The simulation of quantum systems has been a key aim of quantum technologies for decades, and generalization to open systems is necessary to include physically realistic systems. We introduce an approach for quantum simulations of open system dynamics in terms of an environment of minimal size and a time-dependent Hamiltonian. This enables the implementation of a continuous-time simulation with a finite environment, whereas state-of-the-art methods require an infinite environment or only match the simulation at discrete times. We find the necessary and sufficient conditions for this Hamiltonian to be well behaved, and when these are not met, we show that there exists an approximate Hamiltonian that is well behaved and look into its applications.
The simulation of quantum systems has been a key aim of quantum technologies for decades, and generalization to open systems is necessary to include physically realistic systems. We introduce an approach for quantum simulations of open system dynamics in terms of an environment of minimal size and a time-dependent Hamiltonian. This enables the implementation of a continuous-time simulation with a finite environment, whereas state-of-the-art methods require an infinite environment or only match the simulation at discrete times. We find the necessary and sufficient conditions for this Hamiltonian to be well behaved, and when these are not met, we show that there exists an approximate Hamiltonian that is well behaved and look into its applications.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
Linear dissipative differential equation is a fundamental model for a large number of physical systems, such as quantum dynamics with non-Hermitian Hamiltonian, open quantum system dynamics, diffusion process and damped system. In this work, we propose efficient quantum algorithms for simulating linear dissipative differential equations. The key idea of our algorithms is to perform the simulation only over an effective time period when the dynamics has not significantly dissipated yet, rather than over the entire physical evolution period. We conduct detailed analysis on the complexity of our algorithms and show that, while maintaining low state preparation cost, our algorithms can completely eliminate the time dependence. This is a more than exponential improvement compared to the previous state-of-the-art quantum algorithms.
All systems are open to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. In some cases the environment's only effect is to add a weak relaxation mechanism and thus can be ignored for short timescales. In others, however, the presence of the environment can fundamentally alter the behavior of the system. Such is the case in mesoscopic superconductors where the environment can stabilize superconductivity and in spin-boson systems where the environment induces a localization transition. Likewise, in technological applications we are often interested in systems operating far from equilibrium. Here the environment might act as a particle reservoir or strong driving force. In all these examples, we need accurate methods to describe the influence of the environment on the system and to solve for the resulting dynamics or equilibrium states. In this thesis, we develop computational and conceptual approaches to efficiently simulate quantum systems in contact with an environment. Our starting point is the use of numerical renormalization techniques. Thus, we restrict our attention to one-dimensional lattices or small quantum systems coupled to an environment. We have developed several complementary algorithms: a superoperator renormalization algorithm for simulating real-time Markovian dynamics and for calculating states in thermal equilibrium; a blocking algorithm for simulating integro-differential equations with long-time memory; and a tensor network algorithm for branched lattices, which can be used to simulate strongly dissipative systems. Further, we provide support for an idea that to generically and accurately simulate the real-time dynamics of strongly dissipative systems, one has to include all or part of the environment within the simulation. In addition, we discuss applications and open questions.
Abstract Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case, it corresponds to having a time-dependent Hamiltonian. Designing quantum algorithms for time-dependent Hamiltonian is generally much more complicated than time-independent Hamiltonian problems, due to the challenge in handling the time-ordering; moreover, almost all existing such algorithms are developed for qubit-based quantum devices. Here we propose an alternative formalism based on Sambe-Howland’s clock that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e. quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrödingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.
The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.
Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.