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Unitary $t$-designs from $relaxed$ seeds

The capacity to randomly pick a unitary across the whole unitary group is a powerful tool across physics and quantum information. A unitary $t$-design is designed to tackle this challenge in an efficient way, yet constructions to date rely on heavy constraints. In particular, they are composed of ensembles of unitaries which, for technical reasons, must contain inverses and whose entries are algebraic. In this work, we reduce the requirements for generating an $\varepsilon$-approximate unitary $t$-design. To do so, we first construct a specific $n$-qubit random quantum circuit composed of a sequence of, randomly chosen, 2-qubit gates, chosen from a set of unitaries which is approximately universal on $U(4)$, yet need not contain unitaries and their inverses, nor are in general composed of unitaries whose entries are algebraic; dubbed $relaxed$ seed. We then show that this relaxed seed, when used as a basis for our construction, gives rise to an $\varepsilon$-approximate unitary $t$-design efficiently, where the depth of our random circuit scales as $poly(n, t, log(1/\varepsilon))$, thereby overcoming the two requirements which limited 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 ( Brandão, Harrow, and Horodecki; Commun. Math. Phys. (2016) 346: 397) are sufficient.