### Résumé

In this work we reduce the requirements for generating $t$-designs, an important tool for randomisation with applications across quantum information and physics. We show that random quantum circuits with support over families of $relaxed$ finite sets of unitaries which are approximately universal in $U(4)$ (we call such sets $seeds$), converge towards approximate unitary $t$-designs efficiently in $poly(n,t)$ depth, where $n$ is the number of inputs of the random quantum circuit, and $t$ is the order of the design. We show this convergence for seeds which are relaxed in the sense that every unitary matrix in the seed need not have an inverse in the seed, nor be composed entirely of algebraic entries in general, two requirements which have restricited previous constructions. We suspect the result found here is not optimal, and can be improved. Particularly because the number of gates in the relaxed seeds introduced here grows with $n$ and $t$. We conjecture that constant sized seeds such as those in (Brand~ao, Harrow, and Horodecki, Commun. Math. Phys. 2016) are sufficient.

Publication

Unitary $t$-designs from $relaxed$ seeds

In this work we reduce the requirements for generating $t$-designs, an important tool for randomisation with applications across quantum information and physics. We show that random quantum circuits with support over families of $relaxed$ finite sets of unitaries which are approximately universal in $U(4)$ (we call such sets $seeds$), converge towards approximate unitary $t$-designs efficiently in $poly(n,t)$ depth, where $n$ is the number of inputs of the random quantum circuit, and $t$ is the order of the design. We show this convergence for seeds which are relaxed in the sense that every unitary matrix in the seed need not have an inverse in the seed, nor be composed entirely of algebraic entries in general, two requirements which have restricited previous constructions. We suspect the result found here is not optimal, and can be improved. Particularly because the number of gates in the relaxed seeds introduced here grows with $n$ and $t$. We conjecture that constant sized seeds such as those in (Brand~ao, Harrow, and Horodecki, Commun. Math. Phys. 2016) are sufficient.