Limits to black-box amplification in QMA

Quantum 10, 2227 (2026).

https://doi.org/10.22331/q-2026-10-06-2227

We study the limitations of black-box amplification in the quantum complexity class ${sf QMA}$. Amplification is known to boost any inverse-polynomial gap between completeness and soundness to exponentially small error, and a recent result (Jeffery and Witteveen, 2025) shows that completeness can in fact be amplified to be doubly exponentially close to 1. We prove that this is optimal for black-box procedures: we provide a quantum oracle relative to which no ${sf QMA}$ verification procedure using polynomial resources can achieve completeness closer to 1 than doubly exponential, or a soundness which is super-exponentially small. This is proven by making the oracle separation from (Aaronson, 2009) between ${sf QMA}$ and ${sf QMA}_1$ quantitative, using techniques from complex approximation theory.