documenting the sparse matrix example

@qjit

def ansatz(q : qreg, x : List[float]):

    X(q[0])

    # The goal of 'with decompose(..) as MAT' is to define a 

    # the unitary matrix MAT within the scope of the 

    # with statement. Here we use SciPy and Openfermion 

    # to construct the X0Y1 - Y0X1 operator as a matrix. 

    # Programmers must provide MAT as a numpy.matrix, 

    # here todense() maps the sparse operator to a numpy.matrix.

    # Note that if your matrix is dependent on a kernel argument, 

    # you must define it in the depends_on=[..] decompose arg.

    with decompose(q, kak, depends_on=[x]) as u:

        from scipy.sparse.linalg import expm

        from openfermion.ops import QubitOperator

        qubit_sparse = get_sparse_operator(qop)

        u = expm(0.5j * x[0] * qubit_sparse).todense()

# Define the Hamiltonain

H = -2.1433 * X(0) * X(1) - 2.1433 * \

    Y(0) * Y(1) + .21829 * Z(0) - 6.125 * Z(1) + 5.907

# Create the VQE ObjectiveFunction, use a 

# central finite difference gradient strategy

vqe_obj = createObjectiveFunction(ansatz, H, 1, {'gradient-strategy':'central', 'step':1e-1})

# Create a gradient-based optimizer, l-bfgs

optimizer = createOptimizer('nlopt', {'initial-parameters':[.5], 'maxeval':10, 'algorithm':'l-bfgs'})

# Find the ground state via optimization

results = optimizer.optimize(vqe_obj)

# Print the results

print('Results:')

print(results)

print()

# See what the circuit decomposition was at the opt angle

ansatz.print_kernel(qalloc(2), results[1])

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