# Subgroups of the Clifford Group

We recall the definition of a Clifford group (over $n$ qubits) is the set of unitary transformations:

$$\{U: UPU^\dagger\in\mathcal{P}\}$$

where $\mathcal{P}$ denotes the corresponding Pauli group (again over $n$ qubits).

1. What progress has been made in characterizing the subgroups of the Clifford group?

2. In particular, what progress has been made in characterizing those subgroups isomorphic to the Pauli Group?

• Progress, from which starting point? – Niel de Beaudrap Jun 18 '14 at 8:09
• Not sure what you mean. – ruadath Jun 18 '14 at 11:23
• You ask what "progress" there has been made in (etc.), which implies that you are asking about what developments there has been from some starting point. What is it that you already know? Are you just asking whether there exists some characterization? Do you mind if there isn't a characterization, but a description of some subgroups of the Clifford group which are isomorphic to the Pauli group? Do you care if you get an answer which restricts to inner automorphisms of $\mathrm{GL}(2^n)$ or do you want a more complete theory? What base knowledge are you assuming when you say "progress"? – Niel de Beaudrap Jun 18 '14 at 11:49
• I'm just asking if there is some characterization (I have no base knowledge, besides that obviously the Pauli group itself is a subgroup) – ruadath Jun 18 '14 at 12:09