Optimal Bounds, Barriers, and Extensions for Non-Hermitian Bivariate Quantum Signal Processing

TL;DR

This paper advances the understanding of multivariate quantum signal processing for non-Hermitian Hamiltonians, providing tight bounds, analyzing optimization landscapes, and proposing efficient algorithms with near-optimal complexity.

Contribution

It establishes tight bounds on anti-Hermitian query complexity, analyzes the optimization landscape, and introduces improved algorithms and methods for non-Hermitian quantum simulation.

Findings

Anti-Hermitian query complexity is tight at $d_I = \Theta(\beta_I T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$.

Fast-forwarding to $d_I = \mathcal{O}(\sqrt{\beta_I T})$ is impossible, but linear improvements are achievable.

Block peeling reduces angle-finding complexity from $\mathcal{O}(d^3)$ to $\mathcal{O}(d^2)$.

Abstract

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 $d_I = \Theta(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ 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 $d_I = \mathcal{O} \beta_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ 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 (Schr\"odingerization) achieve the walk-operator barrier. A precisely characterized direct-access construction achieves the intrinsic barrier $e^{-2\omega T}$ (with $\omega < \betaI$ for non-commuting Hamiltonians) on a restricted domain, though extension to the full bitorus remains open.