Unitary matrix corruption

Question

Suppose I have got a unitary matrix and I want to introduce random noise to simulate data corruption. How to introduce the noise in a proper way such that the corrupted matrix is also unitary?

Answer 1

If I may sum up your problem :

You have an unitary matrix $U$ enventually complex so $U^*U=UU^*=I$

So if we introduce a multiplicative noise $N$ such that the resulting signal is $U'=NU$.

Then if we required $U'$ to be unitary :

$$U^*N^*NU=NUU^*N^*=I $$

Second term implies $NN^*=I$. It is sufficient so to take $N$ such as $N^*N=I$

Then it is sufficient to take a multiplicative noise which is unitary.

Then using this paper https://case.edu/artsci/math/esmeckes/Meckes_SAMSI_Lecture2.pdf can give you idea on how to practically generate it.