Ground State Energy with QPE#

For a given Hamiltonian $H$ and an eigenstate $\ket{\lambda_j}$ (e.g., the ground state $\ket{\lambda_0}$), the goal is to find the eigenvalue $E_j$ such that

$$e^{iH}\ket{\lambda_j} = e^{iE_j}\ket{\lambda_j}$$

This is achieved by applying quantum phase estimation to the unitary $U=e^{iH}$ with the initial state $\ket{\lambda_j}$. With the equation $e^{iE_j}=e^{2\pi i\theta_j}$, the energy $E_j$ can be calculated from the resulting phase $\theta_j\in [0,1)$. In practice, the exact eigenstate $\lambda_j$ is unknown, and one uses an approximation $\ket{\psi_0}$ which can be expressed in the eigenbasis of the Hamiltonian as

$$\ket{\psi_0}=\sum_j a_j\ket{\lambda_j}$$

If the approximation $\ket{\psi_0}$ is sufficently close to the eigenstate $\ket{\lambda_j}$, the correct phase, and hence the eigenvalue $E_j$, is obtained with high probability.

Example Hydrogen#

We caluclate the ground state energy of the electronic Hamiltonian for 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 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()
print(E0)
# Yields: -1.85727502928823

The $\ket{10}$ state provides a good approximation of the ground state:

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

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. As the results of the phase estimation are modulo $2\pi$ and the searched for eigenvalue is between $-2\pi$ and 0, we subtract $2\pi$.

U = H.trotterization(forward_evolution=False)

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
print(E_qpe)
# Yields: -1.8591847149174