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What are the ways to rewrite a single-qubit gate follow by a signle-qubit error?

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    $\begingroup$ What kind of error are we working with? Incoherent or Coherent? $\endgroup$ May 27, 2020 at 0:18
  • $\begingroup$ Asked and answered in the book Quantum Computing: A Gentle Introduction, chapter 12 exercises; is this a homework question? $\endgroup$
    – Rob
    May 27, 2020 at 5:44

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Let's assume that your single qubit gate is $U$, and it's followed by some noise operator $A$. So, overall, you've implemented $AU$. You want to know if this can be written in the form $UB$ where $B$ is a new noise operator. So, you just want to solve $$ AU=UB. $$ To rearrange for $B$, just pre-multiply by $U^\dagger$: $$ B=U^\dagger AU. $$ As a bonus, if $A$ was unitary, you can easily see that $B$ is unitary. If $A$ is Kraus operator, then you might have a trace preserving property such as $$ \sum_iA_i^\dagger A_i=I, $$ then this also applies to your new description: $$ \sum_iB_i^\dagger B_i=\sum_iU^\dagger A_i^\dagger UU^\dagger A_iU=U^\dagger\left(\sum_iA_i^\dagger A_i\right)U=U^\dagger IU=I. $$

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