Hadamard-conjugated linear oracle ($H^{\otimes (n+1)}{U}_a{H}^{\otimes (n+1)}$)

Let $U_a|x⟩|b⟩=|x⟩|b\oplus (a\cdot x)⟩$ be the Boolean quantum oracle for a linear Boolean function, and put $W=H^{\otimes (n+1)}$. The CNOT gates $C_{j\to y}^{a_j}$ implement $U_a$, where the data wires control the ancilla qubit. CNOT control reversal in the Hadamard basis and $W^2=I$ imply the exact unitary operator identityHere $\oplus$ on strings is bitwise exclusive or. To conjugate a product, insert $W^2=I$ between its factors; each factor reverses its control and target, and the unused Hadamard gates cancel. The global phase is exactly $1$. In particular, the identity holds on arbitrary quantum superpositions, not only on the initialized inputs of the Bernstein-Vazirani algorithm.