In my current use case, I have many binary vectors. In the context of BFV, I want to use each ciphertext to represent one binary vector. On these binary vectors, I need to do a large number of homomorphic multiplications (the multiplicative depth is very low though, like 2-4).

After those multiplications, I need to multiply each vector with a plaintext of integers, that are in a much larger modulo Q.

Thus, I would like to represent the binary vectors using a BFV ciphertext with plaintext modulo P=2, and after all computation on them is done, switch the plaintext modulo to Q and continue with some cheap computation.

I hope, that since the expensive part of the computation is done in a lower plaintext modulo, it can be completed much more quickly.

Is there any way to do this? I am relatively new to FHE, so feel free to keep your answers as thorough as possible or include links for further reading; that would be very appreciated!