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 $(u_j,u_j^*)$, define
$$
\gamma_j=\sum_x\left[u_j(x)a_x+u_j(x)^*a_x^\dagger\right].
$$
Then $\gamma_j^\dagger=\gamma_j$. Exponential localization and $\ell/\xi\to\infty$ make distinct zero-mode wavefunctions orthogonal. Normalizing each one and using the fermionic canonical anticommutation relations gives
$$
\boxed{\gamma_j=\gamma_j^\dagger,
\qquad\{\gamma_i,\gamma_j\}=2\delta_{ij}\mathbb1.}
$$

Pair the $2M$ <Majorana zero modes> into ordinary zero-energy fermions
$$
f_r=\frac12(\gamma_{2r-1}+i\gamma_{2r}),
\qquad r=1,\ldots,M.
$$
Their 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
$$
\boxed{\dim\mathcal L=2^{M-1},
\qquad k=M-1.}
$$

Solution

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

{c}
{title2=MZM dense encoding}

$2M$ Majorana zero modes define $M$ complex fermionic modes. Fixing total <fermion parity> leaves a $2^{M-1}$-dimensional ground space, encoding $M-1$ qubits.

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 Incoming links: Dense encoding with Majorana zero modes