Fully graphical treatment of the quantum algorithm for the Hidden Subgroup Problem

TL;DR

This paper introduces a fully diagrammatic, high-level graphical approach to the abelian Hidden Subgroup Problem, providing new insights and extending the protocol's applicability beyond traditional quantum theory.

Contribution

It presents the first fully diagrammatic proof of correctness for the abelian HSP protocol, emphasizing the role of strongly complementary observables and extending the framework.

Findings

Proof of correctness using diagrammatic methods

Extension to real quantum theory and infinite abelian groups

Demonstration of the importance of strongly complementary observables

Abstract

The abelian Hidden Subgroup Problem (HSP) is extremely general, and many problems with known quantum exponential speed-up (such as integers factorisation, the discrete logarithm and Simon's problem) can be seen as specific instances of it. The traditional presentation of the quantum protocol for the abelian HSP is low-level, and relies heavily on the the interplay between classical group theory and complex vector spaces. Instead, we give a high-level diagrammatic presentation which showcases the quantum structures truly at play. Specifically, we provide the first fully diagrammatic proof of correctness for the abelian HSP protocol, showing that strongly complementary observables are the key ingredient to its success. Being fully diagrammatic, our proof extends beyond the traditional case of finite-dimensional quantum theory: for example, we can use it to show that Simon's problem can be efficiently solved in real quantum theory, and to obtain a protocol that solves the HSP for certain infinite abelian groups.