QSVT robustness, approximation error, and resources

Let $E_{\mathrm{i}\mathrm{n}},E_{\mathrm{o}\mathrm{u}\mathrm{t}}$ be the declared isometric padding maps in [F1], so $\widehat{A}=E_{\mathrm{o}\mathrm{u}\mathrm{t}}A{E}_{\mathrm{i}\mathrm{n}}^{\ast }$ and $E_{\mathrm{i}\mathrm{n}}^{\ast }{E}_{\mathrm{i}\mathrm{n}}=I$. Put $P_{\mathrm{i}\mathrm{n}}=E_{\mathrm{i}\mathrm{n}}{E}_{\mathrm{i}\mathrm{n}}^{\ast }$. By [L3]–[L4], $P_{\mathrm{i}\mathrm{n}}^{\ast }=P_{\mathrm{i}\mathrm{n}}=P_{\mathrm{i}\mathrm{n}}^2$; [L5] makes it an orthogonal projection, and [L6] gives $\mathrm{\parallel }{P}_{\mathrm{i}\mathrm{n}}z\mathrm{\parallel }\le \mathrm{\parallel }z\mathrm{\parallel }$. The adjoint identity [L3] then yields $\mathrm{\parallel }{E}_{\mathrm{i}\mathrm{n}}^{\ast }z{\mathrm{\parallel }}^2=⟨P_{\mathrm{i}\mathrm{n}}z,z⟩=\mathrm{\parallel }{P}_{\mathrm{i}\mathrm{n}}z{\mathrm{\parallel }}^2\le \mathrm{\parallel }z{\mathrm{\parallel }}^2$. Since both padding maps preserve norms by [L2], [F9] gives $\mathrm{\parallel }\widehat{A}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\parallel }{E}_{\mathrm{o}\mathrm{u}\mathrm{t}}A{E}_{\mathrm{i}\mathrm{n}}^{\ast }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$; conversely $\widehat{A}{E}_{\mathrm{i}\mathrm{n}}x=E_{\mathrm{o}\mathrm{u}\mathrm{t}}Ax$ for every $x$, so taking maxima over unit vectors gives equality. Thus $A_0:=\widehat{A}\mathrm{/}\alpha$ is a contraction. The PUE compression $B$ is a contraction by [L1], and [F1] gives $\mathrm{\parallel }B-A_0{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\epsilon }_{\mathrm{B}\mathrm{E}}$. Applying [F3] gives $\mathrm{\parallel }{P}^{\mathrm{S}\mathrm{V}}(B)-P^{\mathrm{S}\mathrm{V}}(A_0){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 4d\sqrt{{\epsilon }_{\mathrm{B}\mathrm{E}}}$.

Choose the SVD bases of $A_0$ from [F8]. For odd parity, the difference $P^{\mathrm{S}\mathrm{V}}(A_0)-f^{\mathrm{S}\mathrm{V}}(A_0)$ sends each right singular vector $v_j$ to $(P({\sigma }_j)-f({\sigma }_j))u_j$ and annihilates $\ker A_0$. Orthonormality gives, for $x={\sum }_j{c}_j{v}_j+x_0$, $\mathrm{\parallel }(P^{\mathrm{S}\mathrm{V}}-f^{\mathrm{S}\mathrm{V}})x{\mathrm{\parallel }}^2={\sum }_j\mathrm{\mid }P({\sigma }_j)-f({\sigma }_j){\mathrm{\mid }}^2\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\le {\eta }_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}^2\mathrm{\parallel }x{\mathrm{\parallel }}^2$. For even parity the same bound follows by diagonal action on the right singular basis, including $P(0)-f(0)$ on the kernel. Taking the maximum over unit vectors by [F6] yields $\mathrm{\parallel }{P}^{\mathrm{S}\mathrm{V}}(A_0)-f^{\mathrm{S}\mathrm{V}}(A_0){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\eta }_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}$.

Replace the ideal phase operations one slot at a time by their implemented unitaries. The product difference is a sum of $d+1$ terms, each containing one phase difference and otherwise unitary factors. By [F9], multiplying by those unitary factors preserves or bounds the operator norm by their norm $1$; [F7] and [F6] then give $\mathrm{\parallel }{\tilde{U}}_{\mathrm{\Phi }}-U_{\mathrm{\Phi }}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\sum }_{\mathrm{\ell }}{\zeta }_{\mathrm{\ell }}$. Compression by the input and output projections is contractive, so the same upper bound holds for the compressed blocks. The exact ideal compressed block is $P^{\mathrm{S}\mathrm{V}}(B)$ by [F10].

The implemented block differs from $f^{\mathrm{S}\mathrm{V}}(A_0)$ by the sum of its phase-implementation, input-block, and scalar-approximation differences. Applying the triangle inequality to steps 1.1, 1.2, and 1.3 yields $\mathrm{\parallel }\tilde{B}-f^{\mathrm{S}\mathrm{V}}(A_0){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\eta }_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}+4d\sqrt{{\epsilon }_{\mathrm{B}\mathrm{E}}}+{\sum }_{\mathrm{\ell }}{\zeta }_{\mathrm{\ell }}$. The resource statement follows from [F4]–[F5]; each absent access, synthesis, data-loading, or compilation operation remains a separate supplied cost.