  Parametrized and random unitary (or orthogonal) $n$-qubit circuits play a
central role in quantum information. As such, one could naturally assume that
circuits implementing symplectic transformation would attract similar
attention. However, this is not the case, as $\mathbb{SP}(d/2)$ -- the group of
$d\times d$ unitary symplectic matrices -- has thus far been overlooked. In
this work, we aim at starting to right this wrong. We begin by presenting a
universal set of generators $\mathcal{G}$ for the symplectic algebra
$i\mathfrak{sp}(d/2)$, consisting of one- and two-qubit Pauli operators acting
on neighboring sites in a one-dimensional lattice. Here, we uncover two
critical differences between such set, and equivalent ones for unitary and
orthogonal circuits. Namely, we find that the operators in $\mathcal{G}$ cannot
generate arbitrary local symplectic unitaries and that they are not
translationally invariant. We then review the Schur-Weyl duality between the
symplectic group and the Brauer algebra, and use tools from Weingarten calculus
to prove that Pauli measurements at the output of Haar random symplectic
circuits can converge to Gaussian processes. As a by-product, such analysis
provides us with concentration bounds for Pauli measurements in circuits that
form $t$-designs over $\mathbb{SP}(d/2)$. To finish, we present tensor-network
tools to analyze shallow random symplectic circuits, and we use these to
numerically show that computational-basis measurements anti-concentrate at
logarithmic depth.
