# Phase Coordinate Uncomputation in Quantum Recursive Fourier Sampling

## TL;DR

This paper explains how quantum advantage in Recursive Fourier Sampling is tied to managing phase coordinates during computation.

## Contribution

A new phase space description of quantum algorithms reveals why uncomputation is essential for quantum advantage in RFS.

## Key findings

- Quantum advantage in RFS is lost without uncomputing phase coordinate garbage.
- Phase space terminology clarifies the necessity of uncomputation in RFS.
- This approach provides deeper insight into the limitations of quantum advantage.

## 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.