Quantum Pushforward Designs

TL;DR

This paper introduces the concept of pushforward designs in quantum information, enabling the creation of new structures from existing ones through mappings, and proposes a framework for quantum channel designs with applications to quantum computing hardware.

Contribution

It presents the novel idea of pushforward designs and the framework of channel [t,k]-designs, extending the tools for averaging in quantum information.

Findings

Introduced pushforward designs for quantum structures.

Defined channel [t,k]-designs for averaging over quantum channels.

Estimated effective environment dimensionality for IBM Kyoto quantum computer.

Abstract

Designs, structures connected to averaging with respect to a given measure using finite sets of points, have proven themselves as invaluable tools across the field of quantum information, finding their uses in state and process tomography, key distribution and others. In this work, we introduce a new concept of pushforward designs, which allows us to obtain new structures from already existing ones by mapping them between the spaces, with specific examples including simplex designs and mixed state designs from complex projective designs. Based on the general concept, we put forward a structure called channel $[t,k]$-design, allowing for averaging over space of quantum channels for systems in contact with an environment of dimension $k$. Based on this notion, we introduce the concept of effective environment dimensionality $k^*$, which we estimate for the IBM Kyoto quantum computer to be below $2.2$ for times up to $350\mu\text{s}$.