Simon's Algorithm, Clebsch-Gordan Sieves, and Hidden Symmetries of Multiple Squares

TL;DR

This paper connects Simon's algorithm to Clebsch-Gordan transforms and demonstrates how these transforms can be used to efficiently find hidden involutions in dihedral groups, advancing quantum algorithm design.

Contribution

It reveals a novel connection between Simon's algorithm and Clebsch-Gordan transforms, enabling efficient quantum solutions for hidden involution problems.

Findings

Clebsch-Gordan transforms can be used to find hidden involutions in dihedral groups.

A new interpretation of Simon's algorithm as a Clebsch-Gordan transform.

Enhanced understanding of symmetries in quantum algorithms.

Abstract

The first quantum algorithm to offer an exponential speedup (in the query complexity setting) over classical algorithms was Simon's algorithm for identifying a hidden exclusive-or mask. Here we observe how part of Simon's algorithm can be interpreted as a Clebsch-Gordan transform. Inspired by this we show how Clebsch-Gordan transforms can be used to efficiently find a hidden involution on the group G^n where G is the dihedral group of order eight (the group of symmetries of a square.) This problem previously admitted an efficient quantum algorithm but a connection to Clebsch-Gordan transforms had not been made. Our results provide further evidence for the usefulness of Clebsch-Gordan transform in quantum algorithm design.