My interest in QC comes from a problem in geometry called the Atiyah problem on configurations of points. In short, there is a nice one-to-one correspondence between quantum states of a single qubit and the $2$-sphere of directions in $3$-space. This can be seen for example using the Hopf map, or using Pauli matrices. Of course, the two approaches are ultimately related.
To go from $S^2$ to $\mathbb{C}P^1$, one can take the dot product of the $3$-vector representing a point on $S^2$ with the $3$-vector of Pauli matrices. One then obtains an hermitian $2$ by $2$ matrix with eigenvalues $\pm 1$. The $1$-eigenspace defines a point on $\mathbb{C}P^1$.
To go from $\mathbb{C}P^1$ to $S^2$, say one has $\psi \in S^3 \subset \mathbb{C}^2$ (the sphere with respect to the standard hermitian inner product on $\mathbb{C}^2$), one then forms $\psi \psi\dagger$. The tracefree part of the latter is a tracefree hermitian $2$ by $2$ matrix which can be decomposed as $$ \frac{1}{2} \vec{v}.\vec{\sigma},$$ for some uniquely determined $\vec{v} \in S^2$, where $\vec{\sigma}$ denotes the $3$-vector of Pauli matrices.
This provides a link between geometry and quantum states of qubits.
In the Atiyah problem of configurations of points, one associates to $n$ distinct points in $3$-space $n(n-1)$ qubits encoding the pairwise directions between these points. Atiyah's linear independence conjecture is essentially that some quantum state of these $n(n-1)$ qubits, after symmetrizing the states of the qubits corresponding to directions from the same point (towards the other points), has a nonzero projection onto the $1$-dimensional quantum state which is the tensor product of all the antisymmetric Bell states corresponding to pairs of qubits associated to pairs of directions in opposite directions (for example the direction from point $\mathbf{x}_i$ to point $\mathbf{x}_j$, for $1 \leq i < j \leq n$ is paired with the direction from point $\mathbf{x}_j$ to point $\mathbf{x}_i$).
I am interested in QC software which would allow me to compute such projections, please. For example, projecting the state of a multi-qubit system onto a subspace obtained by symmetrizing some indices then antisymmetrizing some other indices. Are there some python QC software that can do that please? I am curious about IBM's qiskit for example. I am eager to have your thoughts, pieces of advice, comments etc.
