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Broadly: I have noticed on the qiskit developer (assessment) exam there are many questions asking to calculate the probability after measurement however I generally have no idea how to do this without using code (for any gate).

Specifically: Given this code fragment, what is the probability that a measurement would result in $|1\rangle$ if the state is prepared with the following code:

qc = QuantumCircuit(1) qc.ry(2 * math.pi/4, 0)

I.e.: how do we answer questions like these in an exam setting using our knowledge of cosine and sine?

Note that I am referring to and broadening Sample Question 2 - IBM Quantum Developer Certification which can be seen here: https://www.youtube.com/watch?v=LBaVOE5pcaI

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$Ry$ gate is described by following matrix $$ Ry(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}. $$

If your qubit was in state $|0\rangle$ before application of the gate, operation $Ry(\theta)|0\rangle$ leads to state $$ Ry(\theta)|0\rangle = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} \cos(\theta/2) \\ \sin(\theta/2) \end{pmatrix}= \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle. $$

Since probability of measuring a particular state is square of absolute value of the probability amplitude, we have $$ P(0) = |\cos(\theta/2)|^2 \\ P(1) = |\sin(\theta/2)|^2. $$

In your case $\theta = 2\pi/4 = \pi/2$. So, $P(0) = \cos^2(\pi/4) = 1/2$ and $P(1) = 1/2$.

Try yourself to calculate probabilities $P(0)$ and $P(1)$ for state $Ry(\theta)|1\rangle$.

Simiarly you can do the calculation for application of $Rx$ and $Rz$ gate. See matrices defining the gates here for $Rx$ and here for $Rz$.

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The way I guess the probability and know the correct answer is to imagine a Bloch sphere, and imagine how the gate effect the arrow of the bloch sphere.

also can look at these two cheat sheet from Qiskit Slack # qiskit-cert-exam

I suggest playing with these two websites while learning quantum gate

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  • $\begingroup$ Note that Bloch sphere is useful only for one-qubit state. Try to calculate probabilities after application of e.g. Toffoli gate on three-qubit state. :-) $\endgroup$
    – Martin Vesely
    Jun 18, 2022 at 21:21
  • $\begingroup$ The links for the slack pdfs seem to be broken :( $\endgroup$
    – epsilonolispe
    Jun 19, 2022 at 21:55
  • $\begingroup$ @epsilonolispe just join qiskit slack channel, and looking for the file, if you do, it should show file details at the right side bar., after you click the link $\endgroup$
    – poig
    Jun 20, 2022 at 2:23
  • $\begingroup$ @epsilonolispe you can use qiskit.visualization.visualize_transition for trace animation of Bloch sphere $\endgroup$
    – poig
    Jun 22, 2022 at 5:00

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