QSVT implementation of a bounded matrix function

Inputs. A nonzero finite-dimensional Hermitian matrix $H$, a scale $\alpha >0$ with $\mathrm{\parallel }H\mathrm{\parallel }\le \alpha$, a supplied unit-normalized $(1,a,{\epsilon }_{BE})$ block encoding of $A=H\mathrm{/}\alpha$ in the sense of Finite block encoding, with designated compression $\tilde{A}$ satisfying $\mathrm{\parallel }\tilde{A}-H\mathrm{/}\alpha \mathrm{\parallel }\le {\epsilon }_{BE}$ (and ${\epsilon }_{BE}=0$ for exact access), a function $f$ on $\sigma (H)$, and a caller-supplied extension $\bar{f}:[-\alpha ,\alpha ]\to \mathbb{C}$ agreeing with $f$ on the spectrum. Let $J$ be a nonempty subset of $\{0,1,2,3\}$. For each $j\mathrm{\notin }J$, supply an exact certificate that $g_j$ is identically zero on $[-\alpha ,\alpha ]$. For each $j\in J$, supply a positive scale ${\gamma }_j$ satisfying $\mathrm{\mid }{g}_j(t)\mathrm{\mid }\le {\gamma }_j$ on that interval, a tolerance ${\tau }_j\ge 0$, a nonnegative degree bound $d_j$, and a real polynomial $p_j$ of degree at most $d_j$ and parity $d_j<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$ (including when $p_j=0$), with $\mathrm{\mid }{p}_j(x)\mathrm{\mid }\le 1$ on $[-1,1]$ and $\mid p_j(x)-\frac{g_j(\alpha x)}{{\gamma }_j}\mid \le {\tau }_j(-1\le x\le 1)\mathrm{.}$ For each active component, also supply either a reflected-convention phase-list certificate of length $d_j+1$ for which the real part of the QSP sequence's upper-left entry equals $p_j$ (exactly or within a certified uniform error bound), or, when $p_j$ has rational or exact algebraic coefficients, a positive rational phase-synthesis tolerance ${\nu }_j$ for OP1. The certificate's finite verification procedure and cost are included in the phase-source record. Any certified phase-generation or compilation error is recorded for the component and included in its implementation-error bound ${\kappa }_j$ in Matrix-function transformation error and resource bound.

Here the four real component functions are $g_0(t)=\frac{\mathrm{\Re }\bar{f}(t)+\mathrm{\Re }\bar{f}(-t)}{2},g_1(t)=\frac{\mathrm{\Re }\bar{f}(t)-\mathrm{\Re }\bar{f}(-t)}{2},$ $g_2(t)=\frac{\mathrm{\Im }\bar{f}(t)+\mathrm{\Im }\bar{f}(-t)}{2},g_3(t)=\frac{\mathrm{\Im }\bar{f}(t)-\mathrm{\Im }\bar{f}(-t)}{2}\mathrm{.}$ Thus $g_0,g_2$ are even, $g_1,g_3$ are odd, and $\bar{f}=g_0+g_1+i{g}_2+i{g}_3$. The zero certificates and scale bounds give $\mathrm{\mid }\bar{f}(t)\mathrm{\mid }\le B:={\sum }_{j\in J}{\gamma }_j$; the construction makes no optimal-normalization claim. For an identically zero target with $\bar{f}=0$, one may take $J=\{0\}$, ${\gamma }_0=1$, $d_0=0$, ${\tau }_0=0$, and $p_0=0$; the exact list ${\mathrm{\Phi }}_0=(\pi \mathrm{/}2)$ certifies its phase source since the degree-zero QSP upper-left entry is $e^{i\pi \mathrm{/}2}=i$.

Additional input. An LCU access record that supplies or compiles the coefficient-preparation unitary $P$, its inverse, and the required SELECT from the active component circuits, with all implementation costs declared as in Block-encoding oracle and implementation cost model.

Output. The LCU circuit $W$ constructed with declared normalization $B$ and designated top-left block $W_{00}$. The error bound for $W_{00}$ is given separately by Matrix-function transformation error and resource bound. If a supplied phase certificate fails its stated verification or OP1 returns a non-success result, return that failure and no block-encoding output.

Instructions. For each $j\in J$, verify its supplied phase-list certificate's length, reflected-convention polynomial identity or uniform error bound; if no list is supplied, run Real bounded-polynomial QSP phase synthesis under OP1 on $p_j$ with degree bound $d_j$, required parity $d_j<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$, and tolerance ${\nu }_j$. Propagate any certificate, synthesis or compilation failure without producing a block-encoding output. Apply Quantum singular-value transformation circuit to the supplied block encoding to produce the singular-value transform block for $p_j$, retaining separate ancillas and all inverse, controlled, projector-phase, compilation, and implementation-error costs. Since $p_j$ is real with known parity, the even or odd transform is selected by its parity. Any certified phase or implementation deviation from the ideal $p_j$ transform is included in ${\kappa }_j$.

Run LCU block-encoding construction on the component unitaries indexed by $J$, with unit component normalizations and coefficients $c_j{\gamma }_j$, where $c_0=c_1=1$ and $c_2=c_3=i$. Its LCU normalization is $B={\sum }_{j\in J}{\gamma }_j>0$. Return its designated top-left block as an approximation to $f(H)\mathrm{/}B$. OP1 and the matrix-function error theorem specify the phase and error obligations.

This algorithm does not claim a general efficient approximation theorem, automatic extension of $f$, or a finite bit-time phase compiler beyond the stated exact-algebraic/rational OP1 interface. Charge polynomial descriptions, root/phase generation, state preparation, SELECT, PREPARE and its inverse, all QSVT calls, gate compilation, and all output bits under the separate cost coordinates.