# Magic state distillation, eigenvectors of Clifford gates, and transversal gates

I have some (possibly mistaken) ideas about magic state distillation. Please disabuse me of them?

In section III "Universal Quantum Computation with Magic States" of the original paper on magic states, Bravyi and Kitaev give a protocol using Clifford gates that takes in one qubit of unknown logical information $$|\psi\rangle=\alpha |0\rangle+ \beta |1\rangle\,,$$ in the first register and the eigenstate $$|H\rangle= \cos(\frac{\pi}{8})|0\rangle+\sin(\frac{\pi}{8})|1\rangle$$ of the Hadamard gate $$H$$ in the second register and outputs the gate $$T= \begin{bmatrix} 1& 0 \\ 0 & e^{i \pi/4} \end{bmatrix}\,,$$ applied to $$|\psi\rangle$$ in the first register (the output of the second register is discarded). This is notable because the protocol uses only Clifford gates and the eigenstate $$|H\rangle$$ of the Hadamard gate but is able to apply a non-Clifford gate $$T$$ to the given quantum information.

Later in the same section Bravyi and Kitaev give a protocol using Clifford gates that takes in one qubit of unknown logical information $$|\psi\rangle=\alpha |0\rangle+ \beta |1\rangle\,,$$ in the first register and the eigenstate $$|F\rangle$$ of the facet gate in the second register and outputs the gate $$\begin{bmatrix} 1& 0 \\ 0 & e^{i \pi/6} \end{bmatrix}\,,$$ applied to $$|\psi\rangle$$ in the first register (the output of the second register is discarded). This is notable because the protocol uses only Clifford gates and the eigenstate $$|F\rangle$$ of the facet gate but is able to apply a non-Clifford gate $$\begin{bmatrix} 1& 0 \\ 0 & e^{i \pi/6} \end{bmatrix}$$ to the given quantum information.

How do you get the eigenstate $$|H\rangle$$ of the Hadamard gate? Is it true that you can distill $$|H\rangle$$ from any $$d\geq 2$$ stabilizer code with transversal $$H$$? For example, the $$[\![4,2,2]\!]$$ quantum repetition code or the $$[\![15,7,3]\!]$$ quantum Hamming code?

How do you get the eigenstate $$|F\rangle$$ of the facet gate? Is it true that you can distill $$|F\rangle$$ from any $$d\geq 2$$ stabilizer code with transversal $$F$$? For example, the $$[\![5,1,3]\!]$$ code?

• I am not familiar with magic state distillation, so I probably won't be able to help much. But I think maybe sections V and VI answer some of your questions? I take it that Bravyi and Kitaev use $|T\rangle$ to refer to one of the facet directions, and they indeed use $[[5,1,3]]$ to distill $|T\rangle = |F\rangle$ in section V. Are you OK with any degree of distillation? Like if you needed really good mixed states to see any distillation benefit, you would still call that distillation with that code? Commented Apr 6 at 3:30
• How should I think about distillation with $k>1$? It seems like it would be hard to iterate -- do you throw away some of the qubits after a decoding transformation in order to iterate distillation with $[[15,7,3]]$? Commented Apr 6 at 3:30