Classical computing, quantum computing, and Shor's factoring algorithm

TL;DR

This paper provides an expository overview of classical and quantum computing, focusing on the structure, key algorithms like Shor's factoring, and foundational concepts such as quantum parallelism and complexity theory.

Contribution

It offers a comprehensive, categorical, and conceptual explanation of quantum algorithms, including detailed discussion of Shor's algorithm and foundational quantum subroutines.

Findings

Explanation of quantum parallelism and its implications

Detailed description of quantum subroutines including Fourier transform and Grover's algorithm

Insight into the spectral properties of computable functions

Abstract

This is an expository talk written for the Bourbaki Seminar. After a brief introduction, Section 1 discusses in the categorical language the structure of the classical deterministic computations. Basic notions of complexity icluding the P/NP problem are reviewed. Section 2 introduces the notion of quantum parallelism and explains the main issues of quantum computing. Section 3 is devoted to four quantum subroutines: initialization, quantum computing of classical Boolean functions, quantum Fourier transform, and Grover's search algorithm. The central Section 4 explains Shor's factoring algorithm. Section 5 relates Kolmogorov's complexity to the spectral properties of computable function. Appendix contributes to the prehistory of quantum computing.