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arXiv:2503.21693 · The Journal of Chemical Physics

Numerically efficient quasi-adiabatic propagator path integral approach with two independent non-commuting baths

Roman Ovcharenko, Benjamin Philipp Fingerhut

Published Jul 7, 2025• 3 citations
arXivHTMLPDF

Abstract

Path integral methods, such as the quasi-adiabatic propagator path integral (QUAPI), are widely used in general-purpose and highly accurate numerical benchmark simulations of open quantum systems, particularly in regimes inaccessible to perturbative methods. Nevertheless, the applicability of the QUAPI method to realistic systems of interest is restricted by the exponentially growing computer memory requirements with respect to the size of the quantum system and the time range of non-Markovian correlation effects. This exponential "wall" becomes even more severe for multiple non-commuting fluctuating environments. In the present work, we address the numerical efficiency and accuracy of approximations that have been introduced for the QUAPI method with a single general environment, for the case of two independent non-commuting environments where one of them is considered as a pure dephasing environment. In particular, we consider a sharply defined cutoff of the memory time, path filtering, and mask assisted coarse graining of influence functional coefficients as approximations. We demonstrate that commonly applied numerical techniques, such as path filtering, cannot be straightforwardly transferred to the two-bath case even in the weak-coupling and quasi-Markovian limits. On the other hand, the sharply defined memory cutoff can be accurately handled with the mask assisted coarse graining approach. Our findings demonstrate that if system coupling operators to different baths do not commute, the additive nature of the statistically independent environments may be misleading. In particular, the quasi-Markovian nature of a pure dephasing bath is lost once there simultaneously exists another non-commuting source of fluctuations.

DOI
10.1063/5.0271212

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Path integral methods, such as the quasi-adiabatic propagator path integral (QUAPI), are widely used in general-purpose and highly accurate numerical benchmark simulations of open quantum systems, particularly in regimes inaccessible to perturbative methods. Nevertheless, the applicability of the QUAPI method to realistic systems of interest is restricted by the exponentially growing computer memory requirements with respect to the size of the quantum system and the time range of non-Markovian correlation effects. This exponential "wall" becomes even more severe for multiple non-commuting fluctuating environments. In the present work, we address the numerical efficiency and accuracy of approximations that have been introduced for the QUAPI method with a single general environment, for the case of two independent non-commuting environments where one of them is considered as a pure dephasing environment. In particular, we consider a sharply defined cutoff of the memory time, path filtering, and mask assisted coarse graining of influence functional coefficients as approximations. We demonstrate that commonly applied numerical techniques, such as path filtering, cannot be straightforwardly transferred to the two-bath case even in the weak-coupling and quasi-Markovian limits. On the other hand, the sharply defined memory cutoff can be accurately handled with the mask assisted coarse graining approach. Our findings demonstrate that if system coupling operators to different baths do not commute, the additive nature of the statistically independent environments may be misleading. In particular, the quasi-Markovian nature of a pure dephasing bath is lost once there simultaneously exists another non-commuting source of fluctuations.
The accurate simulation of dissipative quantum dynamics subject to a non-Markovian environment poses persistent numerical challenges, in particular, for structured environments, where sharp mode resonances induce long-time system bath correlations. We present a scalable distributed memory implementation of the mask-assisted coarse graining of influence coefficients (MACGIC)-quasi-adiabatic propagator path integral (-QUAPI) method that exploits the memory resources of multiple compute nodes and mitigates the memory bottleneck of the method via a new premerging algorithm and a hash-based look-up (hMACGIC-QUAPI) while preserving numerical accuracy. The source code of the hMACGIC-QUAPI implementation is available in a publicly accessible Git repository under GPL license. The developed distributed memory implementation spreads the paths over the computing nodes by means of the MPI protocol, and efficient high level path management is achieved via the hash map-based implementation with constant access time. The efficiency of the implementation is demonstrated in large-scale dissipative quantum dynamics simulations that account for the coupling to a structured non-Markovian environment containing a sharp resonance, a setup for which convergence properties are investigated in depth. Broad applicability and the nonperturbative nature of the simulation method is illustrated via the tuning of the mode resonance frequency of the structured environment with respect to the characteristic system frequency. The simulations reveal a splitting of resonances due to a strong system–environment interaction and the emergence of sidebands due to multiexcitations of the bosonic mode that are not accounted for in perturbative approaches. The simulations demonstrate the versatility of the new hMACGIC-QUAPI method in the presence of strong non-Markovian system bath correlations.
Quantum systems are typically subject to various environmental noise sources. Treating these environmental disturbances with a system-bath approach beyond weak coupling, one must refer to numerical methods as, for example, the numerically exact quasi-adiabatic path integral approach. This approach, however, cannot treat baths which couple to the system via operators, which do not commute. We extend the quasi-adiabatic path integral approach by determining the time discrete influence functional for such non-commuting fluctuations and by modifying the propagation scheme accordingly. We test the extended quasi-adiabatic path integral approach by determining the time evolution of a quantum two-level system coupled to two independent baths via non-commuting operators. We show that the convergent results can be obtained and agreement with the analytical weak coupling results is achieved in the respective limits.
The dynamics of a qubit under the decoherence of a two level fluctuator (TLF) in addition to its coupling to a bosonic bath is investigated theoretically. Two different methods are applied and compared for this problem. One is a perturbation method based on a unitary transformation. With the merit of our unitary transformation, non-adiabatic effect can be taken into account efficiently. And the other one is the numerically exact method, namely the quasi-adiabatic path-integral (QUAPI) propagator technique. We find that the analytical method works well for a wide parameter range and show good agreement with QUAPI. On the other hand, The enhancement and the reduction of quantum decoherence of the qubit is checked with varying bath temperature TTT and TLF-bath coupling.
The dynamics of a qubit in a structured environment is investigated theoretically. One point of view of the model is the spin-boson model with a Lorentz shaped spectral density. An alternative view is a qubit coupled to harmonic oscillator (HO), which in turn coupled to a Ohmic environment. Two different methods are applied and compared for this problem. One is a perturbation method based on a unitary transformation. Since the transformed hamiltonian is of rotating wave approximation (RWA) form, we call it the transformed rotating wave approximation (TRWA) method. And the other one is the numerically exact method of the quasi-adiabatic propagator path-integral (QUAPI) method. TRWA method can be applied from the first point of view. And the QUAPI method can applied from both points of views. We find that from the 1st point of view QUAPI only works well for large ΓΓΓ. Since the memory time is too long for the practical evaluation of QUAPI when ΓΓΓ is small. We call this treatment as QUAPI1. And from the 2nd point of view, QUAPI works well for small ΓΓΓ, since the non-adiabatic effect become more important as ΓΓΓ increases, one need smaller time-step and more steps to obtain accurate result which also quickly runs out the computational resources. This treatment is called QUAPI2. We find that the TRWA method works well for the whole parameter range of ΓΓΓ and show good agreement with QUAPI1 and QUAPI2. On the other hand, we find that the decoherence of the qubit can be reduced with increasing coupling between HO and bath. This result may be relevant to the design of quantum computer.
Accurately simulating non-Markovian quantum dynamics in system-bath coupled problems remains challenging. In this work, we present a novel memory truncation scheme for the iterative quasi-adiabatic propagator path integral (iQuAPI) method to improve accuracy. Conventional memory truncation in iQuAPI discards all influence functional beyond a certain time interval, which is not effective for problems with a long memory time. Our proposed scheme selectively retains the most significant parts of the influence functional using the density matrix renormalization group algorithm. We validate the effectiveness of our scheme through simulations of the spin-boson model across various parameter sets, demonstrating faster convergence and improved accuracy compared to the conventional scheme. Our findings suggest that the new memory truncation scheme significantly advances the capabilities of iQuAPI for problems with a long memory time.
Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.