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There seems to be numerous work about the distillation protocol of the $T$-magic state $$ \frac{1}{\sqrt{2}}(|0\rangle+e^{i\pi/4}|1\rangle). $$

Similarly, I am wondering if it is possible to distill a $\theta$-magic state $$ \frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle). $$

Can we achieve it for arbitrary angle $\theta$? If not, what are the conditions for distillable $\theta$?

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    $\begingroup$ Could you comment a little bit more on allowed operations? For example, could I distill $|H\rangle$ magic states and use that those along with Clifford gates give universal quantum computation (and hence I can to any degree of accuracy rotate a qubit to your desired state)? Or is that off-limits? $\endgroup$
    – user196574
    Nov 19, 2023 at 20:48
  • $\begingroup$ @user196574 No strict requirements. Just make sure every step can be achieved (in principle) fault-tolerant. $\endgroup$
    – Yunzhe
    Nov 20, 2023 at 5:06

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