Reassessing the computational advantage of quantum-controlled ordering of gates

TL;DR

This paper reevaluates the computational benefits of quantum-controlled gate ordering, showing that the advantage over causal algorithms is less significant than previously thought, with new algorithms reducing query complexity.

Contribution

It introduces new causal algorithms that solve Fourier promise problems with fewer queries, challenging prior assumptions about quantum advantage in indefinite causal structures.

Findings

Causal algorithms can solve specific FPP with O(n log n) queries.

General FPP can be solved causally with O(n√n) queries.

The advantage of indefinite causal structures is limited for these problems.

Abstract

Research on indefinite causal structures is a rapidly evolving field that has a potential not only to make a radical revision of the classical understanding of space-time but also to achieve enhanced functionalities of quantum information processing. For example, it is known that indefinite causal structures provide exponential advantage in communication complexity when compared to causal protocols. In quantum computation, such structures can decide whether two unitary gates commute or anticommute with a single call to each gate, which is impossible with conventional (causal) quantum algorithms. A generalization of this effect to $n$ unitary gates, originally introduced in M. Ara\'ujo et al., Phys. Rev. Lett. 113, 250402 (2014) and often called Fourier promise problem (FPP), can be solved with the quantum-$n$-switch and a single call to each gate, while the best known causal algorithm so far calls $O(n^2)$ gates. In this work, we show that this advantage is smaller than expected. In fact, we present a causal algorithm that solves the only known specific FPP with $O(n \log(n))$ queries and a causal algorithm that solves every FPP with $O(n\sqrt{n})$ queries. Besides the interest in such algorithms on their own, our results limit the expected advantage of indefinite causal structures for these problems.