Chaotic Roots of the Modular Multiplication Dynamical System in Shor's Algorithm

TL;DR

This paper investigates the paradoxical nature of Shor's quantum modular multiplication operator, revealing it can be expressed as a superposition of chaotic quantum maps, which may explain its integrability despite classical chaos.

Contribution

It demonstrates that Shor's modular multiplication operator can be decomposed into superpositions of chaotic quantum maps, linking integrability with underlying chaotic structures.

Findings

Shor's operator can be expressed as a superposition of quantized chaotic maps.

Signatures of quantum chaos are present in the operator's structure.

The integrability may result from interference among chaotic components.

Abstract

Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the "maximally random" Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an "integrable" quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized A-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other "chaotic" quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity and chaos in and via quantum algorithms.