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arXiv:2411.12049 · Journal of Chemical Theory and Computation

Simulating Non-Markovian Quantum Dynamics on NISQ Computers Using the Hierarchical Equations of Motion

Xiaohan Dan, Eitan Geva, Víctor S. Batista

Published Feb 14, 2025• 22 citations
arXivHTMLPDF

Abstract

Quantum computing offers promising new avenues for tackling the long-standing challenge of simulating the quantum dynamics of complex chemical systems, particularly open quantum systems coupled to external baths. However, simulating such nonunitary dynamics on quantum computers is challenging since quantum circuits are specifically designed to carry out unitary transformations. Furthermore, chemical systems are often strongly coupled to the surrounding environment, rendering the dynamics non-Markovian and beyond the scope of Markovian quantum master equations like Lindblad or Redfield. In this work, we introduce a quantum algorithm designed to simulate non-Markovian dynamics of open quantum systems. Our approach enables the implementation of arbitrary quantum master equations on noisy intermediate-scale quantum (NISQ) computers. We illustrate the method as applied in conjunction with the numerically exact hierarchical equations of motion (HEOM) method. The effectiveness of the resulting quantum HEOM algorithm is demonstrated as applied to simulations of the non-Lindbladian electronic energy and charge transfer dynamics in models of the carotenoid-porphyrin-C 60 molecular triad dissolved in tetrahydrofuran and the Fenna–Matthews–Olson complex.

DOI
10.1021/acs.jctc.4c01565

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In this review we give a comprehensive account of a hierarchical equations of motion (HEOM) approach to the characterization of stationary and dynamic properties of open quantum systems. This approach is rooted at the Feynman-Vernon influence functional path integral formalism, but much more implementable numerically and operationally for the study of various complex molecular dynamics and quantum transport in strongly correlated electronic systems. By construction, HEOM resolves nonperturbatively the combined effects of many-particle interaction, system-bath coupling,and non-Markovian memory. Finally the practicality of HEOM to address physical and chemical problems is exemplified with a model simulation of coherent two-dimensional spectroscopy signals of a biological light-harvesting system and a time-dependent quantum transport system involving dynamic Kondo transition.級聯方程已成為研究量子開放系統的穩態性質和動力學過程的重要方法。本文旨在系統綜述量子耗散和量子輸運的級聯方程組方法的建立、發展以及在理論、算法和應用方面的一些最新進展。級聯方程形式理論的建立以影響泛函路徑積分為基礎,并具有數值上的高效性和應用上的靈活性,可用于研究分子體系的復雜動力學過程以及強關聯電子體系中的量子輸運。其級聯耦合結構以非微擾的方式揭示了多體相互作用、體系-環境耦合、非馬爾可夫記憶等的綜合效應。作為應用示例,我們采用級聯方程模擬了生物光富集體系的二維相干動力學光譜以及含時電子輸運過程中的動態近藤效應。

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Quantum computing offers promising new avenues for tackling the long-standing challenge of simulating the quantum dynamics of complex chemical systems, particularly open quantum systems coupled to external baths. However, simulating such nonunitary dynamics on quantum computers is challenging since quantum circuits are specifically designed to carry out unitary transformations. Furthermore, chemical systems are often strongly coupled to the surrounding environment, rendering the dynamics non-Markovian and beyond the scope of Markovian quantum master equations like Lindblad or Redfield. In this work, we introduce a quantum algorithm designed to simulate non-Markovian dynamics of open quantum systems. Our approach enables the implementation of arbitrary quantum master equations on noisy intermediate-scale quantum (NISQ) computers. We illustrate the method as applied in conjunction with the numerically exact hierarchical equations of motion (HEOM) method. The effectiveness of the resulting quantum HEOM algorithm is demonstrated as applied to simulations of the non-Lindbladian electronic energy and charge transfer dynamics in models of the carotenoid-porphyrin-C 60 molecular triad dissolved in tetrahydrofuran and the Fenna–Matthews–Olson complex.
An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.
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.
Steady states play pivotal roles in many equilibrium and non-equilibrium open system studies. Their accurate evaluations call for exact theories with rigorous treatment of system-bath interactions. Therein, the hierarchical equations-of-motion (HEOM) formalism is a nonperturbative and non-Markovian quantum dissipation theory, which can faithfully describe the dissipative dynamics and nonlinear response of open systems. Nevertheless, solving the steady states of open quantum systems via HEOM is often a challenging task, due to the vast number of dynamical quantities involved. In this work, we propose a self-consistent iteration approach that quickly solves the HEOM steady states. We demonstrate its high efficiency with accurate and fast evaluations of low-temperature thermal equilibrium of a model Fenna-Matthews-Olson pigment-protein complex. Numerically exact evaluation of thermal equilibrium Rényi entropies and stationary emission line shapes is presented with detailed discussion.
We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.
The theory of open quantum systems has many applications ranging from simulating quantum dynamics in condensed phases to better understanding quantum-enabled technologies. At the center of theoretical chemistry are the developments of methodologies and computational tools for simulating charge and excitation energy transfer in solutions, biomolecules, and molecular aggregates. As a variety of these processes display non-Markovian behavior, classical computer simulation can be challenging due to exponential scaling with existing methods. With quantum computers holding the promise of efficient quantum simulations, in this paper, we present a new quantum algorithm based on Kraus operators that capture the exact non-Markovian effect at a finite temperature. The implementation of the Kraus operators on the quantum machine uses a combination of singular value decomposition (SVD) and optimal Walsh operators that result in shallow circuits. We demonstrate the feasibility of the algorithm by simulating the spin-boson dynamics and the exciton transfer in the Fenna-Matthews-Olson (FMO) complex. The NISQ results show very good agreement with the exact ones.
In this review we give a comprehensive account of a hierarchical equations of motion (HEOM) approach to the characterization of stationary and dynamic properties of open quantum systems. This approach is rooted at the Feynman-Vernon influence functional path integral formalism, but much more implementable numerically and operationally for the study of various complex molecular dynamics and quantum transport in strongly correlated electronic systems. By construction, HEOM resolves nonperturbatively the combined effects of many-particle interaction, system-bath coupling,and non-Markovian memory. Finally the practicality of HEOM to address physical and chemical problems is exemplified with a model simulation of coherent two-dimensional spectroscopy signals of a biological light-harvesting system and a time-dependent quantum transport system involving dynamic Kondo transition.級聯方程已成為研究量子開放系統的穩態性質和動力學過程的重要方法。本文旨在系統綜述量子耗散和量子輸運的級聯方程組方法的建立、發展以及在理論、算法和應用方面的一些最新進展。級聯方程形式理論的建立以影響泛函路徑積分為基礎,并具有數值上的高效性和應用上的靈活性,可用于研究分子體系的復雜動力學過程以及強關聯電子體系中的量子輸運。其級聯耦合結構以非微擾的方式揭示了多體相互作用、體系-環境耦合、非馬爾可夫記憶等的綜合效應。作為應用示例,我們采用級聯方程模擬了生物光富集體系的二維相干動力學光譜以及含時電子輸運過程中的動態近藤效應。