Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

Question

I see here in Olivia DeMatteo's notes, she states:

When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible Paulis an equal number of times. Since we have $|C_n|/|P_n|$ possible Cliffords, we get mapped to each Pauli $\frac{|Cn|/|Pn|}{d^2−1}$ times.

This is also stated in this paper (arXiv)

[...] conjugation under the Clifford group maps each nonidentity Pauli element to every other nonidentity Pauli element with equal frequency.

I don't see a proof for this claim anywhere. Does anyone know of a proof for it? I think the proof for the cardinality of the Clifford group can be used to prove it, but I'm sure there is a simpler way to prove it.

Answer 1

Conjugating a non-identity Pauli operator $P$ by Clifford operators yields all non-identity Pauli operators, including $P$, with equal frequency.

Proof Let $G_n$ and $C_n$ denote the $n$-qubit Pauli and Clifford groups, respectively. For $P, Q\in G_n$, let $C_{P\to Q}$ denote the set of Clifford operators that map $P$ to $Q$ under conjugation, i.e.

$$ C_{P\to Q}=\{U\in C_n\,|\,UPU^\dagger=Q\}. $$

For any $P\in G_n$, $C_{P\to P}$ is a subgroup of $C_n$. Indeed, for any $U\in C_{P\to P}$ its inverse$^1$ $U^\dagger\in C_{P\to P}$ and for any $U, V\in C_{P\to P}$ their product $UV\in C_{P\to P}$.

Let $P,Q\in G_n$ and suppose that neither $P$ nor $Q$ is the identity$^2$. Then there exists$^3$ $V\in C_{P\to Q}$. It is easy to see that

$$ C_{P\to Q} = VC_{P\to P} $$

where $VC_{P\to P}$ is the left coset

$$ VC_{P\to P} := \{VU\,|\,U\in C_{P\to P}\} $$

of the subgroup $C_{P\to P}$ in $C_n$. By group theory, all cosets have the same size, so

$$ |C_{P\to Q}| = |C_{P\to P}| $$

for all $P,Q\in G_n$. Conclusion follows from the observation that $|C_{P\to Q}|$ is precisely the number of Cliffords that map $P$ to $Q$ under conjugation.$\square$

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^{$^1$Strictly speaking, we do not need to check for closure under inverses. Explanation why is an exercise for the reader.
$^2$ The assumption is important, because $C_{I\to I}=C_n$ and $C_{I\to R} = C_{R\to I} =\emptyset$ for all non-identity $R\in G_n$.
$^3$ Proof that such $V$ exists is another exercise for the reader.}