The question asks whether there exists an in-place amplification method for QMA machines that preserves all initially valid witnesses. Here's the key analysis:

Key Observations:

1. QMA Amplification Basics: QMA amplification aims to increase the completeness probability (for valid inputs, existence of a witness with high success probability) and decrease the soundness probability (for inv...

Short answer: No in general. You cannot, in general, amplify a QMA verifier on a fixed witness register so that every witness that originally achieved acceptance ≥ a is boosted to acceptance ≥ 1 − 2^{-r}, while simultaneously keeping all witnesses with original acceptance ≤ b down to ≤ 2^{-r}. Marriott–Watrous’ construction is essentially optimal with respect to this “witness-preserving” desiderat...

This is an excellent question about a subtle issue in quantum complexity theory. You've correctly identified a fundamental limitation of the Marriott-Watrous (MW05) amplification technique for QMA.

The Core Issue

You're absolutely right that the MW05 technique does not preserve the validity of all witnesses. The key insight is that their amplification operator acts differently on differen...

Short answer
No. Once the completeness/soundness gap is larger than
(1–a)/(1–b) there is no procedure that

• leaves Merlin’s witness register exactly the same size,
• is computable by an efficient quantum circuit that does not look at the particular witness that is provided, and
• has the property

  ∀ |ψ〉 : 〈ψ|Q|ψ〉 ≥ a  ⇒  〈ψ|Q′|ψ〉 ≥ 1–ε,
        〈ψ|Q|ψ〉 ≤ b  ⇒  〈ψ|Q′|ψ〉 ≤ ...

Better "In-Place" Amplification of QMA

This is an interesting question about quantum complexity theory, specifically about the properties of QMA (Quantum Merlin-Arthur) amplification techniques.

Understanding the Issue

The Marriott-Watrous (MW05) in-place amplification technique for QMA allows Arthur to amplify his success probability without requiring additional copies of Merlin's witness...

No. In fact nothing like a “set‐preserving’’ in‐place amplification is known, and there is a very simple linearity/convexity‐of‐eigenvalues argument showing that no unitary map acting only on the original witness can simultaneously

1. boost all eigenvectors of the QMA‐verifier’s acceptance operator above a threshold (a) up to (say) (1-\varepsilon), and
2. leave every superpositio...

Excellent question. You've hit upon a very subtle but important point in the analysis of in-place QMA amplification.

Your analysis is completely correct. The Marriott-Watrous (MW05) amplification scheme does not guarantee that an arbitrary "good" witness for the original verifier will remain a "good" witness for the amplified verifier.

Let's break down why this is the case and then answer yo...

The original work by Marriott and Watrous (MW05) shows how to perform in-place amplification for QMA, meaning that the witness size does not increase. However, as th...

No – as soon as one insists that

1. Merlin still sends only the original m-qubit witness, and
2. every witness that made the old verifier M accept with probability at least a must make the new verifier M′ accept with probability that is (noticeably) larger,

amplification is impossible in general.
The obstacle is purely information–theoretic and is not an artefact of the Marriot...