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I am interested to have references and comment about pro/cons of the various methods that are used to implement complete gateset in a fault tolerant manner.

Usually the Clifford operations have a standard way to be implemented fault-tolerantly, at least in concatenated code (which I know best), by implementing them transversally. But it has been shown that it is not possible to have a complete gateset only based on transversal gates implementation. For this reason, for the non-clifford operations, other techniques are usually required. There is for what I know:

  • Magic state distillation (ref)
  • Gate teleportation (ref)

I have heard about other techniques like

  • Code switching techniques (two codes $A$ and $B$ are able to implement different transversal gate such that the union of those gates form a complete gateset. You then decode the state from $A$ to encode it to $B$ when you need to perform a transversal gate allowed by $B$)
  • Some codes admit a nice construction for a complete fault-tolerant gateset.

In the answer, I would be interested to know why magic state distillation is so popular those days, given the number of publications in this topic (because there are many other techniques). There is probably some clear advantage over other techniques that I don't see.

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  • $\begingroup$ A nice starting point may be the list in this wikipedia article. $\endgroup$
    – JSdJ
    Sep 13 at 19:39
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Magic state distillation is the most popular because it is (as of yet) the most efficient. Also, magic state distillation fuels gate teleportation, so I wouldn't say it's used instead of gate teleportation. The biggest advantage state distillation has is that it can use error detecting codes instead of error correcting codes, and this has a tendency to double the code distance at no additional cost.

Michael Beverland et al looked at the cost of code switching and found "[no] substantial savings over state distillation" despite using more complex connectivity (a 3d code instead of a 2d code for the code switching).

Caution: you should ignore the orange curves because there's no corresponding "optimistic state distillation" to compare it to.

enter image description here

(The 2d surface code has transversal Clifford gates when you can stack interacting layers, so you would expect state distillation cost to drop if you had access to a 3d topology.)

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  • $\begingroup$ Thank you for your answer. I am not sure to understand why you say that it fuels gate teleportation ? Maybe for some schemes but from what I known fault tolerant gate teleportation was known before magic state distillation so it is not "in principle" necessary to use magic states distillation for that ? Also, it appears that your paper is very recent so I am wondering if there is a "good enough" consensus on the fact state distillation is better of if it is actually still an open question against other schemes. $\endgroup$
    – StarBucK
    2 days ago
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    $\begingroup$ @StarBuck oh it is definitely still an open question! But the best non-Clifford technique for the past decade or so has consistently been magic state distillation, so there's an expectation that is hard to beat. But consider S gates. In the surface code the state of the art for that has switched from distillation to twists. $\endgroup$ 2 days ago

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