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

Symmetry-protected non-Abelian braiding of Majorana Kramers' pairs