# Marriott-Watrous style amplification with a quantum input

$$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$$ In MW05 the authors demonstrate so-called "in-place" amplitude amplification for QMA, exhibiting a method for Arthur to amplify his success probability without requiring any increase in the size of Merlin's witness state. Because QMA is a language of classical bitstrings, this in some sense amplification with a classical input and quantum witness.

Is there an analogue of Mariott-Watrous amplification for when the input is quantum? To me it seems like naively pushing it through fails for the following reason:

In the classical case, if $$x$$ is the input and $$A(x,\ket{w})$$ is the verifier, then Marriott-Watrous amplification relies being able to apply $$A_x := A(x, \cdot)$$ and $$A_x^{\dagger}$$ many times. This is fine because even if we modify $$x$$ throughout the course of computing $$A_x$$, we can just prepare a copy in advance so that we always have $$x$$ accessible to us. However, if the input is instead an arbitrary quantum state $$\ket{x}$$, no-cloning forbids us from doing this. As such, we may corrupt $$\ket{x}$$ over the course of computation and all bets are off.