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Determination of optimal control protocols for Majorana zero modes duringtheir exchange is a crucial step towards the realisation of the topologicalquantum computer. In this paper, we study the finite-time exchange process ofMajorana zero modes on a network formed by coupled $p$-wave superconductingone-dimensional nanowires. We provide scalable computational tools foroptimising such an exchange process relying on deep learning techniques. Toaccomplish the scalability, we derive and implement an analytic formula for thegradient of the quantum infidelity which measures the error in the topologicalquantum gate generation in the Majorana zero modes exchange. Our optimisationstrategy relies on learning the optimised transport protocol via a neural netwhich is followed by direct gradient descent fine tuning. The optimisedexchange protocols in the super-adiabatic regime discover the fact that theMajorana zero modes must necessarily stop before crossing a junction point inthe network. We explain that this is caused by fast changes in the energy gapof the system whenever one of the Majorana zero modes approaches a junctionpoint. In particular, the energy gap exhibits oscillations followed by a sharpjump. We explain this phenomenon analytically in the regime where the Majoranazero modes are completely localised. Finally, we study how the disorder in thequantum nanowire affects the exchange protocols. This shows that understandingthe disorder pattern would allow one to improve quantum gate fidelity by one totwo orders of magnitude.

Optimising the exchange of Majorana zero modes in a quantum nanowire network