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Limits to black-box amplification in QMA

Scott Aaronson, Phillip Harris, Freek Witteveen

TL;DR

This paper establishes tight limits on black-box amplification for the quantum complexity class ${\sf QMA}$. By constructing a quantum oracle and analyzing the maximal acceptance probability as a function of a parameter $\theta$ through trigonometric polynomials, the authors prove that no polynomial-resource, black-box verifier can achieve completeness closer to 1 than $1-2^{-2^{\mathrm{poly}(n)}}$ or a soundness that is super-exponentially small. The key technical tool is a growth bound for trig polynomials, which translates bounded acceptance on a yes-interval into quantitative limits on the entire parameter domain, yielding a sharp completeness barrier and a separate, symmetric soundness barrier. These results, together with a parallel finding for doubly-exponential completeness established by prior work, demonstrate the optimality of black-box amplification in ${\sf QMA}$ and clarify the asymmetry between completeness and soundness under such reductions; they also discuss implications for related classes (e.g., ${\sf PP}$ and ${\sf PreciseQMA}$) and the caveat that infinite-dimensional witnesses can circumvent these barriers.

Abstract

We study the limitations of black-box amplification in the quantum complexity class 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 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 using techniques from complex approximation theory, to make the oracle separation from (Aaronson, 2008), between QMA and QMA with perfect completeness, quantitative.

Limits to black-box amplification in QMA

TL;DR

This paper establishes tight limits on black-box amplification for the quantum complexity class . By constructing a quantum oracle and analyzing the maximal acceptance probability as a function of a parameter through trigonometric polynomials, the authors prove that no polynomial-resource, black-box verifier can achieve completeness closer to 1 than or a soundness that is super-exponentially small. The key technical tool is a growth bound for trig polynomials, which translates bounded acceptance on a yes-interval into quantitative limits on the entire parameter domain, yielding a sharp completeness barrier and a separate, symmetric soundness barrier. These results, together with a parallel finding for doubly-exponential completeness established by prior work, demonstrate the optimality of black-box amplification in and clarify the asymmetry between completeness and soundness under such reductions; they also discuss implications for related classes (e.g., and ) and the caveat that infinite-dimensional witnesses can circumvent these barriers.

Abstract

We study the limitations of black-box amplification in the quantum complexity class 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 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 using techniques from complex approximation theory, to make the oracle separation from (Aaronson, 2008), between QMA and QMA with perfect completeness, quantitative.

Paper Structure

This paper contains 8 sections, 2 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Use of AI:
  3. Preliminaries
  4. Trigonometric polynomials
  5. Limits to black-box QMA amplification
  6. Completeness barrier
  7. Soundness barrier
  8. Discussion

Key Result

Theorem 2

Let $p(\theta)$ be a trigonometric polynomial of degree $d$. Then for $u \in (0,\pi/4]$,

Figures (1)

  • Figure 1: The verifier would like to decide whether $\lvert\theta\rvert \leq 3\pi/4$ (a yes-instance) or $\theta = \pi$ (a no-instance), given black-box access to the oracle defined in \ref{['eq:oracle intro']}. Here we give a sketch of what the maximal acceptance probability, optimized over all choices of witness, should look like, when we have completeness $c = 1 - \delta$ close to 1, and soundness $s = 1/3$.

Theorems & Definitions (4)

  • Definition 1
  • Theorem 2
  • Theorem 3: Limits to black-box amplification
  • proof