[seqfan] Phases in A003956

[Alex Meiburg]

I was trying to find the sequence of sizes of the Clifford group (24
distinct elements of SU(2) on 1 qubit, 11520 distinct elements of SU(4) on
2 qubits, etc..) and could not initially find it.

I found A003956, which instead seems to count all the elements with a
multiplicity of 8. And I'm wondering where this number comes from.

The clifford group typically has two definitions. One is as "the set of all
matrices stabilizing the Pauli group", in which case the set is infinite in
number, because you can multiply a matrix through by an exp(i*theta)
without changing what it stabilizes. The other definition is that you treat
the matrices as part of SU(n), where you consider them equivalent up to
phase. In this case the sequence goes 1,24,11520... . This is also the
numbers given on Wikipedia, which is how I managed to find the OEIS
number: Clifford
gates - Wikipedia <https://en.wikipedia.org/wiki/Clifford_gates#cite_ref-2>

The 8-fold counting is, I assume, due to some convention that the phases
are counted only up to exp(k*i*pi/4). But, this is also strange. With the
clifford gates written in their most common form (H, S, CNOT), the only
phases that ever occur are of the form exp(k*i*pi/2), so I could expect a
four-fold multiplicity if all the matrices are generated from strings of
those matrices.

Is there a reason why this should be natural? Should be there a sequence
with the 24,11520, etc. counting?

-- Alexander Meiburg