Generating quantum channels from functions on discrete sets

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.