  We develop the complete theory for non-Abelian braiding of Majorana Kramers'
pairs (MKPs) in time-reversal (TR) invariant topological superconductors. By
introducing an effective Hamiltonian approach to describe the braiding of MKPs,
we show that the non-Abelian braiding is protected when the effective
Hamiltonian exhibits a new TR like anti-unitary symmetry, which is satisfied if
the system is free of dynamical noise. Importantly, even the dynamical noise
may not cause error in braiding, unless the noise correlation function breaks a
dynamical TR symmetry, which generalizes the TR symmetry protection of MKPs to
dynamical regime. Moreover, the resulted error by noise is shown to be a higher
order effect, compared with the decoherence of Majorana qubits without TR
symmetry protection. These results show that the non-Abelian braiding of MKPs
is observable and may have versatile applications to future quantum computation
technologies.
