How to build Qiskit circuits for Trotterized Hamiltonian simulation

This guide provides some examples of building and transpiling Qiskit circuits for Trotterized time evolution of various Hamiltonians.

import numpy as np
import pyscf
from qiskit.circuit import QuantumCircuit, QuantumRegister
from qiskit.providers.fake_provider import GenericBackendV2
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager

import ffsim

Molecular Hamiltonian in the double-factorized representation

# Build N2 molecule
mol = pyscf.gto.Mole()
mol.build(
    atom=[["N", (0, 0, 0)], ["N", (1.0, 0, 0)]],
    basis="sto-6g",
    symmetry="Dooh",
)

# Define active space
n_frozen = pyscf.data.elements.chemcore(mol)
active_space = range(n_frozen, mol.nao_nr())

# Get molecular data and Hamiltonian
scf = pyscf.scf.RHF(mol).run()
mol_data = ffsim.MolecularData.from_scf(scf, active_space=active_space)
norb, nelec = mol_data.norb, mol_data.nelec
mol_hamiltonian = mol_data.hamiltonian
print(f"norb = {norb}")
print(f"nelec = {nelec}")

# Get Hamiltonian in double-factorized representation
df_hamiltonian = ffsim.DoubleFactorizedHamiltonian.from_molecular_hamiltonian(
    mol_hamiltonian
)

# Construct circuit
# For the initial state, take the Hartree-Fock state
qubits = QuantumRegister(2 * norb, name="q")
circuit = QuantumCircuit(qubits)
circuit.append(ffsim.qiskit.PrepareHartreeFockJW(norb, nelec), qubits)
circuit.append(
    ffsim.qiskit.SimulateTrotterDoubleFactorizedJW(
        df_hamiltonian, time=1.0, n_steps=1, order=0
    ),
    qubits,
)
circuit.measure_all()

# Create a generic backend and transpile the circuit to it
backend = GenericBackendV2(2 * norb, basis_gates=["cp", "xx_plus_yy", "p", "x"])
pass_manager = generate_preset_pass_manager(optimization_level=3, backend=backend)
# Set the pre-initialization stage of the pass manager with passes suggested by ffsim
pass_manager.pre_init = ffsim.qiskit.PRE_INIT
transpiled = pass_manager.run(circuit)

transpiled.count_ops()

converged SCF energy = -108.464957764796
norb = 8
nelec = (5, 5)

OrderedDict([('xx_plus_yy', 1946),
             ('cp', 1506),
             ('p', 462),
             ('measure', 16),
             ('x', 10),
             ('barrier', 1)])

Fermi-Hubbard model with split-operator method

# Build Hubbard model and set number of electrons
norb_x = 4
norb_y = 4
norb = norb_x * norb_y
nelec = (2, 2)
n_alpha, n_beta = nelec
op = ffsim.fermi_hubbard_2d(
    norb_x=norb_x,
    norb_y=norb_y,
    tunneling=1.0,
    interaction=4.0,
    chemical_potential=0.0,
    periodic_x=True,
    periodic_y=False,
)

# Convert the FermionOperator to a DiagonalCoulombHamiltonian
dc_hamiltonian = ffsim.DiagonalCoulombHamiltonian.from_fermion_operator(op)

# Construct circuit
# For the initial state, take the ground state of the one-body part of the Hamiltonian
qubits = QuantumRegister(2 * norb, name="q")
circuit = QuantumCircuit(qubits)
eigs, orbital_rotation = np.linalg.eigh(dc_hamiltonian.one_body_tensor)
circuit.append(
    ffsim.qiskit.PrepareSlaterDeterminantJW(
        norb,
        occupied_orbitals=(range(n_alpha), range(n_beta)),
        orbital_rotation=orbital_rotation,
    ),
    qubits,
)
circuit.append(
    ffsim.qiskit.SimulateTrotterDiagCoulombSplitOpJW(
        dc_hamiltonian, time=1.0, n_steps=1, order=0
    ),
    qubits,
)
circuit.measure_all()

# Create a generic backend and transpile the circuit to it
backend = GenericBackendV2(2 * norb, basis_gates=["cp", "xx_plus_yy", "p", "x"])
pass_manager = generate_preset_pass_manager(optimization_level=3, backend=backend)
# Set the pre-initialization stage of the pass manager with passes suggested by ffsim
pass_manager.pre_init = ffsim.qiskit.PRE_INIT
transpiled = pass_manager.run(circuit)

transpiled.count_ops()

OrderedDict([('xx_plus_yy', 514),
             ('p', 54),
             ('measure', 32),
             ('cp', 16),
             ('x', 4),
             ('barrier', 1)])