Connecting unitary t-designs and epsilon-nets
Mathematical Physics Seminar
31st January 2020, 2:00 pm – 3:00 pm
Fry Building, 2.04
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.