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On Exact Quantum Query Complexity

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Abstract

We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by numerically solving the semidefinite programs characterising quantum query complexity for small problem sizes.

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Notes

  1. We would like to thank Scott Aaronson for stressing this point to us.

  2. In some sense, all quantum query algorithms are nonadaptive, as the choice of unitaries applied in the algorithm does not depend on the input. However, the weight placed on queries to different input bits throughout the algorithm does in general depend on the input.

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Acknowledgements

AM was supported by an EPSRC Postdoctoral Research Fellowship and would like to thank Scott Aaronson and Dan Shepherd for comments. We would also like to thank two anonymous referees for their helpful suggestions. Special thanks to Andris Ambainis and Andrey Vihrov for pointing out an error in the algorithm of Sect. 5.1 in an earlier version of this paper.

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Authors and Affiliations

  1. Centre for Quantum Information and Foundations, DAMTP, University of Cambridge, Cambridge, UK

    Ashley Montanaro, Richard Jozsa & Graeme Mitchison

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  1. Ashley Montanaro
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  2. Richard Jozsa
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  3. Graeme Mitchison
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Corresponding author

Correspondence to Ashley Montanaro.

Appendix: A Source Code

Appendix: A Source Code

The following is an example of how the CVX package [15] can be used to determine quantum query complexity. For full source code, see [23]. In this case, we calculate the minimal error probability over all quantum algorithms using 2 queries to compute some function f:{0,1}3→{0,1} (given as a column vector).

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Montanaro, A., Jozsa, R. & Mitchison, G. On Exact Quantum Query Complexity. Algorithmica 71, 775–796 (2015). https://doi.org/10.1007/s00453-013-9826-8

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