Product block encoding with separate ancillas

Inputs. An integer $n\ge 1$, square matrices $A,B\in {\mathbb{C}}^{n\times n}$, positive normalizations ${\alpha }_A,{\alpha }_B$, and unitaries $U_A,U_B$ giving $({\alpha }_A,a_A,{\epsilon }_A)$ and $({\alpha }_B,a_B,{\epsilon }_B)$ block encodings, with ${\alpha }_A\ge \mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ and ${\alpha }_B\ge \mathrm{\parallel }B{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. Their access and implementation costs are supplied separately under Block-encoding oracle and implementation cost model.

Construction. Put the ancillas in the fixed order $A$-ancilla, $B$-ancilla, system. Let $V_A$ act by $U_A$ on the $A$-ancilla and system and by the identity on the $B$-ancilla. Let $V_B$ act by $U_B$ on the $B$-ancilla and system and by the identity on the $A$-ancilla. Return the unitary $W:=V_A{V}_B$, so $V_B$ acts first, and use all $a_A+a_B$ ancillas for the projected block. No intermediate measurement or reset is performed. The resulting normalization is ${\alpha }_A{\alpha }_B$.

The precise normalized product error bound is proved in Product block-encoding error bound.