A 1.5-Query Lower Bound for the Unitary Synthesis Problem

Abstract: We prove a new lower bound for the unitary synthesis problem in the so-called 1.5-query setting. Our analysis establishes that any attempt to implement arbitrary n-qubit unitaries via limited oracle access requires resources that exceed the fractional query threshold. This result extends the one-query lower bound of Lombardi, Ma, and Wright (2023) to the fractional query regime, and introduces a conservative and chaining-based approach to handle intermediate query complexities. As a consequence, we derive cryptographic implications, showing that pseudorandom quantum states remain secure against adversaries restricted to 1.5 queries. Our work provides both conceptual clarification of fractional-query complexity and practical insights into the design of quantum cryptographic protocols.