How to generate sparse complex-valued unitary matrices with given density?

For my research, I need to generate sparse (complex-values) unitary matrices at random from a uniform distribution. It is not a problem for me to generate the generic unitary matrices using, e.g. this code:

X = (randn(n) + i*randn(n))/sqrt(2);
[Q,R] = qr(X);
R = diag(diag(R)./abs(diag(R)));
U = Q*R;

However, I do not know how to generate the sparse version of the matrix, neither I haven't found in the internet. Could you, please, guide me on that?