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Span-Program-Based Quantum Algorithm for Evaluating Unbalanced Formulas

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Theory of Quantum Computation, Communication, and Cryptography (TQC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6745))

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Abstract

The formula-evaluation problem is defined recursively. A formula’s evaluation is the evaluation of a gate, the inputs of which are themselves independent formulas. Despite this pure recursive structure, the problem is combinatorially difficult for classical computers.

A quantum algorithm is given to evaluate formulas over any finite boolean gate set. Provided that the complexities of the input subformulas to any gate differ by at most a constant factor, the algorithm has optimal query complexity. After efficient preprocessing, it is nearly time optimal. The algorithm is derived using the span program framework. It corresponds to the composition of the individual span programs for each gate in the formula. Thus the algorithm’s structure reflects the formula’s recursive structure.

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Notes

  1. 1.

    Actually, [ACR+10, Section 7] only shows a bound of \(O(\sqrt{n} \, d^{3/2})\) queries, but this can be improved to \(O(\sqrt{n} \, d )\) using the bounds on \(\sigma _\pm (\varphi )\) below [ACR+10, Definition 1].

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Acknowledgements

I thank Andrew Landahl and Robert Špalek for helpful discussions. Research supported by NSERC and ARO-DTO.

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Authors and Affiliations

  1. Institute for Quantum Computing, University of Waterloo, Waterloo, Canada

    Ben W. Reichardt

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  1. Ben W. Reichardt
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Correspondence to Ben W. Reichardt .

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Editors and Affiliations

  1. University of Washington, Seattle, Washington, USA

    Dave Bacon

  2. Universidad Complutense de Madrid, Madrid, Spain

    Miguel Martin-Delgado

  3. NEC Laboratories America, Princeton, New Jersey, USA

    Martin Roetteler

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Reichardt, B.W. (2014). Span-Program-Based Quantum Algorithm for Evaluating Unbalanced Formulas. In: Bacon, D., Martin-Delgado, M., Roetteler, M. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2011. Lecture Notes in Computer Science(), vol 6745. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54429-3_6

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  • DOI: https://doi.org/10.1007/978-3-642-54429-3_6

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