Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 324 / 1 / a / i / Solution

A $Z$ measurement is a quantum measurement in the computational basis. To measure $X$, apply a Hadamard gate, measure $Z$, and apply another Hadamard gate to the measured qubit. Since $HXH=Z$, this gives the $X$ outcome and leaves the qubit in the corresponding $X$ eigenstate.