  Certain quantum operations can be built more efficiently through a procedure
known as Repeat-Until-Success. Differently from other non-deterministic quantum
operations, this procedure provides a classical flag which certifies the
success or failure of the procedure and, in the latter case, a recovery step
allows the restoration of the quantum state to its original condition. The
procedure can then be repeated until success is achieved. After success is
certified, the RUS procedure can be equated to a coherent gate. However, this
is not the case when the operation needs to be conditioned on the state of
other qubits, possibly being in a superposition state. In this situation, the
final operation depends on the failure and success history and introduces a
"distortion" that, even after the final success, depends on the past outcomes.
We quantify the distortion and show that it can be reduced by increasing the
probability of success towards unity. While this can be achieved via oblivious
amplitude amplification when the initial success probability is known, we
propose the use of fixed-point oblivious amplitude amplification to reduce this
unwanted distortions below any given threshold even without knowing the initial
success probability.
