# Classical shadows in calculating fidelity

I’m learning classical shadows and I’ve successfully reconstructed 10 qubits GHZ state.Now I have a shadow state,and I wanna calculate the fidelity between shadow state and standard 10 qubits GHZ state.I wanna know the exact python code for the fidelity computing.

Hello @DonaldTrump,
.
For calculating fidelities, you have two options: qml.qinfo.fidelity and qml.math.fidelity. Let’s look at two examples.

qml.info.fidelity acts on circuits that return a quantum state using qml.state(). First, let’s create two QNodes:

dev = qml.device('default.qubit', wires = 2) #create a two wire device

@qml.qnode(dev)
def circuit_1():
# Creates a two qubit GHZ state
qml.CNOT(wires = [0,1])

return qml.state()

@qml.qnode(dev)
def circuit_2(alpha):
# Creates an entangled state dependent on alpha

qml.RY(alpha, wires = 0)
qml.CNOT(wires = [0,1])

return qml.state()


We are going to find the fidelity between the output states for both circuits, for alpha = \pi/6.
To do this, we run

qml.qinfo.fidelity(circuit_1, circuit_2, wires0=[0,1], wires1=[0,1])(None,np.pi/6)


Note that we specified the wires for the state in both circuits using wires0 and wires1. In this case, we want the whole state for both circuits and not only a reduced state. If we wanted a reduced state, we could specify only some of the wires.

Also note the (None, np.pi/6) argument that is wrapped by the qml.qinfo.fidelity. These are the arguments of circuit_1 and circuit_2 respectively. Since circuit_1 doesn’t depend on any parameters, we pass None.

In the back end, qml.qinfo.fidelity works using qml.math.fidelity. This one is useful if you already have the states as density matrices and don’t care too much about the circuits that prepared the states. It works only on density matrices though, so you have to be careful to convert your state vectors into density matrices using qml.math.dm_from_state_vector.

For example, if I want to find the fidelity between the states \vert 0 \rangle and \vert + \rangle = \frac{1}{\sqrt{2}}(\vert 0 \rangle + \vert 1 \rangle), we can express our states as arrays:

state_0 = [1,0]
state_1 = [1/np.sqrt(2), 1/np.sqrt(2)]


But running qml.math.fidelity in the above will not work, so we transform to density matrices:

state_0 = qml.math.dm_from_state_vector(state_0)
state_1 = qml.math.dm_from_state_vector(state_1)


Finally we can run qml.math.fidelity(state_0, state_1) and get the answer.

Hope one of these options will work for your use case! Let us know if you have additional questions!

Alvaro

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