Measurement#

The previous examples were all carried out using state vector simulations of the quantum circuit. However, in actual quantum hardware, the expectation value of the molecular Hamiltonian for a parameterized quantum circuit have to be estimated by using repeated executions of the circuit, or shots in short. Qibochem provides this functionality using the AbstractHamiltonian.expectation_from_samples method implemented in Qibo.

The example below is taken from the Bravyi-Kitaev transformed Hamiltonian for molecular H₂ in minimal basis of Hartree-Fock orbitals, at 0.70 Angstroms separation between H nuclei, as was done in [1]:

Hamiltonian expectation value#

from qibo import models, gates
from qibo.symbols import X, Y, Z
from qibo.hamiltonians import SymbolicHamiltonian
import numpy as np
from qibochem.measurement.expectation import expectation
from scipy.optimize import minimize

# Bravyi-Kitaev tranformed Hamiltonian for H2 at 0.7 Angstroms
bk_ham_form = -0.4584 + 0.3593*Z(0) - 0.4826*Z(1) + 0.5818*Z(0)*Z(1) + 0.0896*X(0)*X(1) + 0.0896*Y(0)*Y(1)
bk_ham = SymbolicHamiltonian(bk_ham_form)
nuc_repulsion = 0.7559674441714287

circuit = models.Circuit(2)
circuit.add(gates.X(0))
circuit.add(gates.RX(0, -np.pi/2, trainable=False))
circuit.add(gates.RY(1, np.pi/2, trainable=False))
circuit.add(gates.CNOT(1, 0))
circuit.add(gates.RZ(0, theta=0.0))
circuit.add(gates.CNOT(1, 0))
circuit.add(gates.RX(0, np.pi/2, trainable=False))
circuit.add(gates.RY(1, -np.pi/2, trainable=False))

print(circuit.draw())

def energy_expectation_samples(parameters, circuit, hamiltonian, nshots=1024):
    return expectation(circuit, hamiltonian, from_samples=True, n_shots=nshots)

parameters = [0.5]
nshots = 8192
vqe_uccsd = minimize(energy_expectation_samples, parameters, args=(circuit, bk_ham, nshots), method='Powell')
print(vqe_uccsd)
print('VQE UCCSD loss:   ', vqe_uccsd.fun)
print('nuclear repulsion:', nuc_repulsion)
print('VQE UCCSD energy: ', vqe_uccsd.fun + nuc_repulsion)

q0: ─X──RX─X─RZ─X─RX─
q1: ─RY────o────o─RY─
message: Optimization terminated successfully.
success: True
status: 0
    fun: -1.8841124999999999
    x: [ 2.188e+00]
    nit: 2
direc: [[ 1.000e+00]]
    nfev: 23
VQE UCCSD loss:    -1.8841124999999999
nuclear repulsion: 0.7559674441714287
VQE UCCSD energy:  -1.128145055828571

References