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Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

Reflections for quantum query algorithms


We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms.
Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound.
Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f ∘ (g, …, g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.

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cover image Proceedings
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 560 - 569
Editor: Dana Randall, Georgia Institute of Technology, Atlanta, Georgia
ISBN (Print): 978-0-898719-93-2
ISBN (Online): 978-1-61197-308-2


Published online: 18 December 2013



Ben W. Reichardt*
School of Computer Science and Institute for Quantum Computing, University of Waterloo


Research supported by NSERC, ARO/DTO and MITACS.

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