Hamiltonian - Linear combination of unitaries

Codebook
1

I cannot obtain the second-order approximation of exp(-itU). This is my code:

def v0(t):
    """Calculates the first column of the PREPARE matrix, v0.

    Args:
        t (float): the time we evolve for.

    Returns:
        (array): v0 = [v00, v01, v02, v03] normalized
    """
    ##################
    # YOUR CODE HERE #
    ##################
    v00 = 1
    v01 = np.sqrt(t)
    v02 = t/np.sqrt(2)
    v03 = 0
    
    return np.array([v00, v01, v02, v03])/np.sqrt(1+t+1/2*t**2)

    
def exp_U_second(unitaries, t):
    """Implements the second-order approximation to Hamiltonian time evolution. 

    Args:
        unitaries (list): A list containing 4 unitary operations in this order:
        (U0, U1, U2, U3).
        t (float): The Hamiltonian evolution time. 
    """
    ##################
    # YOUR CODE HERE #
    ##################
    qml.MottonenStatePreparation(state_vector=v0(t),wires=range(2))
    select(unitaries)
    qml.MottonenStatePreparation(state_vector=np.transpose(np.conjugate(v0(t))),wires=range(2))
2

Hey @Dhawal_Verma!

I recommend trying to use your prepare function that you had to have completed in H.9.4a. Make sure you use qml.adjoint when applicable! Let me know if this helps :slight_smile:

3

Hi I usedprepare in this code. My former code in which I have to used qml.adjoint, I replaced it with np.transpose(np.conjugate(A)) and it worked every time. Please reply. Thanks.

This is little modification but it is still not working. Am I really vey far way from the true solution.

def v0(t):
“”"Calculates the first column of the PREPARE matrix, v0.

Args:
t (float): the time we evolve for.

Returns:
(array): v0 = [v00, v01, v02, v03] normalized
“”"

4

I didn’t see anything wrong with your v0 function:

def v0(t):
    """Calculates the first column of the PREPARE matrix, v0.

    Args:
        t (float): the time we evolve for.

    Returns:
        (array): v0 = [v00, v01, v02, v03] normalized
    """
    ##################
    # YOUR CODE HERE #
    ##################
    v00 = 1
    v01 = np.sqrt(t)
    v02 = t/np.sqrt(2)
    v03 = 0
    
    return np.array([v00, v01, v02, v03])/np.sqrt(1+t+1/2*t**2)

But your exp_U_second function isn’t right. The algorithm is PREP, SELECT, then PREP ^\dagger, and your prepare function should be this (or very similar) from H.6.4a:

def prepare(coeffs):
    qml.MottonenStatePreparation(normalize(coeffs), wires=aux)

Hope this helps!

1 Like
5

Thanks for this. I wrote my code for exp_U_second as, after the modification,

def exp_U_second(unitaries, t):
prepare(v0(t))
select(unitaries)
qml.adjoint(prepare(v0(t)))

I am getting ans error as: “The object None of type <class ‘NoneType’> is not callable. This error might occur if you apply adjoint to a list of operations instead of a function or template.”
Sorry for bothering you but I think it is very close. Please help me.

6

Hi. After giving it some try, I succeeded. Please ignore the former message. By the way, this worked “qml.adjoint(qml.MottonenStatePreparation(normalize(v0(t)),wires=[0,1]))”

2 Likes
7

Awesome! Glad to hear :slight_smile:

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