Does the secret key in homomorphic encryption schemes like BFV, BGV schemes have to be from {-1,0,1}?

The secret key of BFV, BGV schemes is generated as a random ternary polynomial from R2 ( R2 is the key distribution used to sample polynomials with integer coefficients in {−1,0,1}) Is there any specific reason for it to be a ternary polynomial? can we have it as polynomial from Rq i.e integer coefficients from {0, 1, 2, 3, ..q-1} and still have all the guarantees of being post quantum secure?

Actually, if you sample the secret key (sk) from larger sets, you increase the security of the underlying problem (RLWE).

You can test it by yourself on the Lattice estimator, by keeping N and q fixed, then trying several distributions for sk.

So, it would be better to have an FHE scheme that uses sk sampled uniformly from $\mathcal{R}_q$. However, for schemes like BGV, CKKS, FV, in some operations, we have to divide the ciphertexts by some integer and round, and these operations increase the noise in a way that depends on the norm of sk.

For example, in BGV, there is the modulus switching. You can see that the noise after this operation depends on $||sk||$.

So, to reduce the noise growth due to these operations, we just set $||sk|| = 1$, which we can do by having ternary sk.