  We argue that the chiral $U(1)_A$ symmetry of a Weyl fermion cannot be
implemented by a shallow depth quantum circuit operation in a fermionic lattice
Hamiltonian model with finite dimensional onsite Hilbert spaces. We also extend
this result to discrete ${\mathbb{Z}}_{2N}$ subgroups of $U(1)_A$, in which
case we show that for $N_f$ Weyl fermions of the same helicity, this group
action cannot be implemented with shallow depth circuits when $N_f$ is not an
integer multiple of $2N$.
