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The quantum repetition code is an $ [[n,1,1]] $ stabilizer code with stabilizer generators $ Z_iZ_{i+1} $ for $ i=1, \dots, n-1 $.

The Eastin-Knill theorem states that a $ d >1 $ code cannot have a universal set of transversal gates, in fact it can only have finitely many transversal gates.

However, a quantum repetition code has distance $ d=1 $ so the Eastin-Knill theorem does not restrict the transversal gates.

Indeed any diagonal gate $$ \begin{bmatrix} e^{i \phi} & 0 \\ 0 & e^{i \theta} \end{bmatrix} $$ can be implemented transversally on a quantum repetition code.

Thus the $ [[n,1,1]] $ quantum repetition code is a good example of a $ d=1 $ code with infinitely many transversal gates.

Also the Pauli $ X $ gate is exactly transversal, meaning that $ X^{\otimes n} $ implements logical $ X $.

The $ X $ gate together with the diagonal gates forms the group of all single qubit gates that do not create superpositions, a large subgroup of $ U(2) $. But it turns out that no other transversal gates exist for this code i.e. there is not a universal set of transversal gates, see Does the Eastin-Knill theorem hold for repetition codes? . Essentially the argument is that a transversal logical gate is, by definition, implemented by a physical gate of the form $$ U_1\otimes U_2\otimes \dots \otimes U_n $$ i.e. a non entangling gate. So if some codewords in the codespace are unentangled then it will be impossible to use transversal logical gates to create a superposition and thus a universal set of transversal gates is ruled out.

However this argument hinges on a highly unusual property of the repetition code: there are codewords which are unentangled.

For all $ d>1 $ code that I am aware of, every codeword corresponds to a state in which all qubits are entangled with all other qubits.

My question is, what is an example of a $ d=1 $ code with a universal set of transversal gates?

There is of course a silly example which is the $ [[1,1,1]] $ code with encoding map $$ |\psi \rangle \mapsto |\psi \rangle $$ or its generalization the $ [[n,1,1]] $ code with stabilizer generators $ Z_i $, $ i=2,\dots n $ and encoding map $$ |\psi \rangle \mapsto |\psi \rangle \otimes | 0 \rangle^{\otimes (n-1)} $$

Or is every code with a universal set of transversal gates trivial?

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  • $\begingroup$ I guess it depends on what you mean by trivial. Just spitballing, if you correct any error E, then your transversal gates can't include any gates generated by the infinitesimal generator for E. That means that, restricting to any subsystem, your universal set of transversal gates can't be universal on that subsystem. Imma go with "probably trivial". $\endgroup$
    – squiggles
    Commented Jul 2 at 3:34

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