eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-07-16
3:1
3:15
10.4230/LIPIcs.TQC.2018.3
article
Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems
Belovs, Aleksandrs
1
Rosmanis, Ansis
2
Faculty of Computing, University of Latvia, Raina 19, Riga, Latvia
Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore
In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems.
The 3-shift-sum problem is as follows: given a table of 3 x n elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of Omega(n^{1/3}) and Omega(sqrt n), respectively. The second lower bound is tight.
The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools from [Belovs, 2018].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol111-tqc2018/LIPIcs.TQC.2018.3/LIPIcs.TQC.2018.3.pdf
Adversary Bound
Dual Learning Graphs
Quantum Query Complexity
Representation Theory