  The quantum simulation of fermionic gauge field theories is one of the
anticipated uses of quantum computers in the NISQ era. Recently work has been
done to simulate properties of the fermionic Z2 gauge field theory in (1+1) D
and the pure gauge theory in (2+1) D. In this work, we investigate various
options for simulating the fermionic Z2 gauge field theory in (2+1) D. To
simulate the theory on a NISQ device it is vital to minimize both the number of
qubits used and the circuit depth. In this work we propose ways to optimize
both criteria for simulating time dynamics. In particular, we develop a new way
to simulate this theory on a quantum computer, with minimal qubit requirements.
We provide a quantum circuit, simulating a single first order Trotter step,
that minimizes the number of 2-qubit gates needed and gives comparable results
to methods requiring more qubits. Furthermore, variational approaches are
investigated that allow to further decrease the circuit depth.
