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Quantum Search with In-Place Queries

Authors Blake Holman , Ronak Ramachandran , Justin Yirka

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ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Quantum algorithms
  • query complexity
  • quantum complexity theory
  • quantum search
  • Grover’s algorithm
  • permutation inversion

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Abstract

Quantum query complexity is typically characterized in terms of xor queries |x,y⟩ ↦ |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⟩↦ |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(√N) xor queries but was conjectured to require Ω(N) in-place queries.
We refute this conjecture by designing a quantum algorithm for Permutation Inversion using O(√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 Θ(√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.

Cite As Get BibTex

Blake Holman, Ronak Ramachandran, and Justin Yirka. Quantum Search with In-Place Queries. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.TQC.2025.1

Author Details

Blake Holman
  • Sandia National Laboratories, Albuquerque, NM, USA
  • Purdue University, West Lafeyette, IN, USA
Ronak Ramachandran
  • The University of Texas at Austin, TX, USA
Justin Yirka
  • Sandia National Laboratories, Albuquerque, NM, USA
  • The University of Texas at Austin, TX, USA

Funding

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. This work is supported by a collaboration between the US DOE and other Agencies. BH and JY acknowledge this work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing, Fundamental Algorithmic Research for Quantum Utility, with support also acknowledged from Fundamental Algorithm Research for Quantum Computing.
  • Ramachandran, Ronak: Supported by the NSF AI Institute for Foundations of Machine Learning (IFML).
  • Yirka, Justin: This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator.

Acknowledgements

We especially thank John Kallaugher for helpful conversations and mentorship. RR and JY thank Scott Aaronson for stimulating discussions. JY thanks Bill Fefferman for suggesting his conjecture on Permutation Inversion.

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