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Use one target ancilla qubit initially in $|0⟩$. Apply a CNOT gate from each data qubit $x_j$ to that same target, for $j=1,\dots ,n$. Each CNOT gate adds its control bit modulo two without changing the control. After all $n$ gates the target contains the parity bit $f(x)=x_1\oplus \cdots \oplus x_n$. Thus the required circuit is the $n$-gate parity fan-in:This is parity computation by CNOT gates. The circuit also satisfies $W|x⟩|y⟩=|x⟩|y\oplus f(x)⟩$ for either target value, and $W^{-1}=W$ because all these shared-target CNOT gates commute and individually square to the identity. The action on general superpositions follows by linearity.

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.