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Quantum Advantage and Lower Bounds in Parallel Query Complexity

Authors Joseph Carolan , Amin Shiraz Gilani , Mahathi Vempati

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Author Details

Joseph Carolan
  • University of Maryland, College Park, MD, USA
Amin Shiraz Gilani
  • University of Maryland, College Park, MD, USA
Mahathi Vempati
  • University of Maryland, College Park, MD, USA

Funding

  • Carolan, Joseph: US Department of Energy grant no. DESC0020264.
  • Gilani, Amin Shiraz: U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing and Quantum Testbed Pathfinder programs (award numbers DE-SC0020312 and DE-SC001904).
  • Vempati, Mahathi: US Department of Energy grant no. DESC0020264 and the US Department of Energy Nuclear Energy University Programs award number DE-NE0009417.

Acknowledgements

The authors thank Laxman Dhulipala for suggesting to investigate the relationship between quantum algorithms and parallelism, Luke Schaeffer and Chaitanya Karamchedu for helpful discussions, and Andrew Childs for valuable feedback on an earlier draft. We thank an anonymous reviewer for suggesting the idea for the function in Section 1.1.2, greatly simplifying our original construction. ASG additionally thanks Stacey Jeffery and Ronald de Wolf for helpful discussions.

Cite As Get BibTex

Joseph Carolan, Amin Shiraz Gilani, and Mahathi Vempati. Quantum Advantage and Lower Bounds in Parallel Query Complexity. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.31

Abstract

It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular,  
1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 
2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 
3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 
4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them.  We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Computational complexity theory
  • quantum
  • lower bounds
  • parallel

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