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I'm trying to understand how the clifford + T compiler works in "A Game of Surface codes". enter image description here

How do I move a pauli product rotation block past a pauli product measurement block? More specifically, how do I get from a) to b).

a) enter image description here

b) enter image description here

Would appreciate any help. Thank you!

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Let $\sigma$ and $\tau$ be two tensor products of Paulis. Imagine you have a gate sequence $$ e^{i\theta\tau}\sqrt{\sigma}, $$ then you can always rewrite this as $$ e^{i\theta\tau}=\sqrt{\sigma}e^{i\theta\sqrt{\sigma}^\dagger\tau\sqrt{\sigma}}\sqrt{\sigma}^\dagger $$ such that $$ e^{i\theta\tau}\sqrt{\sigma}=\sqrt{\sigma}e^{i\theta\sqrt{\sigma}^\dagger\tau\sqrt{\sigma}}. $$ In other words, we have moved the $\sqrt{\sigma}$ the other term, but the other term has updated from $\tau$ to $\sqrt{\sigma}^\dagger\tau\sqrt{\sigma}$. If $\tau$ and $\sigma$ commute, this just leaves $\tau$. But if they anti-commute, you get $\tau\sigma$ (up to an $i$ phase that I'm not being very careful about.

This manipulation doesn't care if it's a unitary that $\tau$ is doing, or a measurement, where $\theta$ might as well be $\pi/2$.

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  • $\begingroup$ That makes sense. Thank you so much! $\endgroup$ Commented May 23 at 6:12

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