Even quantum advice is unlikely to solve PP

TL;DR

This work provides a corrected proof that $PP \subseteq BQP/qpoly$ would collapse the Counting Hierarchy to $QMA$, specifically to the oblivious class $YQP^*$, and shows $YQP^* \subseteq APP$, establishing $PP$-lowness. The key technical engine is an in-place Marriott–Watrous error reduction applied to the $YQP^*$ verification circuit, enabling a reduction of the ratio of quantum acceptance probabilities to a $GapP$-based decision with a fixed, small gap. Consequently, the paper strengthens unconditional bounds against quantum circuits of fixed polynomial size even with quantum advice, and clarifies the landscape around non-uniform quantum advice, $YQP^*$-low, and the placement of oblivious witness classes inside $APP$. Overall, the results yield a more robust understanding of how quantum advice interacts with classical complexity classes and the Counting Hierarchy, and they refine the pathway from $PP$-related questions to concrete circuit lower bounds.

Abstract

We give a corrected proof that if PP $\subseteq$ BQP/qpoly, then the Counting Hierarchy collapses, as originally claimed by [Aaronson 2006 arXiv:cs/0504048]. This recovers the related unconditional claim that PP does not have circuits of any fixed size $n^k$ even with quantum advice. We do so by proving that YQP*, an oblivious version of (QMA $\cap$ coQMA), is contained in APP, and so is PP-low.

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof