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The Structure of Promises in Quantum Speedups
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11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC) - License:
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- Publication Date: 2016-09-22
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Shalev Ben-David. The Structure of Promises in Quantum Speedups. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.TQC.2016.7
https://doi.org/10.4230/LIPIcs.TQC.2016.7
Abstract
In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure.
Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger.
Keywords
- Quantum computing
- quantum query complexity
- decision tree complexity
- lower bounds
- quantum adversary method
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References
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