On the Power of Non-adaptive Learning Graphs

Aleksandrs Belovs, Ansis Rosmanis
2014 Computational Complexity  
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
more » ... xists 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, 2012). 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.
doi:10.1007/s00037-014-0084-1 fatcat:3a2lurou7nhd5ilcvv6dxakd4y