Difference between Repeat-until-success circuit and Distillation procedure

Question

On the fault-tolerance quantum computer (FTQC), the errors will be corrected by the error correction code (like surface code). In this case, the small angle rotation gate is difficult to apply on the FTQC. So we need “magic-state distillation” to prepare the accurate T gate. In my perception, Distillation is a procedure to purify the error-prone T gate by measurement. Distillation can construct not only T gate but also the smaller Z-axis rotation gates.

On the other hand, Krysta M. Svore proposed a kind of circuit called “Repeat-until-success (RUS) circuit” in the paper (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.080502). They also tried to use Clifford+T gate set to construct small angle rotations. RUS circuit is similar to the dynamic circuit. RUS circuit also use the measurement on ancilla qubits to decrease T-count.

So I wonder what’s the difference between RUS circuit and Distillation? Can I comprehend that RUS circuit is just a kind of distillation methods? Thanks for answering!

Answer 1

Distillation is used to produce T states, that can be used to perform T gates. The repeat-until-success circuits from the paper you linked use T gates (they consume T states) in order to perform more complex arbitrary gates.

You can't use the repeat-until-success circuits specified in the paper you linked to make T states, because they would consume more T states than they produced.

That said, magic state distillation factories use error detection codes. When an error is detected, the result is discarded and the factory is run again. In other words, the factory is using a repeat-until-success process. So magic state distillation is actually a kind of repeat until success process; just not exactly the same as the one in the paper you linked.