Given a unitary matrix Q and a symmetric matrix B, I am trying to find a permutation matrix P such that

$ || QBQ^{T} - PBP^{T} ||_{F} $

is minimized.

The straightforward method of minimizing $ || Q - P ||_{F} $ does not work.

I was wondering if there would be some way to orthogonally project the orbit of B under conjugation by unitary matrices onto the orbit of B under conjugation by permutation matrices. I don't know precisely how that would work though.

Does anyone have any suggestions?

Thanks,

Charles