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

This paper presents a systematic approach for implementing dissipative quantum dynamics of the form exp(-TH^α) using Quantum Singular Value Transformation (QSVT). By leveraging the Poisson summation formula, the authors decompose the target function into a series of Fourier samples, which are then classically compiled into a single Chebyshev polynomial. This approach avoids the need for a frequency-based linear combination of unitaries (LCU), instead utilizing a direct polynomial eigenvalue transformation.

The study evaluates two distinct oracle access models: a standard block encoding of H/||H|| and a unit-normalized shifted signal 2H/||H|| - I. The authors demonstrate that the shifted signal model allows for a quadratic lift, which effectively transforms the approximation problem. This shift makes every positive integer power entire and improves the fixed-scale approximation error for noninteger powers from O(d^-α) to O(d^-2α). This structural change significantly enhances the efficiency of the quantum circuit for a wide range of dissipative applications.

The paper derives tight degree bounds for these transformations in both large-scale fixed-error and high-precision limits. The resulting polynomial construction is shown to be applicable to various dissipative processes, including heat flow, fractional diffusion, and non-Hermitian dynamics. For time-independent non-Hermitian simulation, the authors establish a noncommutative Weyl-Poisson identity that remains compatible with sinh-sinh quadrature, ensuring efficient implementation without requiring commutativity assumptions. Additionally, the method is applied to amplitude-phase separation, where it realizes controlled dissipative families using the semigroup law.