Simulating many-body open quantum systems by harnessing the power of artificial intelligence and quantum computing Free

Many-body open quantum systems (OQSs) are ubiquitous in physics, chemistry, biology, and materials science. In an OQS, quantum correlations arise from the many-body quantum states formed by couplings between the system and its environment, as well as from the non-Markovian memory originating from the environment’s nontrivial energetic structure. Understanding and leveraging these quantum correlations in both space and time could facilitate advancements in thermoelectric transport in nano-devices,^1–12 molecular spectroscopies^13–20 and chemical reaction dynamics^21–28 in condensed phase, quantum computing and quantum sensing utilizing surface-adsorbed molecular spins,^29–40 etc. With ongoing advancements in these fields, quantum correlations are expected to play a crucial role in driving innovative discoveries and applications across science and technology.

The rapid progress of cutting-edge experiments demands greater accuracy, precision, robustness, and versatility from theoretical methods, especially in comprehensively describing realistic complex systems and relevant experimental conditions. To elucidate the underlying mechanisms behind experimental observations, various numerically exact methods for many-body OQSs have been developed. These methods include the hierarchical equations of motion (HEOM) approach for bosonic and fermionic environments,^41–48 pseudomode theory,^49–54 Davydov’s ansatz and its multistate extensions,^55–60 stochastic equation of motion,^61–70 quantum state diffusion,⁷¹ the hierarchy of stochastic pure states,⁷² and quantum Monte Carlo.⁷³ Despite these advancements in simulating OQS dynamics, the “exponential wall” problem, as put forth by Kohn,⁷⁴ significantly hinders the efficient simulation of increasingly complex many-body OQSs. This challenge arises because the number of degrees of freedom required for practical calculations grows exponentially with system size, many-body correlation strength, and the degree of environmental non-Markovianity.

To address this problem, efficient dimension reduction methods have been developed, including matrix product state (MPS),⁷⁵ the density matrix renormalization group⁷⁶ and its time-dependent extension,⁷⁷ multi-configuration time-dependent Hartree and its multi-layer extension,^78,79 and time-evolving matrix product operator.^80,81 The application of MPS methods has been extended from isolated systems^82–84 to OQSs.^85–99 The combination of MPS and HEOM has enabled the simulation of dissipative dynamics in a large OQS consisting of 21 qubits.⁹² 

In recent years, the growing demand for efficient processing and advanced problem-solving capabilities has led to the emergence of innovative methods based on artificial intelligence (AI)^100–109 and quantum computing (QC).^110–118 These technologies are reshaping the landscape of scientific and technological advancement, providing unprecedented opportunities in fields such as materials science, protein structure prediction, and quantum chemistry simulations.

In 2017, Carleo and Troyer¹⁰⁰ pioneered the use of artificial neural networks to represent high-dimensional many-body wavefunctions of isolated quantum systems. This neural quantum state (NQS) approach enabled accurate calculations of ground-state energies and unitary time evolution, with computational costs increasing polynomially with system size. The NQS method was later extended to represent the reduced density matrices of OQSs, facilitating efficient simulations of quantum dissipative dynamics subject to the Markovian condition.^{101–105,119,120} Chen et al.¹²¹ have established the necessary and sufficient conditions for transforming tensor network states into restricted Boltzmann machines (RBMs) of specific architectures, showing that RBMs possess greater expressive power than tensor network states in representing quantum many-body states. Meanwhile, quantum computers,^122–133 which are built of quantum mechanical elements that adhere to quantum laws, utilize quantum resources to manage complex quantum correlations more effectively. This capability holds great promise for solving problems that are impractical or impossible for classical computers.^{111,113–115,117,134,135}

By harnessing the powerful representation capabilities of AI and the quantum parallelism inherent in QC,¹²⁰ there is a promising prospect of breaking the exponential wall that has long constrained the simulation of many-body OQSs. However, current studies^{101,103–105,136–142} are limited and primarily focus on OQSs with weak quantum correlations in space or time. It is difficult for many existing numerically exact methods to integrate directly with AI and QC technologies. For instance, the hierarchical structure of the HEOM significantly complicates its mapping onto neural network models and quantum circuits. Therefore, there is an urgent need to develop novel theoretical frameworks, such as the recently developed dissipaton-embedded quantum master equation (DQME) theory, to better accommodate emerging AI and QC methods.

This Perspective focuses on AI- and QC-based theories and their applications in simulating many-body OQSs. The remainder of this paper is organized as follows: In Sec. II, a novel theoretical framework, the DQME in second quantization, is introduced for simulating the non-Markovian quantum dissipative dynamics of many-body OQSs. In Sec. III, we briefly discuss the computer software developed based on conventional algorithms for the simulation of OQSs. Sections IV and V discuss the representation of correlated quantum states and dissipative quantum dynamical equations using neural networks and quantum circuits, respectively. This is followed by the design of quantum machine learning algorithms for OQS dynamics in Sec. VI. We conclude by exploring the future prospects of AI- and QC-based approaches to many-body OQSs in Sec. VII.

We begin by briefly introducing the fermionic HEOM theory, which describes the quantum dissipative dynamics of OQSs coupled to non-Markovian environments, as illustrated in Fig. 1(a). For example, consider a single-orbital system coupled to a noninteracting electron reservoir. The extension to more complex OQSs is straightforward. The total Hamiltonian of the system plus environment is H_tot = H_sys + H_env + H_coup, where H_sys and H_env represent the system and the electron reservoir, respectively. The system–environment coupling Hamiltonian is $Hcoup=∑ltld̂l†â+H.c.=F̂†â+H.c.$ Here, $F̂†≡∑ltld̂l†$⁠, with $d̂l†$ creating an electron in the lth band of the reservoir, $â$ annihilating an electron from the system’s orbital, and t_l representing the hopping integral between the reservoir’s lth band and the system’s orbital.

The HEOM theory was first proposed by Tanimura and Kubo,⁴¹ and the detailed formalism for bosonic environments has been elaborated in Refs. 42–44 and 153–158. Both the fermionic and bosonic HEOM approaches have been used to benchmark results obtained from other methods for OQSs.^{44,52,53,68,159–161} As coupled differential equations, the HEOM typically involves a large number of ADOs that are distinct in their indices and parities. This inherent complexity makes it challenging to express the HEOM as a compact dynamical equation.

In 2014, Yan introduced the concept of the “dissipaton,”¹⁶² a statistical quasiparticle that captures the environment’s non-Markovian memory. This concept has led to the dissipaton equation of motion (DEOM) for describing OQS dynamics,^{157,158,162–164} where the dissipaton density operators serve as dynamical variables. The DEOM can be viewed as a generalization of the HEOM, as it allows for the evaluation of both the system’s observables and the hybrid system–environment response properties. However, like the HEOM, the complex hierarchical structure of the DEOM complicates its mapping onto neural networks and quantum circuits.

To address the limitations of the DQME-FQ, a second quantization formulation was developed that characterizes the exchange of particles and energy between the system and environment as the creation and annihilation of dissipatons.¹⁶⁶  Figure 1(c) illustrates this concept, where the state of the environment is represented by a collection of dissipaton configurations, analogous to electron configurations representing the electronic structure of a chemical system. An “elementary” dissipation process is defined as the excitation from one dissipaton configuration to another, with each ADO in the HEOM/DEOM corresponding to such a process. This approach leads to the DQME in second quantization (DQME-SQ), which governs transitions between different dissipaton configurations. The theoretical construction of DQME-SQ is briefly outlined below.

The RDO of the system is obtained by projecting $ρ̃$ onto the dissipaton vacuum state.¹⁶⁶ It is straightforward to generalize Eqs. (6) and (7) to cases where the system is coupled to the environment through multiple channels.¹⁶⁶ The DQME-SQ theory is formally exact if the environment follows Gaussian statistics and the exponential decomposition of hybridization correlation function holds.

We highlight the advantages of DQME-SQ as follows: First, the quasiparticle description for environments allows for the treatment of both system and environment degrees of freedom in a unified manner. Second, compared to the continuous-variable representation of the RDT in DQME-FQ [cf. Eq. (4)], the dissipaton degrees of freedom are discrete in DQME-SQ, enabling easier encoding into neural quantum states or qubit states. Third, the DQME-SQ is formulated in a time-local structure and takes a much simpler form of a single differential equation rather than a set of coupled equations, as compared to HEOM. Finally, by utilizing the inherent symmetry and sparsity of RDT, its mapping to neural networks and quantum circuits and the associated sampling procedure can be greatly simplified. These advantages make DQME-SQ particularly suitable for representation in neural quantum states or quantum circuits [Fig. 1(d)] for AI- or QC-based simulations of many-body OQSs.

The pseudomode theory was first proposed by Garraway⁴⁹ and has since evolved into a formally exact approach to non-Markovian OQS dynamics, based on the exponential decomposition of the hybridization correlation function.^50–54 In the framework of pseudomode theory, the exponential decomposition leads to a finite set of discrete, unphysical pseudomodes. Formulated as a Lindblad quantum master equation (QME), the pseudomode approach has been used to simulate the non-Markovian dynamics of both bosonic⁵⁰ and fermionic OQSs.¹⁶⁷ The relationship and differences between the DQME-SQ and pseudomode methods are discussed in Ref. 166.

Besides methodological advancements, computer software tools^{47,48,94,168–173} capable of accurately and efficiently calculating the stationary and dynamic properties of OQSs are crucial. Such software tools provide a golden standard for assessing results obtained from AI- or QC-based approaches. For instance, Johansson et al. and Shammah et al. have constructed and developed the Quantum Toolbox in Python (QuTiP), an object-oriented open-source framework, for simulating the dynamics of OQSs.^174–177 Recently, the QuTiP-BoFiN library has been integrated with the QuTiP¹⁶⁹ to implement HEOM calculations for both bosonic and fermionic environments. Guan et al. have developed a Python package mpsqd,¹⁷¹ which utilizes the MPS and the matrix product operator to represent the ADO/RDOs and the generalized Liouvillian of the HEOM, respectively, offering a robust and flexible framework to efficiently and accurately simulate the OQSs dynamics. Furthermore, we have developed a general-purpose program, HEOM for QUantum Impurity with a Correlated Kernel (HEOM-QUICK),^{47,48,147,178} and widely used to investigate thermoelectric transport,^7–12 Kondo effect,^46,179–187 and spin excitations^188,189 in strongly correlated OQSs coupled to fermionic environments.

These software tools rely on conventional algorithms, which limit their simulations due to the exponential wall problem. For example, in the practical implementations of the HEOM-QUICK program, about $4NSNEL$ elements in the ADOs are processed in the computer memory, where N_S denotes the number of single-particle states, N_E denotes the dissipaton states, and L denotes the maximum allowed occupancy of dissipatons. Although various numerical algorithms have been designed to improve the efficiency of HEOM-QUICK,^47,48 simulating OQSs of high degrees of freedom in the ultra-low-temperature regime is still a challenging task. Therefore, there is an urgent need to develop and refine AI- and QC-based algorithms for many-body OQSs, along with their corresponding software implementations.

Data-driven machine learning approaches, which develop and optimize models using large training datasets from conventional methods, have proven effective for simulating the quantum dynamics of OQSs.^190–203 Here, we focus on a different class of machine learning approaches that do not rely on large external datasets but, instead, incorporate physical laws or constraints into their architecture and training process.²⁰⁴ In the context of OQSs, quantum correlations in both space and time are encoded in the models, while the quantum dynamical equation serves as a precise physical constraint guiding the training.^205,206

Time evolution and stationary states are obtained by accurately solving the Lindblad QME using variational or Markov-chain Monte Carlo algorithms.^{103–105,136} The effectiveness and accuracy of these algorithms have been demonstrated by simulating the dissipative dynamics of one- and two-dimensional spin lattice models,^103–105 and the simulation results agree well with those obtained with conventional methods. Recently, autoregressive models have also been employed to represent many-body quantum states,^207–209 enabling an accurate and efficient description of local correlations. However, most existing studies focus on the Markovian dynamics of OQSs in bosonic environments, while the simulation of non-Markovian dynamics is considered the “ultimate toughness”¹²⁰ for AI-based approaches to OQS dynamics and remains largely unexplored.

The DQME-SQ theory, represented by Eqs. (6) and (7), provides an effective framework for simulating the non-Markovian dissipative dynamics of many-body OQSs.^166,210 As illustrated in Fig. 2, a neural network can be constructed to represent the RDT, $ρ̃θ(n⃗,n⃗′,m⃗)$⁠, where $m⃗$ indicates the environment’s dissipaton configuration, while $n⃗$ and $n⃗′$ denote particle configurations within the system. In contrast to the NQS approach for Markovian dynamics,^{103–105,136} the neural network model depicted in Fig. 2 explicitly incorporates non-Markovian memory through the dissipaton configurations. Moreover, accurately recovering system–environment correlations requires careful design of the interconnections between nodes; see Ref. 210 for more details.

In contrast to the HEOM/DEOM approach, where the number of ADOs grows exponentially with increasing system size or non-Markovianity of the environment,⁴⁸ the NQS approach is expected to scale polynomially with the complexity of the system or environment. In addition, the efficiency of the NQS approach can be further enhanced by exploiting the sparsity and symmetry of the RDT in the DQME-SQ framework. Consequently, AI-based methods are anticipated to enable the investigation of highly complex many-body NQSs that are currently inaccessible with existing simulation techniques.

The quantum dynamics of OQSs are intrinsically nonunitary, while gate-based operations in quantum computers are unitary. Therefore, novel algorithms are needed to map nonunitary dissipative dynamics into the unitary framework of quantum algorithms. Following the proposal of the first quantum algorithm for simulating Markovian OQS dynamics¹¹⁶ by Kliesch et al., and its subsequent improvement by Childs and Li,²¹¹ substantial progress has been made in recent years.^{139–142,212–218} Various efficient algorithms have been proposed, including those based on imaginary-time evolution,^219–221 the time-dependent variational principle,²²² and duality quantum algorithms.^223,224 General theoretical frameworks for the quantum simulation of OQSs are discussed in Ref. 118.

Some of these algorithms have been implemented on noisy intermediate-scale quantum (NISQ) computers.^{141,218,225–228} However, QC algorithms capable of simulating the non-Markovian dynamics of OQSs remain scarce. Below, we demonstrate that the compact form of the DQME-SQ enables straightforward encoding of correlated quantum states of many-body OQSs into quantum circuits, allowing for digital quantum simulations of non-Markovian OQS dynamics.¹⁶⁶ 

Figure 3 illustrates the design of a quantum circuit for solving the DQME-SQ, which employs the linear combination of unitaries (LCU) method and Trotter decomposition to split the unitary operators U₀(δt) and U_±(δt) into simpler quantum logic units for time propagation. The target bits store the RDT state, which is propagated by the control bit. It is noteworthy that both the number of qubits and the circuit depth increase linearly with the degrees of freedom of the dissipatons. This contrasts with the HEOM/DEOM approach implemented with classical algorithms, where computational costs grow exponentially with the system size or the complexity of the environment. Thus, the favorable scalability of the QC-based DQME-SQ method makes it a viable and efficient strategy for simulating non-Markovian OQS dynamics.

The feasibility of DQME-SQ is demonstrated by implementing the LCU-Trotter algorithm on the Aer simulator of Qiskit.²²⁹ As depicted in Fig. 4, in both the low and high temperature regimes, the population dynamics of a spin-boson model with digital quantum simulations agree remarkably with those obtained with HEOM method using classical algorithms. The exactness and universality of DQME-SQ are further corroborated by simulations of more complex OQSs; see Ref. 166 for more details.

Recently, Hu et al.,^225,226 Wang et al.,²¹⁸ Head-Marsden et al.,²²⁷ and Sun et al.¹⁴¹ have successfully implemented quantum algorithms on NISQ devices to simulate OQSs, ranging from simple^218,225,227 to more realistic model systems.^141,226 The dissipative dynamics simulated are comparable to those obtained using classical numerical methods. In addition, it has been suggested that the intrinsic quantum noise from quantum gates can be harnessed to efficiently model the dissipative environment in a controllable manner.¹⁴¹ Most of these studies focus on the Markovian dynamics of OQSs in bosonic environments, while the simulation of non-Markovian dynamics in fermionic environments remains to be thoroughly explored.

An alternative hybrid quantum–classical algorithm based on the NQS representation to solve Eq. (16) has been proposed by Lee et al.¹³⁹ The OQS is represented by an ensemble of pure state trajectories, whose time evolution is described by the stochastic Schrödinger equation simulated by the time-dependent variational Monte Carlo algorithm. To obtain the accurate expectation value of observables, a large number of trajectories are required to perform the ensemble averaging, resulting in significant computational overhead.

In this Perspective, we have discussed recent advances in simulating many-body OQS dynamics using AI- and QC-based methods. By mapping the non-Markovian environment to statistically independent dissipatons, the DQME-SQ framework is established, represented as a single master equation. This formulation allows for efficient representation through neural quantum states or quantum circuits. These capabilities are exemplified by recent simulations of non-Markovian OQS dynamics based on RBMs and QC algorithms, which offer a compact encoding of the RDT and achieve accuracy comparable to the conventional HEOM/DEOM methods. Furthermore, the RBM representation has been implemented using quantum algorithms to design a quantum NQS representation for simulating the Markovian dynamics of many-body OQSs. This quantum machine learning method circumvents the computational costs associated with classical algorithms and can be extended to simulate non-Markovian dynamics.

Despite the encouraging and promising progress of currently developed AI- and QC-based algorithms in simulating many-body OQSs, the implementations are primarily limited to simple model systems with relatively low degrees of freedom. This limitation arises from the need to sample a large number of dissipaton configurations, which strongly bottlenecks the accuracy and efficiency of the TDVP algorithm. In addition, the depth of quantum circuits remains too large for full quantum simulations. Consequently, these algorithmic limitations severely hinder their versatility in simulating realistic, complex many-body OQSs.

Practical limitations stemming from hardware-related factors also hinder the deployment of QC algorithms for OQS simulations on current quantum devices.^{131,132,231–233} Quantum error correction is not yet feasible on near-term devices due to the high resource overhead required, such as the number of qubits and error rates needed to implement fault-tolerant gates.²³¹ This limits the depth and fidelity of quantum circuits that can be executed reliably. Current quantum devices suffer from high noise levels and short coherence times,¹³¹ which introduce errors and reduce the accuracy of quantum computations. In addition, there are limitations in qubit connectivity and gate fidelity,²³² which can restrict the types of circuits that can be implemented efficiently. Moreover, classical control hardware and software need to be optimized to handle the high overhead of quantum error mitigation techniques for noisy quantum circuits.^132,233 Since the simulations of non-Markovian OQS dynamics have rarely been implemented on real quantum computers, a systemic estimation of the accuracy and scale required for current or near-term devices still remains to be further explored.

Future developments of AI-based simulation methods for many-body OQSs should focus on enhancing the efficiency of neural network parameter determination by employing a more sophisticated variational ansatz and cost functions, exploring alternative representations beyond RBMs, and representing the generalized Liouvillian in Eq. (10) by the quantum neural propagator.^205,206 For QC-based approaches to OQS dynamics, it will be highly desirable to implement the proposed algorithms on real quantum computers in the NISQ regime and beyond. To this end, algorithms should be further simplified to reduce the circuit depth, and systemic errors and measurement costs due to unitary decomposition approximations should be more carefully accommodated. These developments may allow for precise simulations of OQSs with larger system sizes and more complex environments and/or external fields.

The support from the National Natural Science Foundation of China (Grant Nos. 22393912, 22425301, 22203083, 22103073, and 22373091), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0303306), and Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0450101) is gratefully acknowledged.

The authors have no conflicts to disclose.

Lyuzhou Ye: Funding acquisition (equal); Writing – original draft (equal); Writing – review & editing (equal). Yao Wang: Funding acquisition (equal); Writing – original draft (equal); Writing – review & editing (equal). Xiao Zheng: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.