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

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.