Quantum Query Complexity with Matrix-Vector Products

Authors Andrew M. Childs, Shih-Han Hung, Tongyang Li

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Author Details

Andrew M. Childs
  • Joint Center for Quantum Information and Computer Science, Department of Computer Science, and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA
Shih-Han Hung
  • Joint Center for Quantum Information and Computer Science, Department of Computer Science, and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA
Tongyang Li
  • Joint Center for Quantum Information and Computer Science, Department of Computer Science, and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA
  • Center for Theoretical Physics, MIT, Cambridge, MA, USA


We thank Robin Kothari for bringing our attention to work on classical algorithms in the matrix-vector and vector-matrix-vector query models, and for providing feedback on an initial version of this paper. We thank Max Simchowitz and Blake Woodworth for a discussion that clarified aspects of their paper [Braverman et al., 2020], and Jialin Zhang for clarifications of her paper [Sun et al., 2019]. We also thank Ashley Montanaro for pointing out connections to his paper [Montanaro and Shao, 2020], and Joran van Apeldoorn and Sander Gribling for a discussion of their paper [Apeldoorn and Gribling, 2018].

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Andrew M. Childs, Shih-Han Hung, and Tongyang Li. Quantum Query Complexity with Matrix-Vector Products. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Design and analysis of algorithms
  • Quantum algorithms
  • quantum query complexity
  • matrix-vector products


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