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I am getting confused as to how to perform gate operations using matrices and am hoping someone will help me walk through this example.

Say I want to perform a Pauli-X gate on the 3rd qubit in a 3-qubit system. That operation would be:

U = I ⊗ I ⊗ X

Then say I have the state $|001\rangle$ so after applying U the state would become $|000\rangle$. I am trying to use Python and NumPy to calculate this but I believe I am missing something.

i = np.array([[1, 0],
              [0, 1]])

x = np.array([[0, 1],
              [1, 0]])

state_0 = np.array([[1],
                    [0]])
state_1 = np.array([[0],
                    [1]])

x_3 = np.kron(np.kron(i,i),x)

v = np.kron(np.kron(state_0, state_0), state_1)

print(x_3.dot(v))

This code outputs:

[[1]
 [0]
 [0]
 [0]
 [0]
 [0]
 [0]
 [0]]

I am unsure if this output is correct and if it is how would I see this as the state $|000\rangle$? Any clarifications would be very useful! Thank you!

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You can use the same tools you used to get this output to check that it is correct: the state $|000\rangle$ would be represented as tensor product $|0\rangle \otimes |0\rangle \otimes |0\rangle$, which in your Python notation would be np.kron(np.kron(state_0, state_0), state_0). This should give you the same column vector you got from running your code, with the first element 1 and the rest of them 0s.

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    $\begingroup$ Continuing this: do np.kron(np.kron(state_0, state_0), state_1) then np.kron(np.kron(state_0, state_1), state_0) and so on and see they give you other column vectors with only a single nonzero entry somewhere. You should see the relation from where the nonzero is and what the bit string is from that. $\endgroup$ – AHusain Jun 12 at 22:22

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