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

Fermion parity is $P_f=(-1{)}^F$. An operator $O_{+}$ is parity even when $P_f{O}_{+}{P}_f=O_{+}$, equivalently $[P_f,O_{+}]=0$; it is parity odd when $P_f{O}_{-}{P}_f=-O_{-}$, equivalently $\{P_f,O_{-}\}=0$.

Odd operators supported in disjoint spacelike regions anticommute by the canonical anticommutation relations. If each were an observable, their measurement algebras would fail to commute, allowing the order of spacelike separated measurements to affect predictions. Products of an even number of fermionic fields instead commute at spacelike separation. Locality and compatible spacelike measurements therefore require every physical local observable to be fermion-parity even; parity-odd fields can create charged states but are not themselves observables.

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

A local observable is parity even by part (a). In the infinite-separation limit, any nontrivial parity-even product of zero-mode Majoranas that acts on the encoded information has support near at least two separated zero modes and cannot occur in a single local observable. The projection of $O$ onto the fixed-parity ground space is consequently scalar. For every orthonormal ground-space basis,This is Local indistinguishability of separated Majorana zero modes.

At finite but large $\ell /\xi$, zero-mode wavefunctions overlap by $O(e^{-\ell /\xi })$. The Hamiltonian acquires couplings $i{ϵ}_{ij}{\gamma }_i{\gamma }_j$, the ground-state degeneracy is split, and local observables gain logical matrix elements of the same exponential order:

Write $a={\gamma }_{2M}$ and $b={\gamma }_{2M-1}$. The ancilla condition $iab|\psi ⟩=|\psi ⟩$ implies $a|\psi ⟩=ib|\psi ⟩$ and lets every occurrence of $a$ on the state be replaced by $ib$. Define the two parity projectorsTheir factors square to one, so they are valid fermion-parity measurement projectors. Expanding $P_1{P}_2$, using the Majorana anticommutation relations, replacing $a$ with $ib$ on $|\psi ⟩$, and usinggivesThe proportionality absorbs the probability amplitude for obtaining the two displayed measurement outcomes. This realizes the four-Majorana phase gate using one braid and parity measurements.

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 $|\psi ⟩$,