# single-qubit gate followed by a single-qubit error

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

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

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: $$\sum_iB_i^\dagger B_i=\sum_iU^\dagger A_i^\dagger UU^\dagger A_iU=U^\dagger\left(\sum_iA_i^\dagger A_i\right)U=U^\dagger IU=I.$$