change the VQE examples to stop using the deprecated qiskit.algorithms

[1ucian0]

Changes

Fixes #3487
Co-authored-by: @ElePT

As qiskit.algorithms is deprecated, the VQE examples in the landing page needs to be refactor.

Implementation details

How to read this PR

The change was implemented with a script. So, if you are ok with one of the entries, you are ok with all of them. They are literally the same.

Screenshots

[ElePT]

Now that I see the side-by-side comparison with the old example, I am thinking it might make sense to actually keep the hamiltonian and ansatz from the old example (might be easier with those familiar with the code). In that case, the snippet would look like this:

Suggested change

	hamiltonian = SparsePauliOp.from_list(
	[("YZ", 0.3980), ("ZI", -0.3980), ("ZZ", -0.0113), ("XX", 0.1810)]
	)
	# Define VQE ansatz, initial point and cost function
	from qiskit.circuit.library import EfficientSU2
	ansatz = EfficientSU2(hamiltonian.num_qubits)
	initial_point = [0]*16
	def cost_function(params, ansatz, hamiltonian, estimator):
	energy = estimator.run(ansatz, hamiltonian, parameter_values=params).result().values[0]
	return energy
	# Run VQE using SciPy minimizer routine
	from scipy.optimize import minimize
	result = minimize(cost_function, initial_point, args=(ansatz, hamiltonian, estimator), method="cobyla")
	# Print minimum eigenvalue
	print(result.fun)
	hamiltonian = SparsePauliOp.from_list(
	("II", -1.052373245772859),
	("IZ", 0.39793742484318045),
	("ZI", -0.39793742484318045),
	("ZZ", -0.01128010425623538),
	("XX", 0.18093119978423156)
	])
	# Define VQE ansatz, initial point and cost function
	from qiskit.circuit.library import TwoLocal
	ansatz = TwoLocal(num_qubits=2, rotation_blocks="ry", entanglement_blocks="cz")
	initial_point = initial_point = [0] * 8
	def cost_function(params, ansatz, hamiltonian, estimator):
	energy = estimator.run(ansatz, hamiltonian, parameter_values=params).result().values[0]
	return energy
	# Run VQE using SciPy minimizer routine
	from scipy.optimize import minimize
	result = minimize(cost_function, initial_point, args=(ansatz, hamiltonian, estimator), method="cobyla")
	# Print minimum eigenvalue
	print(result.fun)

[1ucian0]

Suggestion added to 6164255 Thanks!

[1ucian0]

Closed in favor of #3531 , which PRs from upstream