Qubitization access model

Let $H$ be a finite-dimensional complex inner-product space, let $K=K^{\ast }$ on $H$, and let $\alpha >0$ satisfy $\mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \alpha$. Put $A=K\mathrm{/}\alpha$. Let $U$ be a unitary on an ancilla-system space $H_A\otimes H$, let $J:H\to H_A\otimes H$ be the isometry $Jv=\mathrm{\mid }{0}^a⟩\otimes v$, and let $\mathrm{\Pi }=J{J}^{\ast }$. An exact Hermitian block encoding is the data $(U,J,\alpha )$ with $J^{\ast }UJ=A$. The matrix $K$ is Hermitian; $U$ itself need not be Hermitian.

The signal-register reflection is $R_{\mathrm{\Pi }}=2\mathrm{\Pi }-I$. The access record lists, separately, each supplied call to $U$, $U^{\ast }$, controlled-$U$, controlled-$U^{\ast }$, and $R_{\mathrm{\Pi }}$, together with their circuit, query, gate, width, ancilla, precision, and classical-generation costs from Block-encoding oracle and implementation cost model and Circuit-family and computation cost model. A state description or block matrix alone supplies none of these calls. A controlled reflection is also separate. Any PREPARE, SELECT, sparse-location, value, or coefficient operation used by a construction is a separately supplied, reversible interface with its inverse and cost recorded.

Here is the standard-form qubiterate for this access instance. Add a flag qubit and define $\tilde{U}=\mathrm{\mid }0⟩⟨0\mathrm{\mid }\otimes U+\mathrm{\mid }1⟩⟨1\mathrm{\mid }\otimes U^{\ast }$, $J^{\mathrm{\prime }}v=\mathrm{\mid }+⟩\otimes Jv$, $S=X\otimes I$, $F=S\tilde{U}$, and $W_{\mathrm{q}}=(2J^{\mathrm{\prime }}{J}^{\mathrm{\prime }\ast }-I)F$. Implementing $\tilde{U}$ requires the separately declared controlled-$U$ and controlled-$U^{\ast }$ calls; $S$, the preparation of $\mathrm{\mid }+⟩$, and the reflection about $\mathrm{ran}(J^{\mathrm{\prime }})$ also have declared costs. Qubitization lifts scalar QSP ↗ proves its compressed block and spectral relation.

For the QSVT simulation sequence, the signal operations are the alternating $U,U^{\ast }$ calls interleaved with the projector phases $e^{i\varphi (2\mathrm{\Pi }-I)}$ from QSP signal matrix and ordered phase sequence. Their exact phase list, parity, real-polynomial extraction, controls, inverse calls, and phase-synthesis precision are part of the supplied circuit description. The scalar polynomial pairs must satisfy Admissible polynomial pair for Hermitian-reflection QSP; singular-value transformations use the convention of Odd and even singular-value transforms.