AI 中文总结
本研究针对幂指数耗散的量子实现问题，提出泊松编译的量子奇异值变换方法，通过平移信号二次提升等改进近似误差，建立相关恒等式并实现受控耗散族。
AI 中文摘要
我们研究了收缩算子$\boldsymbol{\text{exp}(-T H^\boldsymbol{\text{α}})}$的量子实现，其中$H=H^\boldsymbol{\text{†}}\boldsymbol{\text{⪰}}0$且$α>0$。泊松求和给出了精确的目标-混叠-尾部分解，其傅里叶样本经经典编译为单个切比雪夫多项式，因此量子电路采用多项式特征值变换，而非幺正算符的频率线性组合。我们比较了$H/\boldsymbol{\text{norm}}\boldsymbol{\text{{H}}}$与平移信号$2H/\boldsymbol{\text{norm}}\boldsymbol{\text{{H}}}-I$的块编码。在常规单序列QSVT（量子奇异值变换）下，奇偶性迫使前者采用偶延拓，该延拓仅对正偶整数是整函数。针对平移信号的精确二次提升使所有正整数均为整函数，并将非整数幂的固定尺度近似误差在所述访问与奇偶性类别下从$Θ(d^{-α})$改进至$Θ(d^{-2α})$。我们推导了大尺度固定误差与固定尺度高精度极限下的匹配度界，包括输出归一化开销$u_r$。最近邻拉普拉斯算子给出单位归一化的平移信号。我们进一步建立了与LCHS求积兼容的非交换Weyl–Poisson恒等式，并利用相同的多项式构造在振幅-相位分离中实现受控耗散族。
英文摘要
We study quantum implementations of the contraction $\exp(-T H^α)$ for $H=H^\dagger\succeq0$ and $α>0$. 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 $Θ(d^{-α})$ to $Θ(d^{-2α})$ 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.
CommentsThis 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
