Simulating spin-boson dynamics with stochastic Liouville–von Neumann equations

Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.

Simulating physical systems on quantum devices is one of the most promising applications of quantum technology. Current quantum approaches to simulating open quantum systems are still practically challenging on NISQ-era devices, because they typically require ancilla qubits and extensive controlled sequences. In this work, we propose a hybrid quantum-classical approach for simulating a class of open system dynamics called random-unitary channels. These channels naturally decompose into a series of convex unitary evolutions, which can then be efficiently sampled and run as independent circuits. The method does not require deep ancilla frameworks and thus can be implemented with lower noise costs. We implement simulations of open quantum systems up to dozens of qubits and with large channel ranks. Published by the American Physical Society 2024

Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrödinger equation (SSE), and the stochastic Liouville equation (SLE). These approaches replace the explicit system–environment coupling with an effective system‐only dynamics, where dissipative behavior emerges from ensemble averaging. Stochastic Hamiltonians, in particular, have long served as phenomenological tools in physical chemistry to include environmental effects without recourse to an explicit microscopic derivation. In this work, we aim at a self‐contained and accessible presentation of these approaches to further elaborate on their common roots in essential concepts of stochastic calculus and to delineate the conditions under which they are equivalent. We also discuss how different formulations naturally lead to different numerical time‐integration schemes, better suited for either classical simulation platforms, based on finite‐difference approximations, or quantum algorithms, that employ random unitary maps. Our analysis aims at providing a unified perspective and actionable recipes for classical and quantum implementations of stochastic evolution in the simulation of open quantum systems.

The rapid development of quantum computers has enabled demonstrations of quantum advantages on various tasks. However, real quantum systems are always dissipative due to their inevitable interaction with the environment, and the resulting non-unitary dynamics make quantum simulation challenging with only unitary quantum gates. In this work, we present an innovative and scalable method to simulate open quantum systems using quantum computers. We define an adjoint density matrix as a counterpart of the true density matrix, which reduces to a mixed-unitary quantum channel and thus can be effectively sampled using quantum computers. This method has several benefits, including no need for auxiliary qubits and noteworthy scalability. Moreover, some long-time properties like steady states and the thermal equilibrium can also be investigated as the adjoint density matrix and the true dissipated one converge to the same state. Finally, we present deployments of this theory in the dissipative quantum XY model for the evolution of correlation and entropy with short-time dynamics and the disordered Heisenberg model for many-body localization with long-time dynamics. This work promotes the study of real-world many-body dynamics with quantum computers, highlighting the potential to demonstrate practical quantum advantages.

We propose a method that combines the quantum-classical mapping approach to surface hopping with the dissipative quantum dynamics of the Lindblad master equation. Like conventional surface-hopping methods, our approach is based on classical trajectories coupled to the dynamics of a quantum subsystem. However, instead of evolving the subsystem wavefunction according to the time-dependent Schrödinger equation, we use stochastic quantum trajectories derived from secular Redfield theory. This approach enables the simulation of open quantum systems coupled simultaneously to Markovian quantum baths and anharmonic, non-Markovian classical degrees of freedom. Applications to the spin-boson model and to the cavity-enhanced fluorescence of an electronically nonadiabatic molecule show excellent agreement with fully quantum-mechanical benchmarks.

We present a framework to simulate the dynamics of hard probes such as heavy quarks or jets in a hot, strongly coupled quark-gluon plasma (QGP) on a quantum computer. Hard probes in the QGP can be treated as open quantum systems governed in the Markovian limit by the Lindblad equation. However, due to large computational costs, most current phenomenological calculations of hard probes evolving in the QGP use semiclassical approximations of the quantum evolution. Quantum computation can mitigate these costs and offers the potential for a fully quantum treatment with exponential speed-up over classical techniques. We report a simplified demonstration of our framework on IBM Q quantum devices and apply the random identity insertion method to account for cnot depolarization noise, in addition to measurement error mitigation. Our work demonstrates the feasibility of simulating open quantum systems on current and near-term quantum devices, which is of broad relevance to applications in nuclear physics, quantum information, and other fields.

Entanglement phase transitions in hybrid quantum circuits describe individual quantum trajectories rather than the measurement-averaged ensemble, despite the fact that results of measurements are not conventionally used for feedback. Here, we numerically demonstrate that a class of generalized measurements with identical measurement-averaged dynamics give different phases and phase transitions. We show that measurement-averaged destruction of Bell state entanglement is a useful proxy for determining which hybrid circuit yields the lowest-entanglement dynamics. We use this to argue that no unfolding of our model can avoid a volume law phase, which has implications for simulation of open quantum systems.

Methodological aspects of using the driven Liouville-von Neumann (DLvN) approach for simulating dynamical properties of molecular junctions are discussed. As a model system we consider a non-interacting resonant level uniformly coupled to a single Fermionic bath. We demonstrate how a finite system can mimic the depopulation dynamics of the dot into an infinite band bath of continuous and uniform density of states. We further show how the effects of spurious energy resolved currents, appearing due to the approximate nature of the equilibrium state obtained in DLvN calculations, can be avoided. Several ways to approach the wide band limit, which is often adopted in analytical treatments, using a finite numerical model system are discussed including brute-force increase of the lead model bandwidth as well as efficient cancellation or direct subtraction of finite-bandwidth effect. These methodological considerations may be relevant also for other numerical schemes that aim to study non-equilibrium thermodynamics via simulations of open quantum systems.

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.

Relaxation and decoherence processes are key features of the dynamics of open quantum systems [1]. In the standard approach one tries to develop appropriate master equations for the open system's reduced density matrix ρS which is given by the partial trace taken over the environmental variables coupled to the open system. Invoking the weak-coupling assumption one can formulate in many cases of physical interest a Markovian quantum master equation for the reduced density matrix, expressing the dynamical laws for the irreversible motion of the open system.

We discuss the general features of the time-evolving reduced density matrix (RDM) of multistate systems coupled to dissipative environments and show that many important aspects of the dynamics are visualized effectively and transparently through coherence maps, defined as snapshots of the real and imaginary components of the RDM on the square grid of system sites. In particular, the spread, signs, and shapes of the coherence maps collectively characterize the state of the system and the nature of the dynamics, as well as the equilibrium state. The topology of the system is readily reflected in its coherence map. Rows and columns show the composition of quantum superpositions, and their filling indicates the extent of the surviving coherence. Linear combinations of imaginary RDM elements specify instantaneous population derivatives. The main diagonal comprises the incoherent component of the dynamics, while the upper/lower triangular areas give rise to coherent contributions that increase the purity of the RDM. In open systems, the coherence map evolves to a band surrounding the principal diagonal whose width decreases with increasing temperature and dissipation strength. We illustrate these behaviors with examples of 10-site model molecular aggregates with Frenkel exciton couplings, where the electronic states of each monomer are coupled to harmonic vibrational baths.

The dynamics of a coupled electron-boson system is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron-nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.

We study in a unified manner the dissipative dynamics and the transfer of heat in the two-bath spin-boson model. We use the Bloch-Redfield (BR) formalism, valid in the very weak system-bath coupling limit, the noninteracting-blip approximation (NIBA), applicable in the nonadiabatic limit, and iterative, numerically exact path integral tools. These methodologies were originally developed for the description of the dissipative dynamics of a quantum system, and here they are applied to explore the problem of quantum energy transport in a nonequilibrium setting. Specifically, we study the weak-to-intermediate system-bath coupling regime at high temperatures kBT/ħ > ε, with ε as the characteristic frequency of the two-state system. The BR formalism and NIBA can lead to close results for the dynamics of the reduced density matrix (RDM) in a certain range of parameters. However, relatively small deviations in the RDM dynamics propagate into significant qualitative discrepancies in the transport behavior. Similarly, beyond the strict nonadiabatic limit NIBA's prediction for the heat current is qualitatively incorrect: It fails to capture the turnover behavior of the current with tunneling energy and temperature. Thus, techniques that proved meaningful for describing the RDM dynamics, to some extent even beyond their rigorous range of validity, should be used with great caution in heat transfer calculations, because qualitative-serious failures develop once parameters are mildly stretched beyond the techniques' working assumptions.

Quantum coherence dynamics of spin pairs in many-spin cluster

FeynDyn: A MATLAB program for fast numerical Feynman integral calculations for open quantum system dynamics on GPUs