This PDF file includes:

• Section S1. Theoretical details
• Section S2. Experimental details
• Fig. S1. The circuit of one-step cooling algorithm.
• Fig. S2. The process of anticlockwise braiding of MZMs A and C.
• Fig. S3. The process of clockwise braiding of MZMs A and C.
• Fig. S4. The process to anticlockwise braiding of MZMs C and D.
• Fig. S5. The process of clockwise braiding of MZMs C and D.
• Fig. S6. The process for implementing the phase gate based on the dynamics of MZMs.
• Fig. S7. Spatial modes of the output states for the exchange of MZMs A and C.
• Fig. S8. Spatial modes of the output states corresponding to the basis rotation.
• Fig. S9. Spatial modes of the output states for the exchange of MZMs C and D.
• Fig. S10. Spatial modes of the output states for the π8-phase operation.
• Fig. S11. The process to implement the Deutsch-Jozsa algorithm with the braiding of MZMs.
• Fig. S12. Experimental setup for the exchange of MZMs C and D.
• Fig. S13. Experimental setup for the implementation of π8-phase gate and error operations.
• Fig. S14. Experimental setup for the quantum process tomography.
• Fig. S15. Experimental density matrices resulting from the (−π4)-phase gate operation.
• Fig. S16. Experimental density matrices resulting from the gate operations in the full basis.
• Fig. S17. Experimental density matrices resulting from the π8-phase gate in the full basis.
• Fig. S18. The state evolution in the Deutsch-Jozsa algorithm.
• References (38, 39)

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