Homomorphic decryption circuit in integer-based scheme
The circuit looks like any homomorphic version of a usual circuit: you replace the inputs by the corresponding ciphertexts and the operations by the corresponding homomorphic operations.

That means that,


The decryption circuit has as input a ciphertext $c$, which is a vector of $Q$ bits, and the secret key $sk$, a vector of $P$ bits; and it performs a $\mod 2$ operation at the end.
But the homomorphic decryption circuit, let's call $HDec$, has as input a vector of $Q$ ciphertexts ($\bar{c}$) representing the encryption of each bit of $c$ and a vector of $P$ ciphertexts ($\bar{sk}$) (representing the encryption of each bit of $sk$. And $HDec$ does not perform the $\mod 2$ operation, but a homomorphic operation equivalent to the $\mod 2$. Since it is a homomorphic operation, it returns a ciphertext (so, a $Q$-bit value, as expected).


Thus, $Dec$ takes vectors $(c_1, c_1, ..., c_Q)$ and $(sk_1, sk_2, ..., sk_P)$ and returns one bit $b$, while $HDec$ takes vectors $(Enc(c_1),..., Enc(c_Q))$ and $(Enc(sk_1), ..., Enc(sk_P))$ and returns a $Q$-bit value $Enc(b)$.