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Wanted to check on how to mathematically obtain the matrix of an X gate which has fidelity/probability $p$? (i.e. it acts as an $X$ gate with probability $p$)

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  • $\begingroup$ And what does the gate act as with probability $1-p$? $\endgroup$
    – FDGod
    Nov 4, 2023 at 6:22
  • $\begingroup$ It acts as I gate $\endgroup$
    – codeit
    Nov 8, 2023 at 17:27

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The usual way to represent this isn't via a matrix, but via the Kraus representation. That is, a quantum operation can be written as: $$\rho\mapsto\sum_iK_i\rho K_i^\dagger$$ with $\sum\limits_iK_i^\dagger K_i\leqslant I$. Here, the quantum operation you're looking for is the well-known bit-flip channel with probability $p$: $$\rho\mapsto(1-p)\rho+pX\rho X$$ That is, with probability $1-p$, nothing is done on the quantum state, and with probability $p$ an $X$ gate is applied. Here, the Kraus operators describing this operation are $K_0=\sqrt{1-p}I$ and $K_1=\sqrt{p}X$.

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  • $\begingroup$ I was interpreting OP's question as $X$ gate occurs with probability $p$ and erroneous $X$ gate occurs with probability $1-p$, like $e^{-i \theta X}$ with $\theta \neq \frac{\pi}{2}$. $\endgroup$
    – FDGod
    Nov 4, 2023 at 18:29

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