Generating quantum channels from functions on discrete sets

TL;DR

This paper introduces a method to generate quantum channels from functions on discrete sets, enabling new ways to simulate classical dynamics and perform hybrid classical-quantum computations.

Contribution

It presents a novel approach to create quantum channels from discrete functions, linking classical fixed points and orbits to quantum dynamics, with applications in error correction and hybrid algorithms.

Findings

Constructed quantum channels from logistic map functions demonstrating periodic doubling.

Channels can preserve coherence within subspaces using disjoint subsets.

Illustrated application in quantum algorithms for classical iterative computations.

Abstract

Using the recent ability of quantum computers to initialize quantum states rapidly with high fidelity, we use a function operating on a discrete set to create a simple class of quantum channels. Fixed points and periodic orbits, that are present in the function, generate fixed points and periodic orbits in the associated quantum channel. Phenomenology such as periodic doubling is visible in a 6 qubit dephasing channel constructed from a truncated version of the logistic map. Using disjoint subsets, discrete function-generated channels can be constructed that preserve coherence within subspaces. Error correction procedures can be in this class as syndrome detection uses an initialized quantum register. A possible application for function-generated channels is in hybrid classical/quantum algorithms. We illustrate how these channels can aid in carrying out classical computations involving iteration of non-invertible functions on a quantum computer with the Euclidean algorithm for finding the greatest common divisor of two integers.