Oracle Separations for the Quantum-Classical Polynomial Hierarchy

TL;DR

This work proves that the quantum-classical polynomial hierarchy (QCPH) is infinite relative to a random oracle, and that higher levels of PH cannot be captured by lower levels of QCPH under such an oracle, extending classical oracle-separation techniques into the quantum-classical regime. It introduces a novel quantum switching lemma and a robust random-restriction/projection framework to drive depth-hierarchy separations, including a polynomial lower-bound tool via fractional block sensitivity. The results hinge on carefully combining random projections with adaptive hybrids to diagonalize against shallow quantum-classical circuits and extend to the QCMA hierarchy (QCMAH). Overall, the paper advances our understanding of where quantum-classical hierarchies sit relative to classical ones, and provides new tools for proving oracle separations in settings with limited quantum depth but quantum-query gates at the bottom layers.

Abstract

We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any $d$, if we apply a random restriction to a function $f$ with quantum query complexity $\mathrm{Q}(f)\le n^{1/3}$, the restricted function becomes exponentially close (in terms of $d$) to a depth-$d$ decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.

Theorems & Definitions (110)

• Theorem 1.1: HRST17
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 1.8: Collapse theorem for $\mathsf{QCPH}$ agkr_qph, FGN23
• Proposition 1.9: Corollary 3.7 of FR99
• Proposition 1.10: Theorem 32 of AIK22