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For the first time in history, humankind might conceivably begin to imagine itself as a multi-planetary species. This goal will entail technical innovation in a number of contexts, including that of healthcare. All life on Earth shares an evolution that is coupled to specific environmental conditions, including gravitational and magnetic fields. While the human body may be able to adjust to short term disruption of these fields during space flights, any long term settlement would have to take into consideration the effects that different fields will have on biological systems, within the space of one lifetime, but also across generations. Magnetic fields, for example, influence the growth of stem cells in regenerative processes. Circadian rhythms are profoundly influenced by magnetic fields, a fact that will likely have an effect on mental as well as physical health. Even the brain responds to small perturbations of this field. One possible mechanism for the effects of weak magnetic fields on biological systems has been suggested to be the radical pair mechanism. The radical pair mechanism originated in the context of spin chemistry to describe how magnetic fields influence the yields of chemical reactions. This mechanism was subsequently incorporated into the field of quantum biology. Quantum biology, most generally, is the study of whether non-trivial quantum effects play any meaningful role in biological systems. The radical pair mechanism has been used most consistently in this context to describe the avian compass. Recently, however, a number of studies have investigated other biological contexts in which the radical pair might play a role, from the action of anaesthetics and antidepressants, to microtubule development and the proper function of the circadian clock... (full abstract in the manuscript), Comment: 18 pages, 9 figures

The potential role of spin in biological systems is a primary topic in quantum biology. However, much of this research focuses on electron spin. A recent hypothesis suggests that nuclear spin may be better suited to biological processes, being less sensitive to decoherence. The hypothesis details how phosphorus nuclei might be prepared in a spin-entangled state, how this entanglement is protected by assembly into calcium phosphate (Posner) molecules, and how this entanglement might modulate calcium ion production and concomitant neural activation. In this paper, we investigate the robustness of quantum effects such as coherence and entanglement in Posner molecules. We investigate how these effects are directly dependent on specific parameters such as spin-spin coupling strengths and Posner molecule symmetry. We also investigate how lithium isotope-doped Posner molecules differentially modulate quantum resources such as coherence and entanglement and whether this is a viable explanation for lithium's mechanism of action in bipolar disease. Finally, we illustrate how entanglement might possibly be preserved through the exploitation of the biological environment., Comment: 29 pages, 10 figures

The Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm for Near-term Intermediate-Scale Quantum computers (NISQ) providing approximate solutions for combinatorial optimiz\-ation problems. The QAOA utilizes a quantum-classical loop, consisting of a quantum ansatz and a classical optimizer, to minimize some cost function, computed on the quantum device. This paper presents an investigation into the impact of realistic noise on the classical optimizer and the determination of optimal circuit depth for the Quantum Approximate Optimization Algorithm (QAOA) in the presence of noise. We find that, while there is no significant difference in the performance of classical optimizers in a state vector simulation, the Adam and AMSGrad optimizers perform best in the presence of shot noise. Under the conditions of real noise, the SPSA optimizer, along with ADAM and AMSGrad, emerge as the top performers. The study also reveals that the quality of solutions to some 5 qubit minimum vertex cover problems increases for up to around six layers in the QAOA circuit, after which it begins to decline. This analysis shows that increasing the number of layers in the QAOA in an attempt to increase accuracy may not work well in a noisy device., Comment: 13 pages, 9 figures

We compare the performance of three nearest neighbor search algorithms: the Orchard, ball tree, and VP-tree algorithms. These algorithms are commonly used for nearest-neighbor searches and are known for their efficiency in large datasets. We analyze the fraction of distances computed in relation to the size of the dataset and its dimension. For each algorithm we derive a fitting function for the efficiency as a function to set size and dimension. The article aims to provide a comprehensive analysis of the performance of these algorithms and help researchers and practitioners choose the best algorithm for their specific application., Comment: 5 pages Latex

The quantum asymptotically universal multi-feature (QAUM) encoding architecture was recently introduced and showed improved expressivity and performance in classifying pulsar stars. The circuit uses generalized trainable layers of parameterized single-qubit rotation gates and single-qubit feature encoding gates. Although the improvement in classification accuracy is promising, the single-qubit nature of this architecture, combined with the circuit depth required for accuracy, limits its applications on NISQ devices due to their low coherence times. This work reports on the design, implementation, and evaluation of ensembles of single-qubit QAUM classifiers using classical bagging and boosting techniques. We demonstrate an improvement in validation accuracy for pulsar star classification. We find that this improvement is not problem-specific as we observe consistent improvements for the MNIST Digits and Wisconsin Cancer datasets. We also observe that the boosting ensemble achieves an acceptable level of accuracy with only a small amount of training, while the bagging ensemble achieves higher overall accuracy with ample training time. This shows that classical ensembles of single-qubit circuits present a new approach for certain classification problems., Comment: 19 pages, 13 figures

Machine learning with hierarchical quantum circuits, usually referred to as Quantum Convolutional Neural Networks (QCNNs), is a promising prospect for near-term quantum computing. The QCNN is a circuit model inspired by the architecture of Convolutional Neural Networks (CNNs). CNNs are successful because they do not need manual feature design and can learn high-level features from raw data. Neural Architecture Search (NAS) builds on this success by learning network architecture and achieves state-of-the-art performance. However, applying NAS to QCNNs presents unique challenges due to the lack of a well-defined search space. In this work, we propose a novel framework for representing QCNN architectures using techniques from NAS, which enables search space design and architecture search. Using this framework, we generate a family of popular QCNNs, those resembling reverse binary trees. We then evaluate this family of models on a music genre classification dataset, GTZAN, to justify the importance of circuit architecture. Furthermore, we employ a genetic algorithm to perform Quantum Phase Recognition (QPR) as an example of architecture search with our representation. This work provides a way to improve model performance without increasing complexity and to jump around the cost landscape to avoid barren plateaus. Finally, we implement the framework as an open-source Python package to enable dynamic QCNN creation and facilitate QCNN search space design for NAS., Comment: 22 pages, 13 figures

One of the approaches used to solve for the dynamics of open quantum systems is the hierarchical equations of motion (HEOM). Although it is numerically exact, this method requires immense computational resources to solve. The objective here is to demonstrate whether models such as SARIMA, CatBoost, Prophet, convolutional and recurrent neural networks are able to bypass this requirement. We are able to show this successfully by first solving the HEOM to generate a data set of time series that depict the dissipative dynamics of excitation energy transfer in photosynthetic systems then, we use this data to test the models ability to predict the long-time dynamics when only the initial short-time dynamics is given. Our results suggest that the SARIMA model can serve as a computationally inexpensive yet accurate way to predict long-time dynamics.

A well-known approach to describe the dynamics of an open quantum system is to compute the master equation evolving the reduced density matrix of the system. This approach plays an important role in describing excitation transfer through photosynthetic light harvesting complexes (LHCs). The hierarchical equations of motion (HEOM) was adapted by Ishizaki and Fleming (J. Chem. Phys., 2009) to simulate open quantum dynamics in the biological regime. We generate a set of time dependent observables that depict the coherent propagation of electronic excitations through the LHCs by solving the HEOM. We solve the inverse problem using classical machine learning (ML) models as this is a computationally intractable problem. The objective here is to determine whether a trained ML model can perform Hamiltonian tomography by using the time dependence of the observables as inputs. We demonstrate the capability of convolutional neural networks to tackle this research problem. The models developed here can predict Hamiltonian parameters such as excited state energies and inter-site couplings of a system up to 99.28\% accuracy and mean-squared error as low as 0.65., Comment: 9 pages, 4 figures

The SARS-CoV-2 pandemic has added new urgency to the study of viral mechanisms of infection. But while vaccines offer a measure of protection against this specific outbreak, a new era of pandemics has been predicted. In addition to this, COVID-19 has drawn attention to post-viral syndromes and the healthcare burden they entail. It seems integral that knowledge of viral mechanisms is increased through as wide a research field as possible. To this end we propose that quantum biology might offer essential new insights into the problem, especially with regards to the important first step of virus-host invasion. Research in quantum biology often centres around energy or charge transfer. While this is predominantly in the context of photosynthesis there has also been some suggestion that cellular receptors such as olfactory or neural receptors might employ vibration assisted electron tunnelling to augment the lock-and-key mechanism. Quantum tunnelling has also been observed in enzyme function. Enzymes are implicated in the invasion of host cells by the SARS-CoV-2 virus. Receptors such as olfactory receptors also appear to be disrupted by COVID-19. Building on these observations we investigate the evidence that quantum tunnelling might be important in the context of infection with SARS-CoV-2. We illustrate this with a simple model relating the vibronic mode of, for example, a viral spike protein to the likelihood of charge transfer in an idealised receptor. Our results show a distinct parameter regime in which the vibronic mode of the spike protein enhances electron transfer. With this in mind, novel therapeutics to prevent SARS-CoV-2 transmission could potentially be identified by their vibrational spectra., Comment: 32 pages, 7 figures

Finding solutions to systems of linear equations is a common prob\-lem in many areas of science and engineering, with much potential for a speedup on quantum devices. While the Harrow-Hassidim-Lloyd (HHL) quantum algorithm yields up to an exponential speed-up over classical algorithms in some cases, it requires a fault-tolerant quantum computer, which is unlikely to be available in the near term. Thus, attention has turned to the investigation of quantum algorithms for noisy intermediate-scale quantum (NISQ) devices where several near-term approaches to solving systems of linear equations have been proposed. This paper focuses on the Variational Quantum Linear Solvers (VQLS), and other closely related methods. This paper makes several contributions that include: the first application of the Evolutionary Ansatz to the VQLS (EAVQLS), the first implementation of the Logical Ansatz VQLS (LAVQLS), based on the Classical Combination of Quantum States (CQS) method, the first proof of principle demonstration of the CQS method on real quantum hardware and a method for the implementation of the Adiabatic Ansatz (AAVQLS). These approaches are implemented and contrasted., Comment: 13 pages, 15 figures; fixed typo in References

Variational Hybrid Quantum Classical Algorithms (VHQCAs) are a class of quantum algorithms intended to run on noisy intermediate-scale quantum (NISQ) devices. These algorithms employ a parameterized quantum circuit (ansatz) and a quantum-classical feedback loop. A classical device is used to optimize the parameters in order to minimize a cost function that can be computed far more efficiently on a quantum device. The cost function is constructed such that finding the ansatz parameters that minimize its value, solves some problem of interest. We focus specifically on the Variational Quantum Linear Solver (VQLS), and examine the effect of several gradient-free and gradient-based classical optimizers on performance. We focus on both the average rate of convergence of the classical optimizers studied, as well as the distribution of their average termination cost values, and how these are affected by noise. Our work demonstrates that realistic noise levels on NISQ devices present a challenge to the optimization process. All classical optimizers appear to be very negatively affected by the presence of realistic noise. If noise levels are significantly improved, there may be a good reason for preferring gradient-based methods in the future, which performed better than the gradient-free methods with the only shot-noise present. The gradient-free optimizers, Simultaneous Perturbation Stochastic Approximation (SPSA) and Powell's method, and the gradient-based optimizers, AMSGrad and BFGS performed the best in the noisy simulation, and appear to be less affected by noise than the rest of the methods. SPSA appears to be the best performing method. COBYLA, Nelder-Mead and Conjugate-Gradient methods appear to be the most heavily affected by noise, with even slight noise levels significantly impacting their performance., Comment: 17 pages; pre-accepted version

The performances of quantum thermometry in thermal equilibrium together with the output power of certain class of quantum engines share a common characteristic: both are determined by the heat capacity of the probe or working medium. After noticing that the heat capacity of spin ensembles can be significantly modified by collective coupling with a thermal bath, we build on the above observation to investigate the respective impact of such collective effect on quantum thermometry and quantum engines. We find that the precision of the temperature estimation is largely increased at high temperatures, reaching even the Heisenberg scaling - inversely proportional to the number of spins. For Otto engines operating close to the Carnot efficiency, collective coupling always enhances the output power. Some tangible experimental platforms are suggested., Comment: Accepted version, 8 + 4 pages, 5 figures

The entropy production in dissipative processes is the essence of the arrow of time and the second law of thermodynamics. For dissipation of quantum systems, it was recently shown that the entropy production contains indeed two contributions: a classical one and a quantum one. Here we show that for degenerate (or near-degenerate) quantum systems there are additional quantum contributions which, remarkably, can become negative. Furthermore, such negative contributions are related to significant changes in the ongoing thermodynamics. This includes phenomena such as generation of coherences between degenerate energy levels (called horizontal coherences), alteration of energy exchanges and, last but not least, reversal of the natural convergence of the populations toward the thermal equilibrium state. Going further, we establish a complementarity relation between horizontal coherences and population convergence, particularly enlightening for understanding heat flow reversals. Conservation laws of the different types of coherences are derived. Some consequences for thermal machines and resource theory of coherence are suggested., Comment: 8+11 pages, 5 figures. References added. Small changes and adjustments implemented. Accepted in Phys. Rev. A - final author version

Rich quantum effects emerge when several quantum systems are indistinguishable from the point of view of the bath they interact with. In particular, delocalised excitations corresponding to coherent superposition of excited states appear and change drastically the dynamics and steady state of the systems. Such phenomena, which are central mechanisms of superradiance, present interesting properties for thermodynamics and potentially other quantum technologies. Indeed, a recent paper [Phys. Rev. A 99, 052105 (2019)] studies these properties in a pair of indistinguishable two-level systems and points out surprising effects of mitigation and amplification of the bath's action on the energy and entropy of the pair. Here, we generalise the study to ensembles of arbitrary number of spins of arbitrary size (i.e. dimension). We confirm that the previously uncovered mitigation and amplification effects remain, but also that they become more and more pronounced with growing number of spins and growing spin size. Moreover, we find that the free energy variation and the entropy production associated with the bath-driven dissipation are systematically reduced, formalising the idea of mitigation of the bath's action. Most remarkably, the combination of mitigation effects from two baths at different temperatures can result in amplifying their action. This is illustrated with cyclic thermal machines, and leads to large power enhancements. The reduction of irreversibility is also an interesting aspect since irreversibility is known to limit the performance of thermodynamic tasks. The above findings might also lead to interesting applications in collective work extraction, quantum battery charging, state protection, light harvesting devices, quantum biology, but also for the study of entropy production. Moreover, some experimental realisations and observations suggest that such effects are within reach., Comment: 15+12 pages, 10 figures. Accepted for publication in Phys. Rev. Research

Open quantum walks (OQWs) describe a quantum walker on an underlying graph whose dynamics is purely driven by dissipation and decoherence. Mathematically, they are formulated as completely positive trace preserving (CPTP) maps on the space of density matrices for the walker on the graph. Any microscopically derived OQW must include the possibility of remaining on the same site on the graph when the map is applied. We extend the CPTP map to describe a lazy OQW. We derive a central limit theorem for lazy OQWs on a $d$-dimensional lattice, where the distribution converges to a Gaussian. We show that the properties of this Gaussian computed using conventional methods agree with the general formulas derived from our central limit theorem.

One of the stunning consequences of quantum correlations in thermodynamics is the reversal of the arrow of time, recently shown experimentally in [K. Micadei, et al., Nat. Commun. 10:2456 (2019)], and manifesting itself by a reversal of the heat flow (from the cold system to the hot one). Here, we show that contrary to what could have been expected, heat flow reversal can happen without reversal of the arrow of time. Moreover, contrasting with previous studies, no initial correlations between system and bath is required. Instead, the heat flow reversal only relies on internal quantum coherences or correlations, which provides practical advantages over previous schemes: one does not need to have access to the bath in order to reverse the heat flow. The underlying mechanism is explained and shown to stem from the collective system-bath coupling and the impact of non-energetic coherences (coherences between degenerate energy levels) on apparent temperatures. The phenomenon is first uncovered in a broad framework valid for diverse quantum systems containing energy degeneracy. By the end of the paper, aiming at experimental realisations, more quantitative results are provided for a pair of two-level systems. Finally, as a curiosity, we mention that our scheme can be adapted as a correlations-to-energy converter, which have the particularity to be able to operate at constant entropy, similarly to ideal work sources., Comment: 7 pages of main text, 3 figures, 4 pages of appendices and references

The computational cost of preparing a quantum state can be substantial depending on the structure of data to be encoded. Many quantum algorithms require repeated sampling to find the answer, mandating reconstruction of the same input state for every execution of an algorithm. Thus, the advantage of quantum computation can diminish due to redundant state initialization. We present a framework based on quantum forking that bypasses this fundamental issue and expedites a family of tasks that require sampling from independent quantum processes. Quantum forking propagates an input state to multiple quantum trajectories in superposition, and a weighted power sum of individual results from each trajectories is obtained in one measurement via quantum interference. The significance of our work is demonstrated via applications to implementing non-unitary quantum channels, studying entanglement and benchmarking quantum control. A proof-of-principle experiment is implemented on the IBM and Rigetti quantum cloud platforms., Comment: 17 pages, 6 figures

Heat exchanges are the essence of Thermodynamics. In order to investigate non-equilibrium effects like quantum coherence and correlations in heat flows we introduce the concept of apparent temperature. Its definition is based on the expression of the heat flow between out-of-equilibrium quantum systems. Such apparent temperatures contain crucial information on the role and impact of correlations and coherence in heat exchanges. In particular, both behave as populations, affecting dramatically the population balance and therefore the apparent temperatures and the heat flows. We show how seminal results can be re-obtained, offering an interesting alternative point of view. We also present new predictions and suggest a simple experiment to test them. Our results show how quantum and non-equilibrium effects can be used advantageously, finding applications in quantum thermal machine designs and non-equilibrium thermodynamics but also in collective-effect phenomena., Comment: 9 pages of main text, 3 figures, 8 pages of supplemental material. Accepted in Quantum Science and Technology. Close to accepted version

The force estimation problem in quantum metrology with an arbitrary non-Markovian Gaussian bath is considered. No assumptions are made on the bath spectrum and coupling strength with the probe. Considering the natural global unitary evolution of both bath and probe and assuming initial global Gaussian states we are able to solve the main issues of any quantum metrological problem: the best achievable precision determined by the quantum Fisher information, the best initial state and the best measurement. Studying the short time behavior and comparing to regular Markovian dynamics we observe an increase of quantum Fisher information. We emphasize that this phenomenon is due to the ability to perform measurements below the correlation time of the bath, activating non-Markovian effects. This brings huge consequences for the sequential preparation-and- measurement scenario as the quantum Fisher information becomes unbounded when the initial probe mean energy goes to infinity, whereas its Markovian counterpart remains bounded by a constant. The long time behavior shows the complexity and potential variety of non-Markovian effects, somewhere between the exponential decay characteristic of Markovian dynamics and the sinusoidal oscillations characteristic of resonant narrow bands., Comment: 6 pages of main text and 12 pages of Appendix

Since the emergence of oxygenic photosynthesis, living systems have developed protective mechanisms against reactive oxygen species. During charge separation in photosynthetic reaction centres, triplet states can react with molecular oxygen generating destructive singlet oxygen. The triplet product yield in bacteria is observed to be reduced by weak magnetic fields. Reaction centres from plants' photosystem II share many features with bacterial reaction centres, including a high-spin iron whose function has remained obscure. To explain observations that the magnetic field effect is reduced by the iron, we propose that its fast-relaxing spin plays a protective role in photosynthesis by generating an effective magnetic field. We consider a simple model of the system, derive an analytical expression for the effective magnetic field and analyse the resulting triplet yield reduction. The protective mechanism is robust for realistic parameter ranges, constituting a clear example of a quantum effect playing a macroscopic role vital for life., Comment: 25 pages, 6 figure

We give an algorithm for prediction on a quantum computer which is based on a linear regression model with least squares optimisation. Opposed to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on the machine learning task of guessing the output corresponding to a new input given examples of data points. Furthermore, we adapt the algorithm to process non-sparse data matrices that can be represented by low-rank approximations, and significantly improve the dependency on its condition number. The prediction result can be accessed through a single qubit measurement or used for further quantum information processing routines. The algorithm's runtime is logarithmic in the dimension of the input space provided the data is given as quantum information as an input to the routine., Comment: 6 pages, 1 figure

Open Quantum Walks (OQWs) are exclusively driven by dissipation and are formulated as completely positive trace preserving (CPTP) maps on underlying graphs. The microscopic derivation of discrete and continuous in time OQWs is presented. It is assumed that connected nodes are weakly interacting via a common bath. The resulting reduced master equation of the quantum walker on the lattice is in the generalised master equation form. The time discretisation of the generalised master equation leads to the OQWs formalism. The explicit form of the transition operators establishes a connection between dynamical properties of the OQWs and thermodynamical characteristics of the environment. The derivation is demonstrated for the examples of the OQW on a circle of nodes and on a finite chain of nodes. For both examples a transition between diffusive and ballistic quantum trajectories is observed and found to be related to the temperature of the bath., Comment: 13 pages, 7 figures

A microscopic derivation of an open quantum walk on a two node graph is presented. It is shown that for the considered microscopic model of the system-bath interaction the resulting quantum master equation takes the form of a generalized master equation. The explicit form of the "quantum coin" operators is derived. The formalism is demonstrated for the example of a two-level system walking on a two-node graph., Comment: 10 pages, 2 figures

We consider the problem of constructing a "universal set" of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension $d$, we explicitly construct a universal set of semigroup generators, parametrized by $d^2-3$ continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a $d$ dimensional Markovian quantum system is the ability to implement a) quantum channels from the semigroups generated by elements of the universal set of generators, and b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems., Comment: Revised version. Restricted recombination method to first order Suzuki-Lie-Trotter integrators, and added discussion concerning issues with the application of higher order integrators in the open quantum systems setting

It is well known that for certain tasks, quantum computing outperforms classical computing. A growing number of contributions try to use this advantage in order to improve or extend classical machine learning algorithms by methods of quantum information theory. This paper gives a brief introduction into quantum machine learning using the example of pattern classification. We introduce a quantum pattern classification algorithm that draws on Trugenberger's proposal for measuring the Hamming distance on a quantum computer (CA Trugenberger, Phys Rev Let 87, 2001) and discuss its advantages using handwritten digit recognition as from the MNIST database., Comment: 14 pages, 3 figures, presented at the 13th Pacific Rim International Conference on Artificial Intelligence

Perceptrons are the basic computational unit of artificial neural networks, as they model the activation mechanism of an output neuron due to incoming signals from its neighbours. As linear classifiers, they play an important role in the foundations of machine learning. In the context of the emerging field of quantum machine learning, several attempts have been made to develop a corresponding unit using quantum information theory. Based on the quantum phase estimation algorithm, this paper introduces a quantum perceptron model imitating the step-activation function of a classical perceptron. This scheme requires resources in $\mathcal{O}(n)$ (where $n$ is the size of the input) and promises efficient applications for more complex structures such as trainable quantum neural networks., Comment: 11 pages, 6 figures, accepted by Physics Letters A

Quantum walks have been shown to be fruitful tools in analysing the dynamic properties of quantum systems. This article proposes to use quantum walks as an approach to Quantum Neural Networks (QNNs). QNNs replace binary McCulloch-Pitts neurons with a qubit in order to use the advantages of quantum computing in neural networks. A quantum walk on the firing states of such a QNN is supposed to simulate central properties of the dynamics of classical neural networks, such as associative memory. It is shown that a biased discrete Hadamard walk derived from the updating process of a biological neuron does not lead to a unitary walk. However, a Stochastic Quantum Walk between the global firing states of a QNN can be constructed and it is shown that it contains the feature of associative memory. The quantum contribution to the walk accounts for a modest speed-up in some regimes., Comment: 9 pages, 7 figures

The dynamics of two spins coupled to bosonic baths at different temperatures is studied. The analytical solution for the reduced density matrix of the system in the Markovian and Post-Markovian case with exponential memory kernel is found. The dynamics and temperature dependence of spin-spin entanglement is analyzed., Comment: Proceedings of the International Conference on Mathematical Physics and Its Applications (Russian Federation, Samara, September 8-13, 2008); accepted in the Vestnik of Samara State University, Special Issue, 2009; ref. [14] added and typos corrected

The quantum dynamics of a spin chain interacting with multiple bosonic baths is described in a mixed Wigner-Heisenberg representation. The formalism is illustrated by simulating the time evolution of the reduced density matrix of two coupled spins, where each spin is also coupled to its own bath of harmonic oscillators. In order to prove the validity of the approach, an analytical solution in the Born-Markov approximation is found. The agreement between the two methods is shown.