  We review the staircase algorithm to decompose the exponential of a
generalized Pauli matrix and we propose two alternative recursive methods which
offer more efficient quantum circuits. The first algorithm we propose, defined
as the inverted staircase algorithm, is more efficient in comparison to the
standard staircase algorithm in the number of one-qubit gates, giving a
polynomial improvement of n/2. For our second algorithm, we introduce fermionic
SWAP quantum gates and a systematic way of generalizing these. Such fermionic
gates offer a simplification of the number of quantum gates, in particular of
CNOT gates, in most quantum circuits. Regarding the staircase algorithm,
fermionic quantum gates reduce the number of CNOT gates in roughly n/2 for a
large number of qubits. In the end, we discuss the difference between the
probability outcomes of fermionic and non-fermionic gates and show that, in
general, due to interference, one cannot substitute fermionic gates through
non-fermionic gates without altering the outcome of the circuit.
