Phase Coordinate Uncomputation in Quantum Recursive Fourier Sampling

TL;DR

This paper introduces a phase space perspective on Recursive Fourier Sampling, revealing that uncomputing phase coordinate garbage is essential for quantum advantage and explaining the limitations of this advantage.

Contribution

It provides a new phase space framework for understanding quantum algorithms in RFS and clarifies why uncomputation is crucial for quantum advantage.

Findings

Uncomputation of phase coordinate garbage is necessary for quantum advantage in RFS.

Phase space description enhances understanding of quantum computational processes.

Limitations of quantum advantage are linked to uncomputing phase information.

Abstract

Recursive Fourier Sampling (RFS) was one of the earliest problems to demonstrate a quantum advantage, and is known to lie outside the Merlin--Arthur complexity class. This work contains a new description of quantum algorithms in phase space terminology, demonstrating its use in RFS, and how and why this gives a better understanding of the quantum advantage in RFS. Most importantly, describing the computational process of quantum computation in phase space terminology gives a much better understanding of why uncomputation is necessary when solving RFS: the advantage is present only when phase coordinate garbage is uncomputed. This is the underlying reason for the limitations of the quantum advantage.