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The $ [[15,1,3]] $ quantum Reed-Muller code is a CSS code famous for implementing logical $ T $ (strongly) transversally. In particular, logical $ T $ is implemented using the physical unitary $$ \bigotimes_{i=1}^{15} T^\dagger $$ How do we know that the $ [[15,1,3]] $ code doesn't have weakly transversal implementations of other gates, for example $ \sqrt{T} $? In other words, is it possible that there is some (weakly) transversal physical gate $$ \bigotimes_{i=1}^{15} g_i $$ which implements logical $ \sqrt{T} $ on the codespace? Here all the $ g_i $ are in $ U(2) $ but they are not all assumed to be equal.

The obvious first place to look is the paper The Smallest Code with Transversal T which claims that the smallest code with transversal $ \sqrt{T} $ is the $ [31,1,3] $ quantum Reed-Muller code.

Although this paper does not assume strong transversality (all the $ g_i $ equal) it assumes something that is perhaps even more restrictive. Namely, "Assumption 2" of the paper is that every $ g_i $ for a transversal implementation of logical $ \sqrt{T} $ must be either the single qubit unitary $ \sqrt{T} $ itself or it must be some power of the single qubit unitary $ \sqrt{T} $.

For definitions of transversal (what I call weakly transversal here for emphasis) and strongly transversal we quote the paper above

"Definition 2: A logical single qubit unitary is implemented in a transversal manner if it is implemented by individual operations on each qubit $i$." So $$ \bigotimes_{i=1}^{n} g_i $$ "Definition 3: We say that a gate is strongly transversal if the operation on each set of identically labelled qubits is the same for each and every label." So $$ \bigotimes_{i=1}^{n} g_i $$ with all the $ g_i $ equal.

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    $\begingroup$ Have you seen this paper from this week arxiv.org/pdf/2303.15615.pdf which doesn't exactly answer your question, but does provide algorithms to find logical operators for stabilizer codes. $\endgroup$ Apr 2, 2023 at 22:28
  • $\begingroup$ Probably worth noting that assumption 2 is really quite weak. It's relying on a result in an earlier paper that says that all gates can be written as some Clifford operations - tensor product of phase gates of different phases - some Clifford operations. The Cliffords are irrelevant. The paper proves that the phase gates must all have phases $\pi/2^k$. For the [[15,1,3]] code, $k$ could be as large as 3. So the paper just chooses to eliminate the $k=3$ case because the motivation was to use these codes in a concatenated hierarchy. $\endgroup$
    – DaftWullie
    Apr 13, 2023 at 6:37
  • $\begingroup$ There's no point in implementing a transversal gate at one level of the hierarchy if you're building it out of gates which cannot be implemented transversally at the next level of the hierarchy. But true, if you don't care about fault tolerance and only care about error correction, it does leave a hole. $\endgroup$
    – DaftWullie
    Apr 13, 2023 at 6:37

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By Theorem 5/ Example 6 of The disjointness of stabilizer codes and limitations on fault-tolerant logical gates the transversal gates of the $ [[15,1,3]] $ quantum Reed-Muller code are all in the 3rd level of the Clifford hierarchy.

$ \sqrt{T} $ is in the $ 4 $th level of the Clifford hierarchy. So $ \sqrt{T} $ has no weakly transversal implementation for the $ [[15,1,3]] $ quantum Reed-Muller code.

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I don't know the answer to your question, but I think I know something relevant. The space of codes with transversal T gates has relationships to the space of circuits that can distill T states. And the distillation circuits don't all look like "and then I apply a big layer of T gates to all qubits leaving out none of them".

The T state distillation circuit I know that uses the fewest non-Clifford gates is one that originally came from block codes but I don't know if it corresponds to a code anymore. It consumes 10 T states with error rate $p$ and produces 2 T states with error rate $O(p^2)$. It's part of a family of circuits that limit to a 3:1 conversion ratio:

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So the smallest circuits don't particularly look like codes with strongly transversal T gates. I dunno if that says anything about Reed-Muller codes specifically, but I'd expect the smallest code with a constant depth $\sqrt{T}$ to be something very weird looking.

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