Model Reduction for Quantum Systems: Discrete-time Quantum Walks and Open Markov Dynamics

TL;DR

This paper introduces a new algebraic method for reducing complex discrete-time quantum systems to simpler models that exactly reproduce outputs while maintaining physical constraints like complete positivity.

Contribution

It develops a general algebraic framework for exact model reduction of quantum systems, ensuring physical validity and applying it to quantum walks and algorithms.

Findings

Reduced models reproduce original outputs exactly

Models satisfy physical constraints such as complete positivity

Algorithm successfully applied to quantum walk examples

Abstract

A general approach to obtain reduced models for a wide class of discrete-time quantum systems is proposed. The obtained models not only reproduce exactly the output of a given quantum model, but are also guaranteed to satisfy physical constraints, namely complete positivity and preservation of total probability. A fundamental framework for exact model reduction of quantum systems is constructed leveraging on algebraic methods, as well as novel results on quantum conditional expectations in finite-dimensions. The proposed reduction algorithm is illustrated and tested on prototypical examples, including the quantum walk realizing Grover's algorithm.