LCU block-encoding construction

Inputs. Integers $n\ge 1$ and $m\ge 1$, a finite family of square matrices $A_j\in {\mathbb{C}}^{n\times n}$, positive numbers ${\alpha }_j\ge \mathrm{\parallel }{A}_j{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$, and exact $({\alpha }_j,a_j,0)$ block-encoding unitaries $U_j$. Also supply coefficients ${\beta }_j\in \mathbb{C}$ after omitting zero coefficients, with $\mathrm{\Lambda }:={\sum }_{j=0}^{m-1}\mathrm{\mid }{\beta }_j\mathrm{\mid }{\alpha }_j>0$. The component circuits and their controlled access are separate access inputs.

Instructions. Let $a_{\ast }:={\max }_j{a}_j$. Extend each component unitary to the common ordered ancilla register by identities on its unused ancillas, so its all-zero block is unchanged. Let $b:=\lceil {\log }_{2}m\rceil$, $D:=2^b$, and provide a unitary $P$ on the $b$-qubit coefficient register satisfying

$P\mathrm{\mid }{0}^b⟩=\mathrm{\mid }\chi ⟩:={\sum }_{j=0}^{m-1}\sqrt{\frac{\mathrm{\mid }{\beta }_j\mathrm{\mid }{\alpha }_j}{\mathrm{\Lambda }}}\mathrm{\mid }j⟩\mathrm{.}$

The squared amplitudes sum to one by the definition of $\mathrm{\Lambda }$. Define the unitary

$\mathrm{S}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{C}\mathrm{T}:={\sum }_{j=0}^{m-1}\mathrm{\mid }j⟩⟨j\mathrm{\mid }\otimes \frac{{\beta }_j}{\mathrm{\mid }{\beta }_j\mathrm{\mid }}{U}_j+{\sum }_{j=m}^{D-1}\mathrm{\mid }j⟩⟨j\mathrm{\mid }\otimes I,$

where each $U_j$ acts on its padded component ancilla and the data register; an empty second sum is zero. Return $W:=P^{†}\mathrm{S}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{C}\mathrm{T}P$, acting on the coefficient register, common component ancillas, and data system.

The instance must supply or compile $P$, $P^{†}$, and SELECT, and state the cost of each separately. A coefficient table is not itself a state-preparation circuit; table access, synthesis, controlled component calls, phases, precision, and classical preprocessing are charged under Block-encoding oracle and implementation cost model.

If $P$ is synthesized from the explicit exact normalized amplitude table $(\sqrt{w_j}{)}_{j<D}$, where $w_j:=\mathrm{\mid }{\beta }_j\mathrm{\mid }{\alpha }_j\mathrm{/}\mathrm{\Lambda }$ for $j<m$ and $w_j:=0$ for $j\ge m$, the exact-table procedure QC24 State Preparation is an optional construction. Since zero coefficients were omitted and each ${\alpha }_j>0$, $w_0>0$ and the first nonzero amplitude $\sqrt{w_0}$ is positive real. The state-preparation procedure omits only this reference phase, which is $0$ here, so the prepared vector equals $\mathrm{\mid }\chi ⟩$ exactly. Under its ideal continuous-angle model, QC24 State Preparation Rotation Level Upper Bound gives at most $D-1$ prefix rotations, at most $D$ phases, $O((b+1)D)$ bounded-arity gates and serial depth, and at most $\max \{b-1,0\}$ reusable clean work bits. Supply or compile the inverse circuit $P^{†}$ separately and charge it separately. If the table uses exactly $s$ bits for each real and imaginary coordinate, QC24 Explicit State-Preparation Cost Guarantee separately charges its $2Ds$ input bits and full-table preprocessing scan. These suppliers assert no finite-alphabet arbitrary-angle implementation or Turing bit-time bound; a supplied circuit for $P$ instead carries its own declared costs.

The current XOR-valued interface Coherent XOR function oracle does not itself implement this unitary-valued SELECT; if SELECT is assembled from component calls, its separate controlled-call and phase construction must be supplied and costed.

The projected block and normalization of the returned unitary are proved in LCU projected-block correctness.