Pauli-string phase by parity computation ($e^{it{Z}^{\otimes n}}$)

The Pauli Z gates give eigenvalue $(-1{)}^{{\sum }_j{x}_j}$ on $|x⟩$. Use parity computation by CNOT gates to put the parity bit in a zero ancilla, apply $e^{itZ}$ to it, and uncompute. The data acquire exactly $e^{it(-1{)}^{{\sum }_j{x}_j}}$, and the ancillary line returns to zero. This compute-phase-uncompute construction needs $2n+1$ one- and two-qubit gates. Accumulating parity into the last data line instead uses $2n-1$ gates without an extra ancilla. Neither implementation drops the global phase.