Quantum Search with In-Place Queries

2025-04-04

AI Summary

This work investigates the fundamental separation between in-place and XOR query models for permutation inversion—a task equivalent to unstructured search—in quantum query complexity. Prior belief held that in-place queries require Ω(N) queries to invert an N-element permutation, whereas XOR queries achieve O(√N) via Grover’s algorithm. We refute this long-standing conjecture by constructing the first quantum algorithm using only O(√N) in-place queries. We introduce the “function erasure” problem and provide the first rigorous proof that in-place queries are strictly weaker than XOR queries for certain tasks. Moreover, we establish the first exact criterion for separating in-place and XOR query complexities, exhibiting an explicit problem with a tight separation ratio of 1∶Θ(√N). Technically, our approach integrates quantum superposition, amplitude amplification, and a novel adversary method—bypassing reliance on efficient uncomputation or standard oracle reflections.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Other Foundations of Search & Optimization
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Data transparency and provenance

Abstract

Quantum query complexity is typically characterized in terms of XOR queries |x,y>to |x,y+f(x)>or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x>to |f(x)>. Some problems are known to require exponentially fewer in-place queries than XOR queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with $O(sqrt{N})$ XOR queries but was conjectured to require $Omega(N)$ in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using $O(sqrt{N})$ in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and $Theta(sqrt{N})$ in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.

Problem

• Compares in-place and XOR quantum query complexities.
• Solves permutation inversion with O(sqrt(N)) in-place queries.
• Identifies problems needing fewer XOR than in-place queries.

Innovation

• Quantum algorithm for Permutation Inversion
• Uses O(sqrt(N)) in-place queries
• Introduces Function Erasure problem