Poisson-Compiled Quantum Singular Value Transformation for Power-Exponential Dissipation

Abstract:We study quantum implementations of the contraction $\exp(-T H^\alpha)$ for $H=H^\dagger\succeq0$ and $\alpha>0$. Poisson summation provides an exact target--alias--tail decomposition whose Fourier samples are compiled classically into a single Chebyshev polynomial, so the quantum circuit uses polynomial eigenvalue transformation rather than a frequency linear combination of unitaries. We compare block encodings of $H/\norm{H}$ and of the shifted signal $2H/\norm{H}-I$. Under ordinary single-sequence QSVT, parity forces the former to use an even extension, which is entire only for even positive integers. An exact quadratic lift for the shifted signal makes every positive integer entire and improves the fixed-scale approximation error for noninteger powers from $\Theta(d^{-\alpha})$ to $\Theta(d^{-2\alpha})$ within the stated access and parity classes. We derive matching degree bounds in the large-scale fixed-error and fixed-scale high-precision limits, including the output-normalization overhead $u_r$. Nearest-neighbor Laplacians give a unit-normalized shifted signal. We further establish a noncommutative Weyl--Poisson identity compatible with LCHS quadrature, and use the same polynomial construction to implement controlled dissipative families in amplitude--phase separation.

This article supersedes arXiv:2604.02874. The present work substantially reformulates the previous framework, replaces the coherent frequency implementation with a polynomial eigenvalue transformation, and establishes access-dependent approximation and query-complexity results

arXiv:2608.04263 [quant-ph]

(or arXiv:2608.04263v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.04263 arXiv-issued DOI via DataCite (pending registration)