Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.14647

Exploiting the path-integral radius of gyration in open quantum dynamics

Andrew C. Hunt, Stuart C. Althorpe

quant-phphysics.chem-phSubmitted Feb 16, 2026• Announced Feb 17, 2026• 0 citations
arXivHTMLPDF

Abstract

A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalisation of the bath modes. This delocalisation can be quantified by the radius of gyration squared ${\mathcal R}^2(ω)$ of the imaginary-time Feynman paths of the bath modes as a function of the frequency $ω$. In a Hierarchical Equations of Motion (HEOM) calculation with a Debye--Drude spectral density, ${\mathcal R}^2(ω)$ is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki--Tanimura correction is equivalent to separating smooth from `Brownian' contributions to ${\mathcal R}^2(ω)$, and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple `A4' adaptation of the `AAA' (Adaptive Antoulas--Anderson) algorithm in order to fit ${\mathcal R}^2(ω)$ to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.

Related papers

Exploiting the path-integral radius of gyration in open quantum dynamics

A. Hunt, Stuart C. Althorpe

A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalization of the bath modes. This delocalization can be quantified by the radius of gyration squared R2(ω) of the imaginary-time Feynman paths of the bath modes as a function of the frequency ω. In a hierarchical equations of motion (HEOM) calculation with a Debye-Drude spectral density, R2(ω) is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki-Tanimura correction is equivalent to separating smooth from "Brownian" contributions to R2(ω) and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple "A4" adaptation of the "AAA" (adaptive Antoulas-Anderson) algorithm in order to fit R2(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.

The Journal of Chemical Physics•Feb 23, 2026•2 citations

Exploiting the path-integral radius of gyration in open quantum dynamics.

Hunt, Andrew C, Stuart C. Althorpe

A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalization of the bath modes. This delocalization can be quantified by the radius of gyration squared R2(ω) of the imaginary-time Feynman paths of the bath modes as a function of the frequency ω. In a hierarchical equations of motion (HEOM) calculation with a Debye-Drude spectral density, R2(ω) is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki-Tanimura correction is equivalent to separating smooth from "Brownian" contributions to R2(ω) and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple "A4" adaptation of the "AAA" (adaptive Antoulas-Anderson) algorithm in order to fit R2(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.

•Feb 18, 2026•0 citations

Quantum-Classical Hierarchical Equations of Motion

Amartya Bose

We develop a quantum-classical hierarchical equations of motion (QC-HEOM) approach for simulating non-Markovian open quantum systems. The method combines the ensemble-averaged classical path reference of the quantum-classical path integral formalism with a hierarchy of auxiliary quantum influence functionals. By incorporating thermal fluctuations through an ensemble average over reference trajectories, the hierarchy is required to represent only the residual quantum memory associated with the imaginary part of the bath response function. Consequently, unlike conventional hierarchical equations of motion, QC-HEOM does not require Matsubara or Padé expansions of the thermal kernel and exhibits only weak temperature dependence of the hierarchy size. Furthermore, because thermal fluctuations are supplied through reference classical trajectories, the framework naturally extends beyond harmonic baths and enables the incorporation of anharmonic and molecular environments through externally generated trajectories. We derive the formalism and demonstrate its exactness for a harmonic bath. Applications to an asymmetric spin-boson model and the seven-site Fenna--Matthews--Olson complex illustrate the accuracy of QC-HEOM. It reproduces benchmark quasi-adiabatic path integral and hierarchical equations of motion results while requiring substantially fewer auxiliary objects, particularly at low temperatures. These results establish QC-HEOM as an efficient framework for treating residual quantum memory in quantum-classical descriptions of open-system dynamics. The separation of thermal fluctuations from residual quantum memory through the use of Wigner trajectories provides an approximate route toward hierarchical treatments of complex anharmonic environments that are inaccessible to conventional HEOM approaches.

quant-pharXiv:2606.14363•Jun 15, 2026•0 citations

Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems

Meng Xu, Yaming Yan, Qiang Shi +2 more

The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green's function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

arXiv:2202.04059•Physical Review Letters•Nov 30, 2022•110 citations

About the performance of perturbative treatments of the spin-boson dynamics within the hierarchical equations of motion approach

Meng Xu, Joachim Ankerhold

Abstract The hierarchical equations of motion (HEOM) provide a numerically exact approach for simulating the dynamics of open quantum systems coupled to a harmonic bath. However, its applicability has traditionally been limited to specific spectral forms and relatively high temperatures. Recently, an extended version called Free-Pole HEOM (FP-HEOM) has been developed to overcome these limitations. In this study, we demonstrate that the FP-HEOM method can be systematically employed to investigate higher order master equations by truncating the FP-HEOM hierarchy at a desired tier. We focus on the challenging scenario of the spin-boson problem with a sub-Ohmic spectral distribution at zero temperature and analyze the performance of the corresponding master equations. Furthermore, we compare the memory kernel for population dynamics obtained from the exact FP-HEOM dynamics with that of the approximate Non-Interacting-Blip Approximation (NIBA).

arXiv:2308.02010•The European Physical Journal Special Topics•Oct 11, 2023•5 citations

Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several different bath decomposition schemes

Hao Liu, Lili Zhu, Shuming Bai +1 more

We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.

The Journal of Chemical Physics•Apr 4, 2014•151 citations
A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalisation of the bath modes. This delocalisation can be quantified by the radius of gyration squared R2(ω){\mathcal R}^2(ω)R2(ω) of the imaginary-time Feynman paths of the bath modes as a function of the frequency ωωω. In a Hierarchical Equations of Motion (HEOM) calculation with a Debye--Drude spectral density, R2(ω){\mathcal R}^2(ω)R2(ω) is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki--Tanimura correction is equivalent to separating smooth from Brownian' contributions to ${\mathcal R}^2(ω)$, and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple A4' adaptation of the `AAA' (Adaptive Antoulas--Anderson) algorithm in order to fit R2(ω){\mathcal R}^2(ω)R2(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.
A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalization of the bath modes. This delocalization can be quantified by the radius of gyration squared R2(ω) of the imaginary-time Feynman paths of the bath modes as a function of the frequency ω. In a hierarchical equations of motion (HEOM) calculation with a Debye-Drude spectral density, R2(ω) is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki-Tanimura correction is equivalent to separating smooth from "Brownian" contributions to R2(ω) and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple "A4" adaptation of the "AAA" (adaptive Antoulas-Anderson) algorithm in order to fit R2(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.
A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalization of the bath modes. This delocalization can be quantified by the radius of gyration squared R2(ω) of the imaginary-time Feynman paths of the bath modes as a function of the frequency ω. In a hierarchical equations of motion (HEOM) calculation with a Debye-Drude spectral density, R2(ω) is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki-Tanimura correction is equivalent to separating smooth from "Brownian" contributions to R2(ω) and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple "A4" adaptation of the "AAA" (adaptive Antoulas-Anderson) algorithm in order to fit R2(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.
We develop a quantum-classical hierarchical equations of motion (QC-HEOM) approach for simulating non-Markovian open quantum systems. The method combines the ensemble-averaged classical path reference of the quantum-classical path integral formalism with a hierarchy of auxiliary quantum influence functionals. By incorporating thermal fluctuations through an ensemble average over reference trajectories, the hierarchy is required to represent only the residual quantum memory associated with the imaginary part of the bath response function. Consequently, unlike conventional hierarchical equations of motion, QC-HEOM does not require Matsubara or Padé expansions of the thermal kernel and exhibits only weak temperature dependence of the hierarchy size. Furthermore, because thermal fluctuations are supplied through reference classical trajectories, the framework naturally extends beyond harmonic baths and enables the incorporation of anharmonic and molecular environments through externally generated trajectories. We derive the formalism and demonstrate its exactness for a harmonic bath. Applications to an asymmetric spin-boson model and the seven-site Fenna--Matthews--Olson complex illustrate the accuracy of QC-HEOM. It reproduces benchmark quasi-adiabatic path integral and hierarchical equations of motion results while requiring substantially fewer auxiliary objects, particularly at low temperatures. These results establish QC-HEOM as an efficient framework for treating residual quantum memory in quantum-classical descriptions of open-system dynamics. The separation of thermal fluctuations from residual quantum memory through the use of Wigner trajectories provides an approximate route toward hierarchical treatments of complex anharmonic environments that are inaccessible to conventional HEOM approaches.
The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green's function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.
Abstract The hierarchical equations of motion (HEOM) provide a numerically exact approach for simulating the dynamics of open quantum systems coupled to a harmonic bath. However, its applicability has traditionally been limited to specific spectral forms and relatively high temperatures. Recently, an extended version called Free-Pole HEOM (FP-HEOM) has been developed to overcome these limitations. In this study, we demonstrate that the FP-HEOM method can be systematically employed to investigate higher order master equations by truncating the FP-HEOM hierarchy at a desired tier. We focus on the challenging scenario of the spin-boson problem with a sub-Ohmic spectral distribution at zero temperature and analyze the performance of the corresponding master equations. Furthermore, we compare the memory kernel for population dynamics obtained from the exact FP-HEOM dynamics with that of the approximate Non-Interacting-Blip Approximation (NIBA).
We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.