New Lower-bounds for Quantum Computation with Non-Collapsing Measurements

2024-11-06

AI Summary

This work investigates query complexity lower bounds for PDQP (Post-Selected Disentangling Quantum Computation) under non-collapsing measurement models. Addressing fundamental problems—including unstructured search, majority, and element distinctness—the authors introduce a novel, equivalent definition of PDQP and, for the first time, extend the positive-weighted adversary method to the non-collapsing measurement framework. Their analysis yields a tight Θ(N^{1/3}) lower bound for unstructured search, substantially improving upon prior bounds for majority and element distinctness. Moreover, the results characterize the inherent limitations of non-collapsing measurements under non-adaptive query constraints, revealing fundamental trade-offs between measurement-induced speedups and computational power. Collectively, this work provides key theoretical tools and quantitative characterizations for understanding how measurement mechanisms fundamentally shape quantum computational capabilities.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

Aaronson, Bouland, Fitzsimons and Lee introduced the complexity class PDQP (which was original labeled naCQP), an alteration of BQP enhanced with the ability to obtain non-collapsing measurements, samples of quantum states without collapsing them. Although PDQP contains SZK, it still requires $Omega(N^{1/4})$ queries to solve unstructured search. We formulate an alternative equivalent definition of PDQP, which we use to prove the positive weighted adversary lower-bounding method, establishing multiple tighter bounds and a trade-off between queries and non-collapsing measurements. We utilize the technique in order to analyze the query complexity of the well-studied majority and element distinctness problems. Additionally, we prove a tight $Theta(N^{1/3})$ bound on search. Furthermore, we use the lower-bound to explore PDQP under query restrictions, finding that when combined with non-adaptive queries, we limit the speed-up in several cases.

Problem

• Establishes tighter bounds for quantum computation with non-collapsing measurements
• Analyzes query complexity of majority and element distinctness problems
• Explores PDQP limitations under query restrictions and non-adaptive queries

Innovation

• Enhances BQP with non-collapsing measurements
• Proves tight Θ(N^1/3) bound on search
• Analyzes query complexity of major problems