Third moments of qudit Clifford orbits and 3-designs based on magic orbits

When the local dimension $d$ is an odd prime, the qudit Clifford group is only a 2-design, but not a 3-design, unlike the qubit counterpart. This distinction and its extension to Clifford orbits have profound implications for many applications in quantum information processing. In this work we systematically delve into general qudit Clifford orbits with a focus on the third moments and potential applications in shadow estimation. First, we introduce the shadow norm to quantify the deviations of Clifford orbits from 3-designs and clarify its properties. Then, we show that the third normalized frame potential and shadow norm are both $\mathcal{O}(d)$ for any Clifford orbit, including the orbit of stabilizer states, although the operator norm of the third normalized moment operator may increase exponentially with the number $n$ of qudits when $d\neq 2\mod 3$. Moreover, we prove that the shadow norm of any magic orbit is upper bounded by the constant $15/2$, so a single magic gate can already eliminate the $\mathcal{O}(d)$ overhead in qudit shadow estimation and bridge the gap between qudit systems and qubit systems. Furthermore, we propose simple recipes for constructing approximate and exact 3-designs (with respect to three figures of merit simultaneously) from one or a few Clifford orbits. Notably, accurate approximate 3-designs can be constructed from only two Clifford orbits. For an infinite family of local dimensions, exact 3-designs can be constructed from two or four Clifford orbits. In the course of study, we clarify the key properties of the commutant of the third Clifford tensor power and the underlying mathematical structures.