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 coupling strengths inaccessible with the conventional HEOM. 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.

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• Dynamical phase transitions
• Nonequilibrium statistical mechanics
• Open quantum systems
• Non-Markovian processes
• Path-integral methods
• Tensor network methods