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Quantum Algorithms for Learning a Hidden Graph

Authors Ashley Montanaro, Changpeng Shao

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Author Details

Ashley Montanaro
  • School of Mathematics, University of Bristol, UK
  • Phasecraft Ltd., Bristol, UK
Changpeng Shao
  • School of Mathematics, University of Bristol, UK

Funding

This paper was supported by the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) and EPSRC grants EP/R043957/1 and EP/T001062/1. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 817581).

Acknowledgements

We would like to thank João Doriguello and Ryan Mann for helpful discussions on the topic of this work.

Cite AsGet BibTex

Ashley Montanaro and Changpeng Shao. Quantum Algorithms for Learning a Hidden Graph. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.TQC.2022.1

Abstract

We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al. for a "gapped" version of the group testing problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Quantum algorithms
  • query complexity
  • graphs
  • combinatorial group testing

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