Effective Hamiltonian for a non-unitary quantum walk

We construct an effective Hamiltonian for the non-unitary evolution operator of a multi-step quantum walk with a mechanism for gain and loss. This Hamiltonian is expressed in the basis of the identity and Pauli operators to make its spectrum easy to calculate and its symmetries evident. The parametrization we use allows for an efficient evaluation of the phase boundary where the PT-symmetry of the model is known to be spontaneously broken and shows that the effective Hamiltonian, although non-Hermitian, is real.