The modular multiplication operator and the quantized bakers maps

TL;DR

This paper explores the quantum properties of the modular multiplication operator used in Shor's algorithm, revealing its connection to quantum baker's maps and analyzing its chaotic behavior and error dynamics.

Contribution

It demonstrates that the modular multiplication operator can be viewed as a superposition of quantum baker's maps and examines its transition to quantum chaos and error decay.

Findings

Modular multiplication operator is a superposition of two quantum baker's maps.

Perturbations can induce regimes of quantum chaos with spectral properties akin to random matrices.

Fidelity decay due to errors exhibits exponential behavior with characteristic shoulders.

Abstract

The modular multiplication operator, a central subroutine in Shor's factoring algorithm, is shown to be a coherent superposition of two quantum bakers maps when the multiplier is 2. The classical limit of the maps being completely chaotic, it is shown that there exist perturbations that push the modular multiplication operator into regimes of generic quantum chaos with spectral fluctuations that are those of random matrices. For the initial state of relevance to Shor's algorithm we study fidelity decay due to phase and bit-flip errors in a single qubit and show exponential decay with shoulders at multiples or half-multiples of the order. A simple model is used to gain some understanding of this behavior.