Adam Sawicki

Connecting unitary t-designs and epsilon-nets

Unitary transformations are at the core of basically all quantum information processing protocols. However, due to computational and practical limitations, implementation of arbitrary unitary gate is in general a difficult task. It is therefore desirable to consider various notions that can capture the features of the unitary group relevant for specific applications. Two such natural concepts are epsilon-nets and (approximate) t-designs that both find broad applications in quantum information and quantum computing. The former constitute subsets of unitary channels that are epsilon-close to every target unitary channel (in a suitable natural distance ). The latter are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of the unitary channel up to order~t. In my talk I will discuss how to establish quantitative connections between these two seemingly different notions. This is a joint work in progress with Michal Horodecki and Michal Oszmaniec.