  Qubits built out of Majorana zero modes (MZMs) constitute the primary path
towards topologically protected quantum computing. Simulating the braiding
process of multiple MZMs corresponds to the quantum dynamics of a
superconducting many-body system. It is crucial to study the Majorana dynamics
both in the presence of all other quasiparticles and for reasonably large
system sizes. We present a method to calculate arbitrary many-body
wavefunctions as well as their expectation values, correlators and overlaps
from time evolved single-particle states of a superconductor, allowing for
significantly larger system sizes. We calculate the fidelity, transition
probabilities, and joint parities of Majorana pairs to track the quality of the
braiding process. We show how the braiding success depends on the speed of the
braid. Moreover, we demonstrate the topological CNOT two-qubit gate as an
example of two-qubit entanglement. Our work opens the path to test and analyze
the many theoretical implementations of Majorana qubits. Moreover, this method
can be used to study the dynamics of any non-interacting superconductor.
