Quantum Lower Bound for a Tripartite Version of the Hidden Shift Problem
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In this paper, we prove a polynomial lower bound of Ω(n^1/3) on the quantum query complexity of the following rather natural generalisation of both the hidden shift and the 3-sum problem. Given an array of 3× 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 lower bound is proven by a novel application of the dual learning graph framework. Additionally, we state a property testing version of the problem, for which we prove a similar lower bound.
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