The Quantum Query Complexity of Finding a Tarski Fixed Point on the 2D Grid

TL;DR

This paper establishes that the quantum query complexity for finding Tarski fixed points on a 2D grid matches the classical bound, demonstrating a fundamental limit in quantum algorithms for this problem.

Contribution

It proves a tight lower bound of a((\u221a) ) for quantum query complexity, linking fixed point finding to nested ordered search.

Findings

Quantum query complexity is a((a( ))^2) for the problem.

Classical and quantum complexities are asymptotically equal.

The proof connects fixed point search to nested ordered search.

Abstract

Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We specifically consider the two-dimensional lattice $\mathcal{L}^2_n$ on points $\{1, \ldots, n\}^2$ and where $(x_1, y_1) \leq (x_2, y_2)$ if $x_1 \leq x_2$ and $y_1 \leq y_2$. We show that the quantum query complexity of finding a fixed point given query access to a monotone function on $\mathcal{L}^2_n$ is $\Omega((\log n)^2)$, matching the classical deterministic upper bound. The proof consists of two main parts: a lower bound on the quantum query complexity of a composition of a class of functions including ordered search, and an extremely close relationship between finding Tarski fixed points and nested ordered search.