On the Power of Non-adaptive Learning Graphs
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Author(s)
Belovs, Aleksandrs
Rosmanis, Ansis
Date Issued
May 2014
Journal
computational complexity
Publisher
Springer Basel
Citation
Belovs, Aleksandrs, and Ansis Rosmanis. “On the Power of Non-Adaptive Learning Graphs.” Comput. Complex. 23, no. 2 (May 13, 2014): 323–354.
Version
Author's final manuscript
Abstract
We introduce a notion of the quantum query complexity of a certificate structure. This is a formalization of a well-known observation that many quantum query algorithms only require the knowledge of the position of possible certificates in the input string, not the precise values therein.
Next, we derive a dual formulation of the complexity of a non-adaptive learning graph and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it.
For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays and generalizes the quantum query lower bound for the k-sum problem derived recently by Belovs and Špalek (Proceeding of 4th ACM ITCS, 323–328, 2013).
Finally, we use these results to show that the learning graph for the triangle problem by Lee et al. (Proceeding of 24th ACM-SIAM SODA, 1486–1502, 2013) is almost optimal in the above settings. This also gives a quantum query lower bound for the triangle sum problem.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00037-014-0084-1
