Grover search with a nonzero reflection label ($G_y=T_{y}G{T}_y,T_y|x⟩=(-1{)}^{x\cdot y}|x⟩$)

Let $|s⟩=H^{\otimes n}|0^n⟩$, let $U_f$ be the marked-state phase oracle, and replace reflection about $|0^n⟩$ by reflection about a known computational basis vector $|y⟩$. Define $T_y={⨂}_{j=0}^{n-1}{Z}_j^{y_j}$, so $T_y|x⟩=(-1{)}^{x\cdot y}|x⟩$ and $H^{\otimes n}|y⟩=T_y|s⟩$. The resulting Grover iterate obeysbecause the two diagonal operators $T_y,U_f$ commute. Thus the good and bad uniform vectors are replaced by their signed vectors $T_y|g⟩,T_y|b⟩$, with exactly the same Grover rotation angle. Initializing in $H^{\otimes n}|y⟩$ gives $G_y^r|s_y⟩=T_y{G}^r|s⟩$. Computational-basis measurement probabilities are unchanged because $T_y$ only multiplies amplitudes by signs.