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

# !pip install classiq

# import classiq
# classiq.authenticate()

from typing import List, cast

from classiq import *
from classiq import Pauli, PauliTerm

# TODO: Complete Hamiltonian
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),
],
)

@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])
# CX(q[0], q[1])

@cfunc
def cmain() -> None:
res = vqe(
hamiltonian=HAMILTONIAN,
maximize=False,
initial_point=[],
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: -1.0754688476562502
Optimal parameters: {'angles_0': 0.7985412460405645, 'angles_1': 4.821416635411074, 'angles_2': -2.4351340720501597}
Eigenstate: {'11': (0.14986973510352247+0j), '10': (0.33874192794810626+0j), '00': (0.8600849340326803+0j), '01': (0.350780380010057+0j)}


Optimal energy: -1.0711231445312501 Optimal parameters: {'angles_0': -3.0914206855935538, 'angles_1': -0.23729943557563232, 'angles_2': -2.5756826635214636} Eigenstate: {'01': (0.02209708691207961+0j), '11': (0.9997558295653994+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])
U(angles[0], angles[1], angles[2], 0, q[1])
CX(q[0], q[1])
# H(q[0])
# X(q[1])
# CX(q[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: -1.8512822265625002
Optimal parameters: {'angles_0': -2.953247896519252, 'angles_1': 6.104234752928836, 'angles_2': 4.463845345952442}
Eigenstate: {'10': (0.08838834764831845+0j), '11': (0.0855816496101822+0j), '01': (0.992402781762526+0j)}


Optimal energy: -1.8452896484374999 Optimal parameters:

Eigenstate: {'11': (0.08267972847076846+0j), '10': (0.07967217989988726+0j), '01': (0.9933863328282708+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?

With entanglement one gets better results.