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I'm currently trying to understand the T magic state distillation algorithm described in "Universal Quantum Computation with Ideal Clifford Gates and Noisy Ancillas" [1] (Section V starting on Page 6). I need to understand the basics of magic state distillation in order to understand the motivations for optimization procedures used in another paper [2], which I need to give a talk on for a class. I know the distillation procedure used [2] (Bravyi-Haah) is different from the one described in [1], but [1] seems like a more natural starting point.

As this is background for the class presentation I am to give, my goals with this post are to refine my understanding as I continue to try and understand this topic and (most importantly) to make sure I do not spread any misinformation. That is, I don't expect to achieve 100% understanding from the responses to this post.

I would like to verify which points of the following plain-language laymen's explanation of the magic state distillation procedure described in [1] are incorrect or correct.

Imagining the production of T-states for use in a surface code,

1) Magic state distillation is performed within the surface code

2) The initial step of producing many copies of raw noisy T-states is done through the direct use of a non-fault tolerant T-gate

3) Distillation of these raw states is performed through discarding states which cause nontrivial surface code stabilizer measurements (eigenvalue = -1)

4) The raw states with trivial surface code stabilizer measurements (eigenvalue = 1) are transformed into a single qubit magic state.

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1) Magic state distillation is performed within the surface code

If you mean the distillation circuit is implemented with encoded logical qubits instead of raw physical qubits, then yes.

2) The initial step of producing many copies of raw noisy T-states is done through the direct use of a non-fault tolerant T-gate

Yes, the initial T states fed into the process are made with physical gates. Later rounds use T states distilled in the previous round.

3) Distillation of these raw states is performed through discarding states which cause nontrivial surface code stabilizer measurements (eigenvalue = -1)

Yes, you discard states that fail any of various parity checks. In principle you could try to correct errors instead of just detecting them, but this would be significantly less efficient because you get less error suppression (e.g. the 15-to-1 distillation process can correct any single T-gate error, but then you'd get p -> O(p^2) suppression instead of the p -> O(p^3) you get by detecting any pair of errors).

4) The raw states with trivial surface code stabilizer measurements (eigenvalue = 1) are transformed into a single qubit magic state.

Yes, though it could be multiple magic states. In the case of the 15-to-1 factory, it's just one.

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