A Separation Between Types of Quantum Oracle Separations

2026-09-29

AI Summary

This study addresses the lack of a systematic characterization of the separation capabilities of different quantum oracles across complexity classes, which has constrained the meta-complexity classification of relativized complexity. To this end, this work proposes a quantum relativization meta-complexity framework that integrates partially ordered set modeling with diagonal unitary matrix construction techniques to establish an oracle capability partial order model. It systematically analyzes the equivalence or separation relationships between PostBQP and PreciseBQP under various oracles. The primary contribution is the first complete classification of separation boundaries for these specific complexity classes. We prove that PostBQP and PreciseBQP are equivalent under polynomial-dimensional unitary oracles, yet can be separated under superpolynomial-dimensional forward-complex diagonal unitary and single-quantum-state preparation oracles, thereby revealing the fine-grained structure of quantum relativization.

Abstract

Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B. We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes $\mathsf{PostBQP}$ and $\mathsf{PreciseBQP}$, the exponentially precise variant of $\mathsf{BQP}$. Both classes equal $\mathsf{PP}$ in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.

Problem

• quantum oracle
• meta-complexity
• relativization
• query complexity
• complexity classes

Innovation

• Quantum oracle separation
• Meta-complexity
• Quantum relativization
• Oracle pantheons
• PostBQP vs PreciseBQP