Measurement of a Pauli observable

The expectation of a Hermitian Pauli string can be obtained by basis rotation and measurement, or by a Hadamard test with the Pauli string as its controlled unitary.

Prepare an ancilla in $|+⟩$, use it to control a Hermitian Pauli operator $P$, apply a Hadamard gate, and measure the ancilla. Outcome $s$ applies the projector $[I+(-1{)}^{s}P]/2$ to the data register.

Pauli-based computation performs an adaptive sequence of commuting Pauli measurements on nonstabilizer resource states. Classical outcomes determine later measurements and the final output.