Ground State Energy with QPE

Example Hydrogen

We caluclate the ground state energy of the Hydrogen molecule with Quantum Phase Estimation.

Utilizing symmetries, one can find a reduced two qubit Hamiltonian for the Hydrogen molecule. Frist, we define the Hamiltonian, and compute the ground state energy classically.

from qrisp import QuantumVariable, x, QPE
from qrisp.operators.pauli.pauli import X,Y,Z
import numpy as np

# Hydrogen (reduced 2 qubit Hamiltonian)
H = -1.05237325 + 0.39793742*Z(0) -0.39793742*Z(1) -0.0112801*Z(0)*Z(1) + 0.1809312*X(0)*X(1)
E0 = H.ground_state_energy()
# Yields: -1.85727502928823

In the following, we utilize the trotterization method of the QubitOperator to obtain a function U that applies Hamiltonian Simulation via Trotterization. If we start in a state that is close to the ground state and apply Quantum Phase Estimation, we get an estimate of the ground state energy.

# ansatz state
qv = QuantumVariable(2)
x(qv[0])
E1 = H.get_measurement(qv)
E1
# Yields: -1.83858104676077

We calculate the ground state energy with quantum phase estimation. As the results of the phase estimation are modulo \(2\pi\), we subtract \(2\pi\).

U = H.trotterization()

qpe_res = QPE(qv,U,precision=10,kwargs={"steps":3},iter_spec=True)

results = qpe_res.get_measurement()
sorted_results= dict(sorted(results.items(), key=lambda item: item[1], reverse=True))

phi = list(sorted_results.items())[0][0]
E_qpe = 2*np.pi*(phi-1) # Results are modulo 2*pi, therefore subtract 2*pi
E_qpe
# Yields: -1.8591847149174