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arXiv:2603.25708 · Physical Review Research

Provably Efficient Long-Time Exponential Decompositions of Non-Markovian Gaussian Baths

Zhen Huang, Zhiyan Ding, Ke Wang, Jason Kaye, Xiantao Li, Lin Lin

quant-phcond-mat.str-elmath-phSubmitted Mar 26, 2026• Announced Mar 27, 2026• 0 citations
v1 · Mar 26, 2026v2 · Apr 5, 2026
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Abstract

Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval $[0,T]$ by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time $T$, inverse temperature $β$, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of $T$, achieving time-uniform complexity. The $T$-dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.

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Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval [0,T][0,T][0,T] by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time TTT, inverse temperature βββ, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of TTT, achieving time-uniform complexity. The TTT-dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.
Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.
We introduce an extended formulation of the non-Markovian stochastic Schrödinger equation with complex frequency modes (extended cNMSSE), designed for simulating open quantum system dynamics under arbitrary spectral densities. This extension employs non-exponential basis sets to expand the bath correlation functions, overcoming the reliance of the original cNMSSE on exponential decompositions of the spectral density. Consequently, the extended cNMSSE is applicable to environments beyond those characterized by Debye-type spectral densities. The flexibility to employ general basis functions is particularly advantageous for handling spectral densities with higher-order poles, for which exponential decompositions are often inaccurate or unavailable. The extended cNMSSE is implemented in a pseudo-Fock space using conventional ladder operators and solved efficiently via matrix product state techniques, preserving the favorable linear-scaling and wavefunction-based nature of the original method. Benchmark simulations across four representative cases, including discrete spectral density, Ohmic spectral density with exponential and algebraic cutoffs, and critically damped Brownian spectral density, demonstrate excellent agreement with results of hierarchy of forward-backward stochastic Schrödinger equations and extended hierarchical equation of motion.
We present an error-controlled framework for the quantum simulation of non-Markovian open-system dynamics in Gaussian environments. For a system coupled linearly to a Gaussian environment with zero mean, the bath's influence on the reduced dynamics is completely determined by its two-time correlation kernel. This allows us to replace the microscopic environment with an equivalent bath that has the same kernel. On a finite simulation interval, a rank-NNN approximation of the kernel yields an effective bath of NNN bosonic modes with time-dependent couplings to the system. To encode these modes in qubits, we first truncate each mode to its lowest ddd occupation levels. When stronger couplings lead to large occupations, we introduce coupling dilution, which replaces each mode with several more weakly coupled copies while preserving the bath correlation kernel. This distributes excitations across more modes and trades additional modes for smaller local dimensions. We derive explicit reduced-state error bounds and establish polynomial circuit width and depth in simulation time and inverse accuracy under stated spectral-decay and coupling-regularity assumptions, provided elementary unitaries admit polynomial-cost implementations.
We study the bath correlation functions (BCFs) of open quantum systems interacting with thermal baths, in case the spectral densities (SDs) exhibit removable logarithmic singularities at low frequencies and are arbitrarily shaped at higher frequencies. The singularities consist in arbitrarily positive or negative powers of logarithmic functions, as additional factors for the power laws of the Ohmic-like SDs. If the SD vanishes sufficiently fast at high frequencies the short time behavior of the BCF is algebraic. The long time behavior of the BCF exhibits a variety of relaxations that involve inverse power laws and arbitrary powers of logarithmic forms. The imaginary part of the BCF shows over long times regular dependence on the low frequency structure of the SD, except for certain conditions where the ohmicity parameter takes odd natural values. Same dependence holds for the real part of the BCF at non-vanishing temperatures. At zero temperature the real part of the BCF exhibits over long times the same regular relationship with the low frequency structure of the SD, except for certain conditions involving even natural values of the ohmicity parameter. The exceptional conditions provide relaxations that are faster than those obtained via the regular dependence. In this way, various long time relaxations of the BCF that are slower than exponential decays and arbitrarily faster or slower than inverse power laws, can be interpreted in terms of removable logarithmic singularities in the low frequency structure SD of an open quantum system.
Decoherence is often modeled using Markovian master equations that predict exponential suppression of coherence and are frequently used as effective bounds on quantum behavior in complex environments. Such descriptions, however, correspond to the singular physical limit of vanishing environmental memory. Here we formulate decoherence using a general time-nonlocal decoherence functional determined solely by the environmental force correlation function, with Markovian dynamics recovered explicitly as a limiting case. For arbitrary stationary environments with finite temporal correlations, we show that the decoherence functional exhibits quadratic short-time growth that is model-independent within the finite-memory class considered. Consequently, the decoherence time defined operationally-without assuming exponential decay-scales as the square root of the environmental correlation time, independent of the detailed form of the bath correlation kernel. These results are illustrated analytically for Gaussian-correlated, soft power-law, and Ornstein-Uhlenbeck environments. In the Ornstein-Uhlenbeck case, the non-Markovian dynamics admit an exact analytical closure, yielding a closed evolution equation for the coherence. Exact numerical simulations based on a pseudomode mapping confirm the predicted scaling and show that exponential decoherence emerges only in the memoryless limit. Beyond coherence decay, we distinguish decoherence rates from observable loss of quantum signatures by analyzing purity and von Neumann entropy dynamics. We show that suppression of a specific coherence element need not coincide with irreversible entropy production. Finally, we introduce an inferred-memory perspective in which the environmental correlation time is treated as an operationally extractable parameter from dynamical data.
The Born-Markov approximation is widely used to study the dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to a noninteracting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has nonanalyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power-law tails. The Green's functions show a short-time ``quasi''-Markovian exponential decay before crossing over to a power-law tail governed by the nonanalyticity of the spectral function. We study a system of bosons (fermions) hopping on a one-dimensional lattice, where each site is coupled linearly to an independent bath of noninteracting bosons (fermions). We obtain exact expressions for the Green's functions of this system, which show power-law decay $\ensuremath{\sim}|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3/2}$. We use these to calculate the density and current profile, as well as unequal-time current-current correlators. While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long-time power-law tails $|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3}$ characteristic of non-Markovian dynamics. We show that the power-law decays survive in the presence of interparticle interaction in the system, but the crossover time scale is shifted to larger values with increasing interaction strength.