Although (A'*B) can be done quickly, the answer is no longer sparse. It is just some complex matrix. So then you are basically multiplying two full 3200x3200 complex matrices which will necessarily take a long time. There is no way around this. You can use the fact that A = [W 0; 0 W] to speed things up by roughly a factor of two, by doing the multiplication blockwise.
%%Make some random A and B
X = randn(1600) + 1i*randn(1600);
A = blkdiag(X,X);
B = sprand(3200,3200,0.01);
%%Do the multiplication blockwise
tic;
W = A(1:1600,1:1600);
B11 = B(1:1600,1:1600);
B12 = B(1:1600,1601:3200);
B21 = B(1601:3200,1:1600);
B22 = B(1601:3200,1601:3200);
F1 = [W'*B11*W W'*B12*W; W'*B21*W W'*B22*W];
toc;
tic; F2 = A'*B*A; toc;
%%Confirm that both yield the same answer
E = F1-F2;
max(abs(E(:))) %<-- Very small
If B has some special characteristics, you maybe be able to exploit them to improve this further.
