Quantum Simulation of Non-Unitary Dynamics via Amplitude-Phase Separation

TL;DR

The paper introduces Amplitude-Phase Separation (APS), a novel framework for simulating non-unitary quantum dynamics by separating coherent and dissipative effects, improving understanding and efficiency.

Contribution

APS provides a unified decomposition framework that isolates unitary and Hermitian components, offering new insights and advantages for non-unitary quantum simulations.

Findings

APS captures complementary advantages in different regimes.

Benchmarks confirm crossover between phase-driven and amplitude-driven methods.

APS clarifies bottlenecks in non-unitary quantum simulation.

Abstract

Linear non-unitary dynamics arise in open quantum systems, non-Hermitian models, and numerical evolution problems, yet current quantum algorithms do not cleanly separate coherent and dissipative effects at the design level. We introduce Amplitude-Phase Separation (APS), a decomposition framework with two complementary forms: phase-driven APS isolates the unitary component and maps the remainder to a Hermitian problem, whereas amplitude-driven APS extracts the Hermitian component and treats the remaining interaction separately. For time-independent dynamics, the two routes capture complementary advantages within one framework: phase-driven APS yields additive rather than multiplicative tolerance dependence, while amplitude-driven APS yields square-root dissipative scaling in multiscale regimes. APS also provides a unified interpretation of representative methods, including LCHS (Linear Combination of Hamiltonian Simulation) and NDME (Non-Diagonal Density Matrix Encoding), and clarifies where coherent and dissipative bottlenecks enter non-unitary simulation. The benchmarks confirm the predicted crossover between phase-driven and amplitude-driven advantages in advection-diffusion and Bloch-relaxation models.