Loop Composition in Quantum Algorithms

TL;DR

This paper explores how incorporating loops into quantum algorithm composition improves modeling and complexity, addressing limitations of the traditional straight-line circuit model.

Contribution

It introduces a modified branching composition that includes looping, enabling better modeling of quantum algorithms with control flow structures.

Findings

Branching composition alone is insufficient for modeling loops in quantum algorithms.

Including loops in the composition aligns complexity with previous results.

Proper control flow modeling is crucial for designing efficient quantum algorithms.

Abstract

The quantum circuit model essentially treats every quantum algorithm as a straight-line program. While this view is universal, recent work has shown that it is inconvenient for using different-length quantum subroutines in superposition. Using the quantum walk formalism of quantum algorithms, it is possible to model such branching behaviour, and get better composition in this setting. We apply the above branching composition to Grover's algorithm, which gives a variable-time quantum search algorithm that is worse than previous work. The reason it is worse is because branching composition does not take into account another deviation from straight-line programs: looping. We show that by modifying branching composition to also include looping, we can get a complexity that matches previous work. This highlights the importance of properly modeling the program control flow when designing quantum algorithms.