Qiskit gates: Support PermutationGate

[q-inho]

Part of #369

Summary

This PR add support Qiskit’s PermutationGate and adds tests accordingly.

Spinless Case

For a permutation $\pi$ acting on spinless orbitals, the simulator decomposes $\pi$ into fermionic swaps and applies them sequentially.
On a computational‑basis state $|n_0 n_1 \dots n_{m-1} \rangle$ with occupations $n_i \in \lbrace 0 , 1 \rbrace$, the gate implements

$$ \hat P_{\pi}=\left(\prod_{(k,j)\in T_\pi}\hat S_{k,j}\right) $$

where​ $T_\pi$ is any sequence of transpositions generating $\pi$ and $\hat{S}_{k,j}$ is the fermionic swap exchanging orbitals $k$ and $j$.

Spinful Case

With spinful orbitals split into $\alpha$ and $\beta$ sectors, the simulator enforces that a permutation never mixes spins.
If $\pi$ separates into permutations $\pi_\alpha$​ and $\pi_\beta$​ acting within the $\alpha$ and $\beta$ sectors, then for a basis state $|n^\alpha; n^\beta\rangle = |n_0^\alpha \dots n_{n-1}^\alpha; n_0^\beta \dots n_{n-1}^\beta\rangle$

$$ \hat P_{\pi} = \left(\prod_{(k,j)\in T_{\pi_alpha}}\hat S_{k,j}^\alpha\right)\left(\prod_{(k,j)\in T_{\pi_\beta}}\hat S_{k,j}^\beta\right) $$

[kevinsung]

I think this can be improved by factoring out a generator function that yields the swap pairs of the permutation decomposition. Then this can look something like

for i, j in _decompose_permutation(perm):
    vec = _apply_swap(vec, (qs[i], qs[j]), ... )

[kevinsung]

This will introduce many "swap defect" sign fixing operations. I wonder if we can use FSWAPS instead of SWAPS, and then only fix the signs a single time at the end. Or maybe it is not straightforward ...

[q-inho]

Thank you for pointing this out!
I have to switch the implementation to use FSWAPs for each transposition so the $−1$ on $|11\rangle$ is handled locally. So it can removes those repeated sign corrections.

[kevinsung]

If it's possible to use FSWAPs instead of SWAPs, then the entire permutation can actually be performed as a single orbital rotation.