Given a $2n \times 2n$ square matrix $B$, I would like to find the square matrix $U$ which is closest to $B$ in a suitable norm (say Frobenius or 2-norm), while satisfying two conditions:

\begin{equation} U U^{\dagger} = \mathbf{1}_{2n} \tag{1} \label{eq1} \end{equation} \begin{equation} U^{*} = \gamma U \gamma, \quad\text{with} \quad \gamma = \begin{pmatrix}\mathbf{0} _{n}& \mathbf{1} _{n}\\ \mathbf{1}_{n} &\mathbf{0} _{n}\end{pmatrix} \tag{2} \label{eq2} \end{equation}

Here $\dagger$ and $^{*}$ denote the conjugate transpose and complex conjugation respectively.

On its own Eq. \eqref{eq1} just says that $U$ should live in $\mathrm{U}(2n)$. Eq. \eqref{eq2} imposes additional structure which means that the set of matrices satisfying these two conditions forms a group isomorphic to $\mathrm{O}(2n)$.

If we only required Eq. \eqref{eq1} then $U$ would be given by the polar decomposition of $B$, namely $U = B H^{-1}$ with $H = \sqrt{B^{\dagger} B}$. Is there a similar procedure to find the closest $U$ to $B$ that satisfies both equations \eqref{eq1} and \eqref{eq2}? It would be preferable if such a procedure were efficient in the sense of computational complexity.


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