Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 3 / b / Solution

The Bogoliubov--de Gennes Hamiltonian has particle-hole symmetry. A locally nondegenerate zero mode can therefore be chosen particle-hole invariant. If its Nambu wavefunction is $(u_j,u_j^{\ast })$, defineThen ${\gamma }_j^{†}={\gamma }_j$. Exponential localization and $\ell /\xi \to \infty$ make distinct zero-mode wavefunctions orthogonal. Normalizing each one and using the fermionic canonical anticommutation relations gives

Pair the $2M$ Majorana zero modes into ordinary zero-energy fermionsTheir occupations produce $2^M$ states. Physical operations preserve total fermion parity, so choosing one parity sector imposes one binary constraint and leaves $2^{M-1}$ states. Thus Dense encoding with Majorana zero modes gives