The Topology of Quantum Algorithms

TL;DR

This paper employs categorical topological semantics to analyze quantum algorithms, uncovering hidden structures, providing new proofs, and proposing generalizations that enhance understanding and performance.

Contribution

It introduces a topological framework for quantum algorithms, offering novel insights, proofs, and generalizations not accessible through traditional algebraic methods.

Findings

Revealed hidden structures in quantum algorithms using topology

Provided new proofs of correctness via topological operations

Proposed a new generalization of the single-shot Grover algorithm

Abstract

We use a categorical topological semantics to examine the Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. This reveals important structures hidden by conventional algebraic presentations, and allows novel proofs of correctness via local topological operations, giving for the first time a satisfying high-level explanation for why these procedures work. We also investigate generalizations of these algorithms, providing improved analyses of those already in the literature, and a new generalization of the single-shot Grover algorithm.