I have a code $C1$ that admits native fault-tolerant gates for any element in the Clifford group (for instance via transversal implementation), but that then doesn't have an easy way to perform the $T$ gate.
In practice, to implement the $T$-gate we can use state injection and create a logical ancilla state in the state $|A\rangle=T |+\rangle$.
A way to create the $|A\rangle$ state in a reliable manner is to make use of a code that has a native fault-tolerant $T$ gate. The $15$ qubit code is such example (the $T$-gate is transversal).
In practice, to prepare $|A\rangle$, we then:
- Regroup $15$ logical qubits of the code $C1$ and encode with them the logical $|+_L\rangle$ state of the $15$ qubit code (hence each qubit of the $15$ qubit code is a logical qubit of $C1$, itself composed of physical qubits).
- Apply a transversal $T$ to this logical qubit (it is equal to a logical $T$ thanks to the $15$ qubit code properties)
- Decode the $15$ qubit encoding: we now have a state $|A\rangle$ encoded with the code $C1$, we can perform state injection.
(Of course we have to repeat the procedure to have a reliable state but I skip this un-necessary detail for my question).
For $C_1$ we do not have a fault-tolerant $T$ gate. However, we need to do a $T$ gate within this code for step 2 of the procedure I described. How is it done? Do we use either one of the following methods? Another thing I am not thinking of?
- We implement a non-fault-tolerant $T$ gate within $C_1$ (hence super noisy, but it is not "really" a problem thanks to the logic of magic state distillation). In this case, I would be interested to know which operations are performed on the physical qubits for the surface code to perform this "non fault-tolerant logical $T$ gate" (it probably deserves another question).
- We basically do another state injection technique to perform the $T$ gates. This time we will then use an encoding with the $15$ qubit code but each qubit within this code will now be a physical qubit (we are basically at a "lower level" of encoding).