Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems

TL;DR

This paper establishes quantum query lower bounds for tripartite generalizations of the hidden shift and set equality problems, revealing fundamental complexity limits using advanced mathematical techniques.

Contribution

It introduces new quantum lower bounds for the 3-shift-sum and 3-matching-sum problems, employing dual learning graphs and representation theory.

Findings

Lower bound of Ω(n^{1/3}) for 3-shift-sum

Lower bound of Ω(√n) for 3-matching-sum, which is tight

Novel application of dual learning graph framework and representation theory

Abstract

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 $3\times n$ 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.