  Unitary $k$-designs are probabilistic ensembles of unitary matrices whose
first $k$ statistical moments match that of the full unitary group endowed with
the Haar measure. In prior work, we showed that the automorphism group of
classical $\mathbb{Z}_4$-linear Kerdock codes maps to a unitary $2$-design,
which established a new classical-quantum connection via graph states. In this
paper, we construct a Markov process that mixes this Kerdock $2$-design with
symplectic transvections, and show that this process produces an
$\epsilon$-approximate unitary $3$-design. We construct a graph whose vertices
are Pauli matrices, and two vertices are connected by directed edges if and
only if they commute. A unitary ensemble that is transitive on vertices, edges,
and non-edges of this Pauli graph is an exact $3$-design, and the stationary
distribution of our process possesses this property. With respect to the
symmetries of Kerdock codes, the Pauli graph has two types of edges; the
Kerdock $2$-design mixes edges of the same type, and the transvections mix the
types. More precisely, on $m$ qubits, the process samples
$O(\log(N^5/\epsilon))$ random transvections, where $N = 2^m$, followed by a
random Kerdock $2$-design element and a random Pauli matrix. Hence, the
simplicity of the protocol might make it attractive for several applications.
From a hardware perspective, $2$-qubit transvections exactly map to the
M{\o}lmer-S{\o}rensen gates that form the native $2$-qubit operations for
trapped-ion quantum computers. Thus, it might be possible to extend our work to
construct an approximate $3$-design that only involves such $2$-qubit
transvections.
