# 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.

## 1 Answer

QMA verification includes three registers, with appropriate bounds on the length of each register:

• $$x$$, a (classical) description of the (quantum) circuit used for verification, along with the promised gap;
• $$\vert w\rangle$$, the (quantum) witness provided by Merlin; and
• $$\vert\eta\rangle$$, the ancilla workspace used by Arthur.

It seems as if you are asking what problems may arise (for example, in the strong error-reduction of Marriott-Watrous), if $$x$$ where quantum rather than classical (for example, if $$\vert x\rangle$$ where in some superposition of states).

But the classical description of the circuit is meant to be agreed-upon by Arthur and Merlin a-priori. If Merlin were to send $$\vert x\rangle$$ in superposition, Arthur could initially start his verification procedure by measuring $$\vert x\rangle$$ to get a classical string $$x$$. Merlin doesn't gain anything by sending $$\vert x\rangle$$ itself in superposition, and cannot improve his completeness/soundness.

Sevag Gharibian has a very good lecture (clocking in at about 2 hours and 40 minutes) on QMA and MW here. I've been enjoying this lecture series very much.