# H₂ Molecule Homework Assignment

### Quantum Software Development Journey: From Theory to Application with Classiq - Part 3

• Similarly to what we have done in class, in this exercise we will implement the VQE on H2 molecule.

• This time instead of using the built-in methods and functions (such as Molecule and MoleculeProblem) to difne and solve the problem, you will be provided with a two qubits Hamiltonian.

## Submission

• Submit the completed Jupyter notebook and report via GitHub. Ensure all files are correctly named and organized.

• Use the Typeform link provided in the submission folder to confirm your submission.

## Important Dates

• Assignment Release: 22.5.2024

• Submission Deadline: 3.6.2024 (7 A.M GMT+3)

Happy coding and good luck!

### Part 1

Given the following Hamiltonian:

$\hat{H} = -1.0523 \cdot (I \otimes I) + 0.3979 \cdot (I \otimes Z) - 0.3979 \cdot (Z \otimes I) - 0.0112 \cdot (Z \otimes Z) + 0.1809 \cdot (X \otimes X)$

Complete the following code

from typing import List

from classiq import *

HAMILTONIAN = QConstant(
"HAMILTONIAN",
List[PauliTerm],
[
PauliTerm([Pauli.I, Pauli.I], -1.0523),
PauliTerm([Pauli.I, Pauli.Z], 0.3979),
PauliTerm([Pauli.Z, Pauli.I], -0.3979),
PauliTerm([Pauli.Z, Pauli.Z], -0.0112),
PauliTerm([Pauli.X, Pauli.X], 0.1809),
],
)  # TODO: Complete Hamiltonian

@qfunc
def main(q: Output[QArray[QBit]], angles: CArray[CReal, 3]) -> None:
# TODO: Create an ansatz which allows each qubit to have arbitrary rotation

allocate(2, q)
U(angles[0], angles[1], angles[2], 0, q[0])
U(angles[0], angles[1], angles[2], 0, q[1])

@cfunc
def cmain() -> None:
res = vqe(
HAMILTONIAN,  # TODO: complete the missing argument
False,
[],
optimizer=Optimizer.COBYLA,
max_iteration=1000,
tolerance=0.001,
step_size=0,
skip_compute_variance=False,
alpha_cvar=1.0,
)
save({"result": res})

qmod = create_model(
main, classical_execution_function=cmain
)  # TODO: complete the line, use classical_execution_function
qprog = synthesize(qmod)
# show(qprog)

execution = execute(qprog)
res = execution.result()
# execution.open_in_ide()
vqe_result = res[0].value  # TODO: complete the line

print(f"Optimal energy: {vqe_result.energy}")
print(f"Optimal parameters: {vqe_result.optimal_parameters}")
print(f"Eigenstate: {vqe_result.eigenstate}")

Optimal energy: -1.0808041015624998
Optimal parameters: {'angles_0': -2.1474722251077907, 'angles_1': -1.564983663323016, 'angles_2': 2.237208921683396}
Eigenstate: {'10': (0.41339864235384227+0j), '00': (0.213096589015404+0j), '01': (0.4244711783501914+0j), '11': (0.7768626809160033+0j)}


Does it similar to the optimal energy we calculated in class? \ Does it similar to the total energy we calculated in class?

### Part 2

Now, we want to have a more interesting ansatz in our main.
Add one line of code to the main function you created in Part 1 that creates entanglement between the two qubits.
Which gate should you use?

@qfunc
def main(q: Output[QArray[QBit]], angles: CArray[CReal, 3]) -> None:
# TODO: Create an ansatz which allows each qubit to have arbitrary rotation
allocate(2, q)
U(angles[0], angles[1], angles[2], 0, q[0])
CX(control=q[0], target=q[1])
U(angles[0], angles[1], angles[2], 0, q[1])

@cfunc
def cmain() -> None:
res = vqe(
HAMILTONIAN,  # TODO: complete the missing argument
False,
[],
optimizer=Optimizer.COBYLA,
max_iteration=1000,
tolerance=0.001,
step_size=0,
skip_compute_variance=False,
alpha_cvar=1.0,
)
save({"result": res})

qmod = create_model(
main, classical_execution_function=cmain
)  # TODO: complete the line, use classical_execution_function
qprog = synthesize(qmod)
# show(qprog)

execution = execute(qprog)
res = execution.result()
# execution.open_in_ide()
vqe_result = res[0].value  # TODO: complete the line

print(f"Optimal energy: {vqe_result.energy}")
print(f"Optimal parameters: {vqe_result.optimal_parameters}")
print(f"Eigenstate: {vqe_result.eigenstate}")

Optimal energy: -1.84725869140625
Optimal parameters: {'angles_0': 3.3288938597226614, 'angles_1': 4.422407264758315, 'angles_2': 3.2771087880496634}
Eigenstate: {'10': (0.07654655446197431+0j), '11': (0.08267972847076846+0j), '01': (0.9936320684740404+0j)}


Does it similar to the optimal energy we calculated in class? \ Does it similar to the total energy we calculated in class? \ What can we learn about the provided form this result Hamiltonian?