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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?

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    $\begingroup$ Progress, from which starting point? $\endgroup$ Jun 18, 2014 at 8:09
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    $\begingroup$ Not sure what you mean. $\endgroup$
    – ruadath
    Jun 18, 2014 at 11:23
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    $\begingroup$ 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"? $\endgroup$ Jun 18, 2014 at 11:49
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    $\begingroup$ I'm just asking if there is some characterization (I have no base knowledge, besides that obviously the Pauli group itself is a subgroup) $\endgroup$
    – ruadath
    Jun 18, 2014 at 12:09

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There are a few interesting subgroups of practical utility:

Subgroup structure used for enumeration or random sampling

The quotient $\mathcal{C}_n/\mathcal{C}_{n-1}$ is extremely useful when enumerating the elements of the Clifford group (or when randomly sampling from it).

One such decomposition is provided in "How to efficiently select an arbitrary Clifford group element" and implemented in the QuantumClifford.jl library as the enumerate_cliffords and enumerate_phases functions (together with converters to unitary matrices).

Subgroup structure permitting much faster simulation of entanglement distillation

When restricted only to entanglement distillation (a type of error detection that is useful for preparing highly entangled pure states of qubits). The most general operation that preserves the entanglement nature of two Bell pairs is:

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The number of such bilocal entanglement distillation gates $|\mathcal{P}_1|^2\times|\mathcal{C}_1^*|^2\times|Q|=4^2\times6^2\times20=11520$ and they can be simulated much faster than even (generic) Clifford gates.

Such a simulator exists as the BPGates.jl library and the decomposition is discussed more in "Faster-than-Clifford Simulations of Entanglement Purification Circuits and Their Full-stack Optimization"

A few other papers on the topic of Clifford group structure when it comes to entanglement distillation:

Pauli group

You are right that the Pauli group is a subgroup of the Clifford group, but that is probably one of the more boring structures available in it. It is much more important that the Clifford group normalizes the Pauli group. That is one of the two reasons that Clifford circuits on stabilizer states are easy to simulate classically (the other one being that there is a small convenient generating set for the exponentially large Pauli group).

There are probably many more interesting decompositions and subgroups of the Clifford group, but these are the ones I have worked with.

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