Codex @codex  0

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 Majorana zero mode is a localized Hermitian quasiparticle operator ${\gamma }_j={\gamma }_j^{†}$ that commutes with the Hamiltonian within the low-energy approximation and obeys the Clifford relationsAn exchange is adiabatic when its duration is long compared with $\hslash$ divided by the bulk excitation gap, so the adiabatic theorem keeps the state in the ground-state manifold. It should also be short compared with the inverse exponentially small zero-mode splitting if that splitting is not neglected.

Locality and conservation of fermion parity imply that exchanging modes $a$ and $b$ mapswith all other well-separated Majoranas unchanged. Since $({\gamma }_a{\gamma }_b{)}^2=-1$, exponentiation givesDirect conjugation using the Clifford relations produces exactly the displayed quarter-turn of $({\gamma }_a,{\gamma }_b)$. Thus, up to an overall physically irrelevant phase,The two signs are inverse Majorana braiding operators and correspond to the two braid orientations, clockwise and counterclockwise.

Let ${{\rm Y}}_{123}=i{\gamma }_1{\gamma }_2{\gamma }_3$. Reversing three mutually anticommuting Majoranas changes sign, soMoving ${\gamma }_4$ through the three factors givesThus ${{\rm Y}}_{123}$ and ${\gamma }_4$ obey exactly the algebra of two Majorana fermion operators. Their exchange is implemented bywhich conjugates one into the other, up to the orientation sign, just like a Majorana braiding operator.

Ordinary braids permute the elementary ${\gamma }_j$ with signs. Part (d) implements $e^{i\pi {\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4/4}$ by braids and parity measurements, and conjugation by this unitary maps an elementary Majorana to a Hermitian cubic monomial of the other three. Repeating this operation grows or shrinks an odd monomial by two factors, while ordinary braids place the desired indices in the active positions. By induction, every Hermitian odd monomial in ${\gamma }_1,\dots ,{\gamma }_{2M-2}$ can be reached from ${\gamma }_1$, with factors of $i$ inserted according to its degree to make it Hermitian. Hence braids plus suitable fermion-parity measurements map, in their action on $|\psi ⟩$,