Hidden Gauge Freedom in Complex-Pole Hierarchical Equations of Motion

While complex-pole hierarchical equations of motion (HEOM) have dramatically expanded the reach of numerically exact quantum dynamics simulations of open quantum systems, they suffer from numerical instabilities rooted in the non-Hermitian structure of their Liouvillian. Yet, the origin of this structure remains obscure. Here, we report a previously unknown gauge freedom in complex-pole HEOM: a continuous family of analytically equivalent Liouvillians, all encoding the same bath correlation function, whose numerical properties vary dramatically. This gauge controls both the eigenspectrum and non-normality of the hierarchy generator, revealing spectral divergence and non-normal error amplification as two distinct instability mechanisms. By optimizing this gauge, we introduce GO–HEOM, which eliminates divergences in strongly coupled Brownian oscillator environments and extends numerically exact simulations of sub-Ohmic dynamics – including through the delocalized-to-localized quantum phase transition – to previously inaccessible coupling strengths. Because this gauge transformation is independent of the bath-correlation decomposition scheme, our GO–HEOM becomes a general, broadly compatible strategy for accessing numerically exact quantum dynamics of open quantum systems over arbitrary coupling and highly non-Markovian regimes.