Epsilon-nets, unitary designs and random quantum circuits

Understand

Epsilon-nets and approximate unitary $t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing.

• The former constitute subsets of unitary channels that are epsilon-close to any unitary channel in the diamond norm.
• The latter are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitary channels up to order $t$.
• In this work we establish quantitative connections between these two notions.