# Are all powers $g^m$ in the Clifford hierarchy if $g$ is?

It is already known that the Clifford hierarchy is not closed under arbitrary products, see this post which shows that the product $$THT$$ is not in any level of the hierarchy.

What about products of a gate with itself? If $$g$$ is a gate in the hierarchy then are all powers $$g^m$$ in the hierarchy as well?

Certainly this is true for all $$k=1,2$$ since these are groups. Also any power of a diagonal gate in the Clifford hierarchy is also in the Clifford hierarchy since diagonal gates in $$\mathcal{C}^{(k)}$$ form a group and thus are closed under powers.

A counter example might be something like a gate $$g \in \mathcal{C}^{(3)}$$ such that $$g^2$$ is not in $$\mathcal{C}^{(3)}$$.

TL;DR: Clifford hierarchy is not closed under raising to integer powers. One suitable counterexample is $$g:=TH$$ which resides in the third level $$\mathcal{C}^{(3)}$$ of the Clifford hierarchy, but whose square $$g^2=THTH$$ is altogether outside the hierarchy $$\mathcal{C}=\cup_{i=1}^\infty\mathcal{C}^{(i)}$$.

## Right-multiplication by Cliffords fixes high levels of the hierarchy

The first level $$\mathcal{C}^{(1)}$$ of the $$n$$-qubit Clifford hierarchy is the $$n$$-qubit Pauli group and the $$k$$th level is defined recursively as $$\mathcal{C}^{(k)}:=\{U\in U(2^n)\,|\,\forall P\in\mathcal{C}^{(1)}\quad UPU^\dagger\in\mathcal{C}^{(k-1)}\}.\tag1$$ In particular, conjugation by any Clifford $$C\in\mathcal{C}^{(2)}$$ sends any Pauli operator $$P\in\mathcal{C}^{(1)}$$ to a Pauli operator $$Q\in\mathcal{C}^{(1)}$$ $$CPC^\dagger=Q\in\mathcal{C}^{(1)}.\tag2$$ Similarly, every $$U\in\mathcal{C}^{(k)}$$ for $$k\geqslant 2$$ sends every Pauli operator $$Q\in\mathcal{C}^{(1)}$$ to a unitary in $$\mathcal{C}^{(k-1)}$$ $$UQU^\dagger=V\in\mathcal{C}^{(k-1)}.\tag3$$ Combining $$(2)$$ and $$(3)$$, we see that for every Clifford $$C\in\mathcal{C}^{(2)}$$ and every $$U\in\mathcal{C}^{(k)}$$ $$UCP(UC)^\dagger=UCPC^\dagger U^\dagger=UQU^\dagger=V\in\mathcal{C}^{(k-1)}.\tag4$$ Thus, we obtain the equivalence $$U\in\mathcal{C}^{(k)}\iff UC\in\mathcal{C}^{(k)}\tag5$$ for every Clifford $$C\in\mathcal{C}^{(2)}$$ and any integer $$k\geqslant 2$$.

## Third level is not closed under squares

Finally, $$T\in\mathcal{C}^{(3)}$$, so by $$(5)$$ we have $$g:=TH\in\mathcal{C}^{(3)}$$. But, we know that $$THT$$ is not in the Clifford hierarchy $$\mathcal{C}$$, so using $$(5)$$ again we obtain$$^1$$ that $$g^2=THTH\notin\mathcal{C}$$. Thus, $$g$$ is an operator in the third level of the Clifford hierarchy whose square is not in any level of the hierarchy.

$$^1$$ The equivalence in $$(5)$$ only implies that $$g^2$$ is not in the second or higher level of the Clifford hierarchy, but $$THTH$$ is a $$2\times 2$$ matrix without zero elements, so it is not in $$\mathcal{C}^{(1)}$$, either.