Composing Quantum Algorithms

TL;DR

This paper explores the complexities of composing quantum algorithms, highlighting differences from classical algorithms, especially the advantages in the bounded-error setting where composition can be more efficient.

Contribution

It clarifies how quantum algorithms can be composed effectively in the bounded-error setting, contrasting with the challenges in zero-error quantum algorithm composition.

Findings

Bounded-error quantum algorithms compose more efficiently than classical algorithms.

Zero-error quantum algorithms do not compose straightforwardly.

Quantum composition can avoid the log factor in success probability amplification.

Abstract

Composition is something we take for granted in classical algorithms design, and in particular, we take it as a basic axiom that composing ``efficient'' algorithms should result in an ``efficient'' algorithm -- even using this intuition to justify our definition of ``efficient.'' Composing quantum algorithms is a much more subtle affair than composing classical algorithms. It has long been known that zero-error quantum algorithms \emph{do not} compose, but it turns out that, using the right algorithmic lens, bounded-error quantum algorithms do. In fact, in the bounded-error setting, quantum algorithms can even avoid the log factor needed in composing bounded-error randomized algorithms that comes from amplifying the success probability via majority voting. In this article, aimed at a general computer science audience, we try to give some intuition for these results: why composing quantum algorithms is tricky, particularly in the zero-error setting, but why it nonetheless works \emph{better} than classical composition in the bounded-error setting.