculate an approximation to the ground state energy of an ethene molecule.

First, let’s build the molecule.

import pyscf
import pyscf.mcscf

import ffsim

# Build an ethene molecule
bond_distance = 1.339
a = 0.5 * bond_distance
b = a + 0.5626
c = 0.9289
mol = pyscf.gto.Mole()
mol.build(
    atom=[
        ["C", (0, 0, a)],
        ["C", (0, 0, -a)],
        ["H", (0, c, b)],
        ["H", (0, -c, b)],
        ["H", (0, c, -b)],
        ["H", (0, -c, -b)],
    ],
    basis="sto-6g",
    symmetry="d2h",
)

# Define active space
active_space = range(mol.nelectron // 2 - 2, mol.nelectron // 2 + 2)

# Get molecular data and molecular Hamiltonian (one- and two-body tensors)
scf = pyscf.scf.RHF(mol).run()
mol_data = ffsim.MolecularData.from_scf(scf, active_space=active_space)
norb = mol_data.norb
nelec = mol_data.nelec
mol_hamiltonian = mol_data.hamiltonian

# Compute FCI energy
mol_data.run_fci()

print(f"norb = {norb}")
print(f"nelec = {nelec}")

converged SCF energy = -77.8266321248745
Parsing /tmp/tmp9c8z1d5x
converged SCF energy = -77.8266321248744
CASCI E = -77.8742165643862  E(CI) = -4.02122442107772  S^2 = 0.0000000
norb = 4
nelec = (2, 2)

Overwritten attributes  get_hcore get_ovlp  of <class 'pyscf.scf.hf_symm.SymAdaptedRHF'>
/home/runner/work/ffsim/ffsim/.tox/docs/lib/python3.13/site-packages/pyscf/gto/mole.py:1300: UserWarning: Function mol.dumps drops attribute energy_nuc because it is not JSON-serializable
  warnings.warn(msg)
/home/runner/work/ffsim/ffsim/.tox/docs/lib/python3.13/site-packages/pyscf/gto/mole.py:1300: UserWarning: Function mol.dumps drops attribute intor_symmetric because it is not JSON-serializable
  warnings.warn(msg)

General UCJ ansatz

Since our molecule has a closed-shell Hartree-Fock state, we’ll use the spin-balanced variant of the UCJ ansatz, UCJOpSpinBalanced. We’ll initialize the ansatz from t2 amplitudes obtained from a CCSD calculations. We’ll first demonstrate the general UCJ ansatz, without adding the locality constraints of the LUCJ ansatz just yet.

The following code cell initializes the ansatz operator, applies it to the Hartree-Fock state, and computes the energy of the resulting state.

import numpy as np
from pyscf import cc

# Get CCSD t2 amplitudes for initializing the ansatz
ccsd = cc.CCSD(
    scf, frozen=[i for i in range(mol.nao_nr()) if i not in active_space]
).run()

# Construct UCJ operator
n_reps = 2
operator = ffsim.UCJOpSpinBalanced.from_t_amplitudes(ccsd.t2, t1=ccsd.t1, n_reps=n_reps)

# Construct the Hartree-Fock state to use as the reference state
reference_state = ffsim.hartree_fock_state(norb, nelec)

# Apply the operator to the reference state
ansatz_state = ffsim.apply_unitary(reference_state, operator, norb=norb, nelec=nelec)

# Compute the energy ⟨ψ|H|ψ⟩ of the ansatz state
hamiltonian = ffsim.linear_operator(mol_hamiltonian, norb=norb, nelec=nelec)
energy = np.real(np.vdot(ansatz_state, hamiltonian @ ansatz_state))
print(f"Energy at initialization: {energy}")

E(CCSD) = -77.8742153637403  E_corr = -0.04758323886584176
Energy at initialization: -77.8716002481628

To variationally optimize the ansatz, we’ll take advantage of methods for conversion to and from real-valued parameter vectors. In the following code cell, we define an objective function that takes a parameter vector as input and outputs the energy of the associated ansatz state. We then optimize this objective function using scipy.optimize.minimize, with an initial guess obtained from the operator we initialized previously from t2 amplitudes.

import scipy.optimize


def fun(x):
    # Initialize the ansatz operator from the parameter vector
    operator = ffsim.UCJOpSpinBalanced.from_parameters(
        x, norb=norb, n_reps=n_reps, with_final_orbital_rotation=True
    )
    # Apply the ansatz operator to the reference state
    final_state = ffsim.apply_unitary(reference_state, operator, norb=norb, nelec=nelec)
    # Return the energy ⟨ψ|H|ψ⟩ of the ansatz state
    return np.real(np.vdot(final_state, hamiltonian @ final_state))


result = scipy.optimize.minimize(
    fun, x0=operator.to_parameters(), method="L-BFGS-B", options=dict(maxiter=10)
)

print(f"Number of parameters: {len(result.x)}")
print(result)

Number of parameters: 88
  message: STOP: TOTAL NO. OF ITERATIONS REACHED LIMIT
  success: False
   status: 1
      fun: -77.87387316400324
        x: [-4.772e-01  1.609e-04 ... -1.330e-04 -1.251e-04]
      nit: 10
      jac: [ 3.979e-05 -3.411e-05 ... -4.690e-05 -2.274e-05]
     nfev: 1157
     njev: 13
 hess_inv: <88x88 LbfgsInvHessProduct with dtype=float64>

LUCJ ansatz

Now, let’s add locality constraints to simulate the LUCJ ansatz. We’ll restrict same-spin interactions to a line topology, and opposite-spin interactions to those within the same spatial orbital. As explained in The local unitary cluster Jastrow (LUCJ) ansatz, these constraints allow the ansatz to be simulated directly on a square lattice.

In the following code cell, we demonstrate the optimization of the ansatz with these constraints imposed. Notice that with the same number of ansatz repetitions, the number of parameters of the ansatz has decreased from 88 to 62.

pairs_aa = [(p, p + 1) for p in range(norb - 1)]
pairs_ab = [(p, p) for p in range(norb)]
interaction_pairs = (pairs_aa, pairs_ab)


def fun(x):
    operator = ffsim.UCJOpSpinBalanced.from_parameters(
        x,
        norb=norb,
        n_reps=n_reps,
        interaction_pairs=interaction_pairs,
        with_final_orbital_rotation=True,
    )
    final_state = ffsim.apply_unitary(reference_state, operator, norb=norb, nelec=nelec)
    return np.real(np.vdot(final_state, hamiltonian @ final_state))


result = scipy.optimize.minimize(
    fun,
    x0=operator.to_parameters(interaction_pairs=interaction_pairs),
    method="L-BFGS-B",
    options=dict(maxiter=10),
)
print(f"Number of parameters: {len(result.x)}")
print(result)

Number of parameters: 62
  message: CONVERGENCE: RELATIVE REDUCTION OF F <= FACTR*EPSMCH
  success: True
   status: 0
      fun: -77.8736342618358
        x: [-4.773e-01  3.145e-05 ... -4.922e-06 -9.234e-05]
      nit: 6
      jac: [ 2.416e-05  1.279e-05 ...  8.527e-06  4.263e-06]
     nfev: 567
     njev: 9
 hess_inv: <62x62 LbfgsInvHessProduct with dtype=float64>

Optimize with the linear method

ffsim includes an implementation of the “linear method” for optimization of a variational wavefunction. The linear method often converges faster than a standard optimization algorithm like L-BFGS-B. The interface is similar to that of scipy.optimize.minimize, the main difference being that instead of passing a callable that directly returns the function value to be optimized, you pass two objects: a callable that returns the wavefunction, and the Hamiltonian representing the energy to be optimized as a LinearOperator. The code cell below shows how to use the linear method to optimize the LUCJ ansatz from the previous example. It also shows how you can use an optional callback function to save intermediate results of the optimization.

from collections import defaultdict

from ffsim.optimize import minimize_linear_method


# Define function that converts a list of parameters to the corresponding state vector
def params_to_vec(x: np.ndarray) -> np.ndarray:
    operator = ffsim.UCJOpSpinBalanced.from_parameters(
        x,
        norb=norb,
        n_reps=n_reps,
        interaction_pairs=interaction_pairs,
        with_final_orbital_rotation=True,
    )
    return ffsim.apply_unitary(reference_state, operator, norb=norb, nelec=nelec)


# Define a callback function used to save optimization information (this is optional)
info = defaultdict(list)


def callback(intermediate_result: scipy.optimize.OptimizeResult):
    # The callback function is called after each iteration. It accepts
    # an OptimizeResult object storing the parameters and function value at
    # the current iteration, and possibly other information
    info["x"].append(intermediate_result.x)
    info["fun"].append(intermediate_result.fun)
    if hasattr(intermediate_result, "jac"):
        info["jac"].append(intermediate_result.jac)
    if hasattr(intermediate_result, "regularization"):
        info["regularization"].append(intermediate_result.regularization)
    if hasattr(intermediate_result, "variation"):
        info["variation"].append(intermediate_result.variation)


# Optimize with the linear method
result = minimize_linear_method(
    params_to_vec,
    hamiltonian,
    x0=operator.to_parameters(interaction_pairs=interaction_pairs),
    maxiter=10,
    callback=callback,
)

# Print some information
print(f"Number of parameters: {len(result.x)}")
print(result)
print()
for i, (fun, jac, regularization, variation) in enumerate(
    zip(info["fun"], info["jac"], info["regularization"], info["variation"])
):
    print(f"Iteration {i + 1}")
    print(f"    Energy: {fun}")
    print(f"    Norm of gradient: {np.linalg.norm(jac)}")
    print(f"    Regularization hyperparameter: {np.linalg.norm(regularization)}")
    print(f"    Variation hyperparameter: {np.linalg.norm(variation)}")

Number of parameters: 62
 message: Convergence: Relative reduction of objective function <= ftol.
 success: True
     fun: -77.87421564188935
       x: [-5.043e-01  5.016e-02 ...  2.268e-03 -1.365e-03]
     nit: 8
     jac: [ 5.313e-06 -4.480e-06 ...  1.814e-06  2.268e-06]
    nfev: 1441
    njev: 9
  nlinop: 883

Iteration 1
    Energy: -77.87362796891745
    Norm of gradient: 0.0018590488589383023
    Regularization hyperparameter: 0.0017995680852471736
    Variation hyperparameter: 0.9884767357094786
Iteration 2
    Energy: -77.87363431545435
    Norm of gradient: 0.0001237135851606635
    Regularization hyperparameter: 0.0017995680852471736
    Variation hyperparameter: 0.9884767357094786
Iteration 3
    Energy: -77.87363531334196
    Norm of gradient: 0.0001588191629325453
    Regularization hyperparameter: 0.0017995680784933692
    Variation hyperparameter: 0.9884767357091488
Iteration 4
    Energy: -77.87384100078171
    Norm of gradient: 0.013205057893690248
    Regularization hyperparameter: 0.00021008211183626177
    Variation hyperparameter: 0.9884881731562698
Iteration 5
    Energy: -77.8741816773828
    Norm of gradient: 0.005232716355327388
    Regularization hyperparameter: 0.0012270267367815912
    Variation hyperparameter: 0.988519838721456
Iteration 6
    Energy: -77.87421385988726
    Norm of gradient: 0.0012748110907321963
    Regularization hyperparameter: 0.00026421804084141324
    Variation hyperparameter: 0.9885625686311561
Iteration 7
    Energy: -77.87421539729331
    Norm of gradient: 0.00017951420482045173
    Regularization hyperparameter: 0.00026421820154174855
    Variation hyperparameter: 0.9885625687801975
Iteration 8
    Energy: -77.87421564188935
    Norm of gradient: 0.00010078059951467942
    Regularization hyperparameter: 0.00026421820154174855
    Variation hyperparameter: 0.9885625687801975