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arXiv:2610.37953

Continuous Representation Theory of the Hierarchical Equations of Motion

Tianchu Li, Pranay Venkatesh, Andrés Montoya−Castillo

Published Oct 27, 2026• 0 citations
arXivHTMLPDF

Abstract

The numerical stability of the hierarchical equations of motion (HEOM) depends on the representation of their Liouvillian. Here, we introduce a continuous gauge freedom through a generalized Bogoliubov transformation with the parameter $r$. It subsumes earlier phase space stabilization methods as special cases and explains both their stabilizing mechanism and their limitations. We find that gauge optimization balances the suppression of hierarchy growth against errors in reconstructing the reduced density operator, and show that the factor $e^{2r}-1$ controls the feedback strength, with earlier methods located at the boundary for reconstruction-error amplification, $r=\frac{1}{2}\ln 2$. For spin-boson models, this boundary gauge recovers known stabilizations for undamped and weakly damped modes. In the weakly damped case, an interior gauge further reduces the trace error by more than an order of magnitude. For the Holstein model, interior gauges reduce trace drift by about an order of magnitude relative to the previously adopted fixed representation at the same bond dimension. This improvement stabilizes error-sensitive calculations of equilibrium autocorrelation functions. Our formulation thus extends fixed stabilization methods to a continuous family of HEOM representations generated by linear canonical transformations, and establishes a framework for improving numerical stability and accuracy through continuous gauge selection.

DOI
10.48550/arxiv.2610.37953

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The numerical stability of the hierarchical equations of motion (HEOM) depends on the representation of their Liouvillian. Here, we introduce a continuous gauge freedom through a generalized Bogoliubov transformation with the parameter rrr. It subsumes earlier phase space stabilization methods as special cases and explains both their stabilizing mechanism and their limitations. We find that gauge optimization balances the suppression of hierarchy growth against errors in reconstructing the reduced density operator, and show that the factor e2r−1e^{2r}-1e2r−1 controls the feedback strength, with earlier methods located at the boundary for reconstruction-error amplification, r=12ln⁡2r=\frac{1}{2}\ln 2r=21​ln2. For spin-boson models, this boundary gauge recovers known stabilizations for undamped and weakly damped modes. In the weakly damped case, an interior gauge further reduces the trace error by more than an order of magnitude. For the Holstein model, interior gauges reduce trace drift by about an order of magnitude relative to the previously adopted fixed representation at the same bond dimension. This improvement stabilizes error-sensitive calculations of equilibrium autocorrelation functions. Our formulation thus extends fixed stabilization methods to a continuous family of HEOM representations generated by linear canonical transformations, and establishes a framework for improving numerical stability and accuracy through continuous gauge selection.
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.
The hierarchical equations of motion (HEOM) approach is one of the most powerful methods avail- able to simulate the dynamics of open quantum systems. Although the HEOM approach has been shown to be stable and efficient in many parameter regimes and physical scenarios, it is also well known that the unavoidable finite truncation of the hierarchy can lead to fundamental numeri- cal instabilities for bosonic environments. In this work, we apply recently developed techniques to analyze these issues. We use an auxiliary Fock-space picture of the hierarchy to first identify the unbalanced hierarchy-raising terms responsible for the strong transient amplification associated with the non-normal algebraic structure of the HEOM generator. We then apply a non-unitary similarity transformation that converts these terms into balanced combinations of hierarchy-raising and hierarchy-lowering terms, thereby suppressing their numerical impact. We first demonstrate the efficacy of this approach via a spin-boson model with a simple Brownian oscillator spectral den- sity and subsequently extend the analysis to a structured spectral density, showing for both cases that the transformed HEOM approach allows simulations of much larger system-bath coupling than would otherwise be possible.
The hierarchical equations of motion (HEOMs) have developed into an important tool in simulating quantum dynamics in condensed phases. Yet, it has recently been found that the HEOM may become numerically unstable in simulations using discrete harmonic oscillator modes [I. S. Dunn, et al., J. Chem. Phys. 150, 184109 (2019)]. In this paper, a new set of equations of motion are obtained based on the equivalence between the HEOM for discrete harmonic oscillator modes and the mixed quantum-classical Liouville equation. The new set of equations can thus be regarded as the expansion of the same phase space partial differential equation using different basis sets. It is shown that they have similar structures as the original HEOM but are free from the problem of numerical instability. The new set of equations are also incorporated into the matrix product state method, where it is found that the trace of the reduced density operator is not well conserved during the propagation. A modified time-dependent variational principle is then proposed to achieve better trace conservation.
Abstract The Hierarchical Equations of Motion (HEOM) method has become one of the cornerstones in the simulation of open quantum systems and their dynamics. It is commonly referred to as a non-perturbative method. Yet, there are certain instances, where the necessary truncation of the hierarchy of auxiliary density operators seems to introduce errors which are not fully controllable. We investigate the nature and causes of this type of critical error both in the case of pure decoherence, where exact results are available for comparison, and in the spin-boson system, a full system-reservoir model. We find that truncating the hierarchy to any finite size can be problematic for strong coupling to a dissipative reservoir, in particular when combined with an appreciable reservoir memory time.
The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. We also discuss issues associated with the non-normality of HEOM.
An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model. In this approach, a complete set of orthonormal functions are used to expand an arbitrary bath correlation function. As a result, a complete dynamic basis set is constructed by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element. The reliability of the extended HEOM is demonstrated by comparison with the stochastic Hamiltonian approach under room-temperature classical ohmic and sub-ohmic noises and the multilayer multiconfiguration time-dependent Hartree theory under zero-temperature quantum ohmic noise. Upon increasing the order in the hierarchical expansion, the result obtained from the extended HOEM systematically converges to the numerically exact answer.