Classical and Quantum Query Complexity of Boolean Functions under Indefinite Causal Order

TL;DR

This paper explores how indefinite causal order can reduce the classical and quantum query complexities of Boolean functions, demonstrating polynomial and quantum separations in computational efficiency.

Contribution

It introduces a generalized classical-deterministic framework with causal indefiniteness and shows how it can lead to complexity reductions, including quantum advantages.

Findings

Causal indefiniteness preserves polynomial and certificate lower bounds.

A Boolean function exhibits reduced query complexity due to causal indefiniteness.

Quantum query complexity can be lowered by causally indefinite computations.

Abstract

Computational models typically assume that operations are applied in a fixed sequential order. In recent years several works have looked at relaxing this assumption, considering computations without any fixed causal structure and showing that such ''causally indefinite'' computations can provide advantages in various tasks. Recently, the quantum query complexity of Boolean functions has been used as a tool to probe their computational power in a standard complexity theoretic framework, but no separation in exact query complexity has thus-far been found. In this paper, we investigate this problem starting with the simpler and fully classical notion of deterministic query complexity of Boolean functions, and using classical-deterministic processes -- which may exhibit causal indefiniteness -- as a generalised computational framework. We first show that the standard polynomial and certificate lower bounds of deterministic query complexity also hold in such generalised models. Then, we formulate a Boolean function for which causal indefiniteness permits a reduction in query complexity and show that this advantage can be amplified into a polynomial separation. Finally, with the insights gained in the classical-deterministic setting, we give a Boolean function whose quantum query complexity is reduced by causally indefinite computations.