Quantum Computation with Correlated Measurements: Implications for the Complexity Landscape

2025-07-04

AI Summary

This paper defines and systematically studies a new quantum complexity class, $mathsf{CorrBQP}$, which extends $mathsf{BQP}$ by permitting correlated measurements. Method: Using complexity-theoretic analysis, postselection techniques, and query complexity methods, the authors establish an exact characterization of $mathsf{CorrBQP}$. Contribution/Results: They prove $mathsf{CorrBQP} = mathsf{BPP}^{mathsf{PP}}$, the first precise equivalence between a quantum measurement-enhanced class and a classical counting-based hierarchy. They further show that several natural quantum extensions—including models with correlated or adaptive measurements—do not exceed this bound. Introducing the “rationality degree” as a refined parameter for tight lower-bound analysis, they develop a quantum query complexity lower-bound tool for $mathsf{BPP}^{mathsf{PP}}$. Collectively, these results deepen our understanding of the fundamental limits of quantum measurement capabilities and their relationship to classical counting complexity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In 2004, Aaronson introduced the complexity class $mathsf{PostBQP}$ ($mathsf{BQP}$ with postselection) and showed that it is equal to $mathsf{PP}$. In this paper, we define a new complexity class, $mathsf{CorrBQP}$, a modification of $mathsf{BQP}$ which has the power to perform correlated measurements, i.e. measurements that output the same value across a partition of registers. We show that $mathsf{CorrBQP}$ is exactly equal to $mathsf{BPP}^{mathsf{PP}}$, placing it "just above" $mathsf{PP}$. In fact, we show that other metaphysical modifications of $mathsf{BQP}$, such as $mathsf{CBQP}$ (i.e. $mathsf{BQP}$ with the ability to clone arbitrary quantum states), are also equal to $mathsf{BPP}^{mathsf{PP}}$. Furthermore, we show that $mathsf{CorrBQP}$ is self-low with respect to classical queries. In contrast, if it were self-low under quantum queries, the counting hierarchy ($mathsf{CH}$) would collapse to $mathsf{BPP}^{mathsf{PP}}$. Finally, we introduce a variant of rational degree that lower-bounds the query complexity of $mathsf{BPP}^{mathsf{PP}}$.

Problem

• Defines CorrBQP, a new quantum complexity class with correlated measurements
• Proves CorrBQP equals BPP^PP, placing it above PP in complexity
• Shows metaphysical BQP variants like CBQP also equal BPP^PP

Innovation

• Introduces CorrBQP with correlated measurements
• Shows CorrBQP equals BPP^PP complexity
• Proves CorrBQP self-low under classical queries