Subgroups of the Clifford Group

Question

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).

What progress has been made in characterizing the subgroups of the Clifford group, and in particular, what progress has been made in characterizing those subgroups isomorphic to the Pauli Group?

This post imported from StackExchange Physics at 2014-06-19 11:26 (UCT), posted by SE-user ruadath

Comments

Progress, from which starting point?

This post imported from StackExchange Physics at 2014-06-19 11:26 (UCT), posted by SE-user Niel de Beaudrap

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Not sure what you mean.

This post imported from StackExchange Physics at 2014-06-19 11:26 (UCT), posted by SE-user ruadath

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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"?

This post imported from StackExchange Physics at 2014-06-19 11:26 (UCT), posted by SE-user Niel de Beaudrap

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I'm just asking if there is some characterization (I have no base knowledge, besides that obviously the Pauli group itself is a subgroup)

This post imported from StackExchange Physics at 2014-06-19 11:26 (UCT), posted by SE-user ruadath