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In the below code snippet, what is the probability of measuring $|1\rangle$?

 qc = QuantumCircuit(1) 
 qc.rx(3*math.pi/4, 0)
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  • $\begingroup$ Have a look at the matrix of Rx gate, then multiply the matrix with vector describing state $|1\rangle$. Finally take square of abs. value of the second member of the resulting vector. This is the probability you look for. $\endgroup$ Commented Oct 8, 2022 at 6:50
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    $\begingroup$ Welcome to QC exchange. Let us know what your initial ideas/doubts for such questions please. Also, first attempts might be helpful as well. $\endgroup$
    – R.W
    Commented Oct 8, 2022 at 8:50

1 Answer 1

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The gate $R_x(\theta)$ is defined by the matrix $$\begin{pmatrix}\cos \frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} &\cos\frac{\theta}{2}\end{pmatrix}$$ and the state $|1\rangle$ is the column-matrix $\begin{pmatrix}0 \\ 1\end{pmatrix}$. Multiply them and you will find the resulting state and thus the amplitude of $|1\rangle$.

Alternatively, you can use the fact that $$R_x(\theta) \equiv \exp\left(\frac{-i\theta X}{2}\right)=\cos\frac{\theta}{2}I-i\sin\frac{\theta}{2}X$$ where $I$ is the identity gate and $X$ the Pauli $X$ gate. The second identity follows from the fact that the $X$ gate is idempotent ($X^2=I$) and the definition of the exponential of an operator.

Either way should give you the same result.

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