Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We resolve several of these problems here. We find the anti-Hermitian query complexity $d_I = Θ(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert~$W$ inversion. Fast-forwarding to $d_I = \mathcal{O}(\sqrt{\betaI T})$ is impossible in the bivariate polynomial model, though a linear state-dependent improvement to $d_I = \mathcal{O} β_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ is achievable. The optimization landscape of M-QSP admits spurious local minima, but a warm-start basin guarantee ensures the two-stage algorithm converges. CRC-exploiting block peeling reduces angle-finding from $\mathcal{O}(d^3)$ to $\mathcal{O}(d^2)$ classical operations, and optimized error allocation yields a leading constant of approximately~$2$ relative to the information-theoretic lower bound. A constant-ratio condition extends to non-identical signal operators, enabling time-dependent non-Hermitian simulation with query complexity $\mathcal{O}(\int_0^T(\alphaR(s) + \betaI(s))\,ds + \log(1/\varepsilon)/\log\log(1/\varepsilon))$. Block-encoding overhead $e^{-2\betaI T}$ holds across all function classes within the walk-operator oracle model, and dilational methods (Schrödingerization) achieve the walk-operator barrier. A precisely characterized direct-access construction achieves the intrinsic barrier $e^{-2ωT}$ (with $ω< \betaI$ for non-commuting Hamiltonians) on a restricted domain, though extension to the full bitorus remains open.

翻译：多变量量子信号处理（M-QSP）最近被证明可应用于非厄米哈密顿量模拟，这引发了关于优化景观、角度寻找和常数因子分析的若干问题。本文解决了其中多个问题。我们发现反厄米查询复杂度 $d_I = Θ(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ 是紧致的，该结果通过切比雪夫系数界、修正贝塞尔函数渐近和朗伯~$W$ 反演建立。在双变量多项式模型中，快进到 $d_I = \mathcal{O}(\sqrt{\betaI T})$ 是不可能的，但可达到依赖于状态的线性改进 $d_I = \mathcal{O}(\beta_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$。M-QSP的优化景观存在伪局部极小值，但暖启动盆地保证确保了两阶段算法的收敛性。利用CRC-块剥离技术，角度寻找从 $\mathcal{O}(d^3)$ 经典操作减少到 $\mathcal{O}(d^2)$，优化的误差分配使得相对于信息论下界的主导常数约为~$2$。常数比值条件可扩展到非相同的信号算子，从而支持具有查询复杂度 $\mathcal{O}(\int_0^T(\alphaR(s) + \betaI(s))\,ds + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ 的时变非厄米模拟。块编码开销 $e^{-2\betaI T}$ 在走步算子预言机模型内适用于所有函数类，而膨胀方法（薛定谔化）实现了走步算子障碍。一种精确表征的直接访问构造在受限域上实现了固有障碍 $e^{-2ωT}$（对非对易哈密顿量有 $ω< \betaI$），但扩展到全双环面仍是一个开放问题。