In this paper (lemma 3) it is claimed that any column of a unitary matrix that has two entries at zero can be approximated to $\mathbb{Z}[i,1/\sqrt2]$ by solving the Diophantine equation $a^2 + b^2 + c^2 + d^2 = n$ (the original unitary is decomposed in several reflections, each one of them corresponding to a column). However, while this process preserves the unit norm of the individual columns, it does not preserve the orthogonality of the original matrix (as far as I could check numerically). Is there a method to approximate an unitary matrix to $\mathbb{Z}[i,1/\sqrt2]$ such that the approximation is also unitary?
2026-03-26 07:58:29.1774511909
Approximating an unitary matrix to $\mathbb{Z}[i,1/\sqrt2]$ while keeping unitarity
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