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On the Quantum Time Complexity of Divide and Conquer

Authors Jonathan Allcock , Jinge Bao , Aleksandrs Belovs , Troy Lee, Miklos Santha

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum Computing
  • Quantum Algorithms
  • Divide and Conquer

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Abstract

In this work, we initiate a systematic study of the time complexity of quantum divide and conquer (QD&C) algorithms for classical problems, and propose a general framework for their analysis. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms are amenable to quantum speedup, and apply these theorems to various problems involving strings, integers, and geometric objects. These include Longest Distinct Substring, Klee's Coverage, several optimization problems on stock transactions, and k-Increasing Subsequence. For most of these problems our quantum time upper bounds match the quantum query lower bounds, up to polylogarithmic factors. 
We give a structured framework for describing and classifying a wide variety of QD&C algorithms so that quantum speedups can be more easily identified and applied, and prove general statements on QD&C time complexity covering a range of cases, accounting for the time required for all operations. In particular, we explicitly account for memory access operations in the commonly used QRAM (read-only) and QRAG (read-write) models, which are assumed to take unit time in the query model, and which require careful analysis when involved in recursion.
Our generic QD&C theorems have several nice features.  
1) To apply them, it suffices to come up with a classical divide and conquer algorithm satisfying the conditions of the theorem. The quantization of the algorithm is then completely handled by the theorem. This can make it easier to find applications which admit a quantum speedup, and contrast with dynamic programming algorithms which can be difficult to quantize due to their highly sequential nature. 
2) As these theorems give bounds on time complexity, they can be applied to a greater range of problems than those based on query complexity, e.g., where the best-known quantum algorithms require super-linear time. 
3) It can handle minimization problems as well as boolean functions, which allows us to improve on the query complexity result of Childs et al. [Childs et al., 2025] for k-Increasing Subsequence by a logarithmic factor.

Cite As Get BibTex

Jonathan Allcock, Jinge Bao, Aleksandrs Belovs, Troy Lee, and Miklos Santha. On the Quantum Time Complexity of Divide and Conquer. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.9

Author Details

Jonathan Allcock
  • Tencent Quantum Laboratory, Hong Kong, China
Jinge Bao
  • School of Informatics, University of Edinburgh, UK
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Aleksandrs Belovs
  • Center for Quantum Computing Science, Faculty of Science and Technology, University of Latvia, Riga, Latvia
Troy Lee
  • Centre for Quantum Software and Information, University of Technology Sydney, Ultimo, Australia
Miklos Santha
  • Centre for Quantum Technologies, National University of Singapore, Singapore
  • CNRS, IRIF, Paris, France

Funding

This project was supported by the National Research Foundation, Singapore through the National Quantum Office, hosted in A*STAR, under its Centre for Quantum Technologies Funding Initiative (S24Q2d0009). A.B. is supported by the QuantERA ERA-NET project QOPT. T.L. is supported in part by the ARC DP grant DP200100950. J.B. is supported by the Quantum Advantage Pathfinder (QAP) project of the UKRI Engineering and Physical Sciences Research Council (Grant No. EP/X026167/1), and was funded by a CQT PhD Scholarship when the work was initialized.

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