Dissipationless topological quantum computation for Majorana objects in the sparse-dense mixed encoding process

Topological quantum computation based on Majorana objects is subject to a significant challenge because at least some of the two-qubit quantum gates rely on the fermion (either charge or spin) parity of the qubits. This dependency renders the quantum operations involving these gates probabilistic when attempting to advance quantum processes within the quantum circuit model. Such an approach leads to significant information loss whenever measurements yield the undesired fermion parity. To resolve the problem of wasting information, we devise topological operations that allow for the non-dissipative correction of information from undesired fermion parity to the desired one. We will use the sparse-dense mixed encoding process for the controlled-NOT gate as an example to explain how corrections can be implemented without affecting the quantum information carried by the computational qubits. This correction process can be applied {to} either the undesired input qubits or the fermion parity-dependent quantum gates, and it works for both Majorana-zero-mode-based and Majorana-edge-mode-based topological quantum computation.