Quadratic fermion Hamiltonian

A quadratic fermion Hamiltonian is bilinear in creation and annihilation operators. In a Majorana basis it is determined, up to a constant, by a real antisymmetric matrix.

A Bogoliubov--de Gennes Hamiltonian acts on a Nambu particle-hole spinor and describes quadratic fermion hopping and pairing.

Particle-hole symmetry relates $H_{\mathrm{BdG}}(k)$ to $-H_{\mathrm{BdG}}^{\ast }(-k)$. Consequently every energy $E$ at momentum $k$ is paired with energy $-E$ at momentum $-k$.

A Majorana fermion operator is self-adjoint and obeys $\{c_j,c_k\}=2{\delta }_{jk}$.

A Majorana zero mode is a spatially localized Majorana operator that commutes with a gapped quadratic Hamiltonian, exactly or up to exponentially small finite-size corrections.

Two counterpropagating Majorana fields coupled by a real mass $m(x)$ bind a Majorana zero mode when $m$ changes sign. Its envelope is proportional to $\exp [-{\int }^{x}m(s)ds/(\hslash v)]$ up to orientation conventions, so a constant asymptotic mass gives localization length $\hslash v/|m|$.

For the long-wavelength Bogoliubov--de Gennes Hamiltonian $H=-\mu (x){\tau }_z-i\hslash \Delta {\tau }_x{\partial }_x$, a sign change from $\mu >0$ in a one-dimensional p-wave superconductor to $\mu <0$ in the vacuum binds a zero mode of width $\xi =\hslash \Delta /|\mu |$. Neglecting the quadratic kinetic term is valid when $|\mu |\ll m{\Delta }^2$ up to an inessential numerical factor.

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

When Majorana zero modes have disjoint exponentially localized supports, every local parity-even observable acts as a scalar on the fixed-parity ground space. Finite overlap changes this by terms exponentially small in separation divided by localization length.

The unitary $R_{ab}=\exp (\pi {\gamma }_a{\gamma }_b/4)$ exchanges two Majorana operators by conjugation, sending one to the other and the other to its negative, with signs fixed by braid orientation.

Pairwise fermion-parity measurements involving an ancillary Majorana mode can reproduce a Majorana braiding operator without physically moving the zero modes. Projecting the ancilla back to its initial parity sector turns the product of projectors into the braid unitary up to normalization and known outcome-dependent signs.

A forced Majorana parity measurement alternates an undesired pairwise measurement with a reference parity measurement until the desired outcome occurs. Anticommuting parity observables give either result with probability one half, so the protocol terminates almost surely with a geometric waiting time.

Fermion parity is $(-1{)}^F$. It acts as plus one on bosonic states and minus one on fermionic states. Even operators commute with it and odd operators, including supersymmetry generators, anticommute with it. Local observables are parity even because observable algebras of spacelike separated regions must commute.

A fermion-parity measurement projects onto an eigenspace of the parity of a chosen collection of fermionic modes. For two Majorana fermion operators ${\gamma }_a$ and ${\gamma }_b$, the projectors are $(1\pm i{\gamma }_a{\gamma }_b)/2$.