Hamiltonian-simulation polynomial target

For a finite Hermitian $H$ with exact block-encoding normalization $\alpha >0$, $\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \alpha$, and real time $t$, put $\tau =\alpha t$. On $x\in [-1,1]$ the scalar target is $f_{\tau }(x)=\exp (-i\tau x)$; if $H{\psi }_{\lambda }=\lambda {\psi }_{\lambda }$, then $f_{\tau }(\lambda \mathrm{/}\alpha )=\exp (-it\lambda )$, the eigenvalue of the finite matrix evolution Finite Hermitian matrix dynamics.

Define the Chebyshev polynomials by $T_0(x)=1$, $T_1(x)=x$, and $T_{k+1}(x)=2x{T}_k(x)-T_{k-1}(x)$ for $k\ge 1$. For $k\ge 0$, define the Bessel coefficients used here by the series $J_k(\tau )={\sum }_{j=0}^{\mathrm{\infty }}\frac{(-1{)}^j(\tau \mathrm{/}2{)}^{2j+k}}{j!(j+k)!}\mathrm{.}$ For $\tau \mathrm{\ne }0$, the ratio of consecutive absolute summands is ${\tau }^2\mathrm{/}[4(j+1)(j+k+1)]\to 0$, so the series converges absolutely. If $\tau =0$, at most its $j=0,k=0$ summand is nonzero, so it also converges absolutely. Thus the series defines $J_k(\tau )$ before any estimate is used. The exponential uses the series convention of The complex exponential by its power series, whose absolute convergence is supplied by The complex exponential series converges absolutely for every complex argument. This definition specifies the target and coefficient series; it does not assert a Jacobi–Anger expansion or a truncation bound. A real QSP or QSVT theorem alone does not implement this full complex, non-parity target; the finite even/odd combination, complementary-polynomial certificates, normalization, and controlled relative phase are separate operations in the binding simulator.