Lindblad Equation Research Articles

Qocttools: A program for quantum optimal control calculations

The qocttools code performs optimization calculations on quantum systems using optimal control theory. The problem that it solves is the following: given a system governed by a Hamiltonian whose precise form can be tuned through the variation of some control parameters, and given a user-defined merit function of the system trajectory, what are the values of those parameters that lead to the time evolution that maximizes the merit function? The code permits to work on generic systems, and uses either Schrödinger's or Lindblad's equation in order to deal with closed or open systems, respectively. It may also use the tools of Floquet formalism when dealing with periodic perturbations. It is written in Python, and the user can interface with it by preparing driver Python scripts that define the model, and load the appropriate qocttools modules. It is open and free software, and also relies on open and free widely used libraries and packages, such as QuTiP (to handle the quantum model definition and manipulation), and NLopt (to perform the optimizations). Program summaryProgram Title:qocttoolsCPC Library link to program files:https://doi.org/10.17632/49z3vydmwk.1Developer's repository link:https://gitlab.com/acbarrigon/qocttoolsCode Ocean capsule:https://codeocean.com/capsule/7560215Licensing provisions: GPLv3Programming language: PythonNature of problem: When a quantum system can be driven by external perturbations, one may be interested in finding the form of those perturbations that lead to a certain behavior. For example, driving the system from its ground to a given excited state, maximizing the value of some observable, etc. This can be interpreted as an inverse spectroscopy: the goal is not to predict the reaction of a piece of matter to an external probe or perturbation (e.g. an electromagnetic field), but instead to predict the precise shape of the perturbation that leads to a given predefined reaction. This problem may be posed for closed systems evolving coherently, or in the context of open quantum systems, in the presence of an environment. Also, when considering periodic perturbations, the problem may be formulated in the language of quantum Floquet theory.Solution method: This code implements the fundamental equations of quantum optimal control theory [1], the mathematical framework that addresses the problem described above. It (1) handles both closed or open quantum systems; (2) allows the user to specify any form for the target function (the one that is to be optimized, and encodes the desired system behavior), as it may be specified through any user-defined Python function; (3) allows the user to specify general parameterized forms for the control functions, and to set bounds and constraints on those parameters; (4) implements formulas, based on the adjoint method in most cases, for the computation of the gradient of the target functions with respect to the control parameters, as these gradients are the key tools for an efficient optimization, and (5) interfaces to a generic optimization library (NLopt), that permits to perform the optimization with numerous different algorithms, allowing for the use of bounds and linear or nonlinear constraints.

Rényi negativities in non-equilibrium open free-boson chains

In this paper, we consider the dynamics of Rényi negativities after a quantum quench in the free-boson chain with homogeneous dissipation. Initially we prepare the system in the squeezed thermal state, and then let it evolves under the tight-binding bosonic Hamiltonian with local linear dissipation. We use the Lindblad equation to solve the time evolution of the covariance matrix, from which one can obtain the time dependence of Rényi negativities. We are interested in the weak dissipation hydrodynamic limit where a quasi-particle picture emerges. In this limit, exact results of non-equilibrium dynamics of Rényi negativities can be obtained using the stationary phase method. We consider the Rényi negativities between both adjacent and disjoint regions in a infinite chain. We numerically test our analytical predictions and perfect matches have found.

Thermal State Preparation via Rounding Promises

A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator. In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state.

A time-dependent regularization of the Redfield equation

We introduce a new regularization of the Redfield equation based on a replacement of the Kossakowski matrix with its closest positive semidefinite neighbor. Unlike most of the existing approaches, this procedure is capable of retaining the time dependence of the Kossakowski matrix, leading to a completely positive divisible quantum process. Using the dynamics of an exactly-solvable three-level open system as a reference, we show that our approach performs better during the transient evolution, if compared to other approaches like the partial secular master equation or the universal Lindblad equation. To make the comparison between different regularization schemes independent from the initial state, we introduce a new quantitative approach based on the Choi-Jamiołkowski isomorphism.

Memory Tensor for Non-Markovian Dynamics with Random Hamiltonian

In the theory of open quantum systems, the Markovian approximation is very widespread. Usually, it assumes the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation for density matrix dynamics and quantum regression formulae for multi-time correlation functions. Nevertheless, now, quantum non-Markovianity is being actively studied, especially the non-Markovianity of multi-time correlations. In this work, we consider dynamics with a random Hamiltonian, which can lead to GKSL dynamics of the density matrix for some special cases, but correlation functions generally do not satisfy the quantum regression formulae. Despite the fact that random Hamiltonians have been actively studied, dynamics with such Hamiltonians has been little discussed from the viewpoint of multi-time correlations. For specific models with a random Hamiltonian, we provide the formulae for multi-time correlations which occur instead of the usual regression formulae. Moreover, we introduce and calculate the memory tensor, which characterizes multi-time correlations against the Markovian ones. We think that, despite being applied to specific models, the methods developed in this work can be used in a much broader setup.

Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials

In this paper, we investigate the problem of simulating open system dynamics governed by the well-known Lindblad master equation. In our prequel paper [Patel and Wilde, Open Sys. & Info. Dyn., 30:2350010 (2023)], we introduced an input model in which Lindblad operators are encoded into pure quantum states, called program states, and we also introduced a method, called wave matrix Lindbladization, for simulating Lindbladian evolution by means of interacting the system of interest with these program states. Therein, we focused on a simple case in which the Lindbladian consists of only one Lindblad operator and a Hamiltonian. Here, we extend the method to simulating general Lindbladians and other cases in which a Lindblad operator is expressed as a linear combination or a polynomial of the operators encoded into the program states. We propose quantum algorithms for all these cases and also investigate their sample complexity, i.e., the number of program states needed to simulate a given Lindbladian evolution approximately. Finally, we demonstrate that our quantum algorithms provide an efficient route for simulating Lindbladian evolution relative to full tomography of encoded operators, by proving that the sample complexity for tomography is dependent on the dimension of the system, whereas the sample complexity of wave matrix Lindbladization is dimension independent.

Efficient Quantum Algorithms for Testing Symmetries of Open Quantum Systems

Symmetry is an important and unifying notion in many areas of physics. In quantum mechanics, it is possible to eliminate degrees of freedom from a system by leveraging symmetry to identify the possible physical transitions. This allows us to simplify calculations and characterize potentially complicated dynamics of the system with relative ease. Previous works have focused on devising quantum algorithms to ascertain symmetries by means of fidelity-based symmetry measures. In our present work, we develop alternative symmetry testing quantum algorithms that are efficiently implementable on quantum computers. Our approach estimates asymmetry measures based on the Hilbert–Schmidt distance, which is significantly easier, in a computational sense, than using fidelity as a metric. The method is derived to measure symmetries of states, channels, Lindbladians, and measurements. We apply this method to a number of scenarios involving open quantum systems, including the amplitude damping channel and a spin chain, and we test for symmetries within and outside the finite symmetry group of the Hamiltonian and Lindblad operators.

Explicit expressions for stationary states of the Lindblad equation for a finite state space

The Lindblad equation describes the time evolution of a density matrix of a quantum mechanical system. Stationary solutions are obtained by time-averaging the solution, which will in general depend on the initial state. We provide an analytical expression for the steady states of the Lindblad equation using the quantum jump unraveling, a version of an ergodic theorem, and the stationary probabilities of the corresponding discrete-time Markov chains. Our result is valid when the number of states appearing the in quantum trajectory is finite. The classical case of a Markov jump-process is recovered as a special case, and differences between the two are discussed.

Commutation simulator for open quantum dynamics

Recent progress in quantum simulation and algorithms has demonstrated a rapid expansion in capabilities. The search continues for new techniques and applications to exploit quantum advantage. Here we propose an innovative method to investigate directly the properties of a time-dependent density operator . Using quantum commutation simulators, we can directly compute the expectation value of the commutation relation and thus of the rate of change of . The approach can be utilized as a quantum eigen-vector solver for the von Neumann equation and a decoherence investigator for the Lindblad equation, by using just the statistics of single-qubit measurements. A simple but important example is demonstrated in the single-qubit case and we discuss extension of the method for practical quantum simulation with many qubits, towards investigation of more realistic quantum systems.

The role of dephasing for dark state coupling in a molecular Tavis-Cummings model.

The collective coupling of an ensemble of molecules to a light field is commonly described by the Tavis-Cummings model. This model includes numerous eigenstates that are optically decoupled from the optically bright polariton states. Accessing these dark states requires breaking the symmetry in the corresponding Hamiltonian. In this paper, we investigate the influence of non-unitary processes on the dark state dynamics in the molecular Tavis-Cummings model. The system is modeled with a Lindblad equation that includes pure dephasing, as it would be caused by weak interactions with an environment, and photon decay. Our simulations show that the rate of pure dephasing, as well as the number of two-level systems, has a significant influence on the dark state population.

Effect of Quantum Decoherence on Collective Neutrino Oscillations

The effect of the quantum decoherence of neutrino mass states on collective oscillations of neutrinos has been studied for the case of three flavors using a method based on the stability analysis of the Lindblad equation with the neutrino evolution Hamiltonian including the effects of the self-interaction. New analytical conditions for the appearance of collective neutrino oscillations in supernova explosions have been obtained taking into account the quantum decoherence of neutrinos.

DESCRIPTION OF THE SIMPLEST NON-MARKOV PROCESS USING A DIFFERENTIAL EQUATION FOR THE QUANTUM STATE VECTOR

The Jaynes-Cummings model with one atom and a photon is considered. A photon leaks out of the cavity (optical resonator). An atom can be in an excited and ground state. Usually, the dynamics of the probability of finding a photon in a cavity is considered using the basic quantum Lindblad equation, in which the density matrix acts as an unknown function. The Lindblad equation describes a quantum Markov random process. The article attempts to replace the equation from the density matrix with an ersatz of the Lindblad equation, which is a differential equation from the state wave vector. The quantum master equation involves the use of a matrix with a dimension equal to the dimension of the state space, which increases the complexity of the calculations, since it requires a quadratically large memory. For example, for the dimension of the main space equal to a billion, the memory required to solve the basic quantum equation will be about a quintillion, which is a problem even for supercomputers. Whereas a billion-long column fits easily into the memory of a personal computer and can be easily processed on a personal laptop. The ersatz of the quantum master equation, which we are constructing, cannot accurately describe the dynamics of the density matrix and therefore cannot serve as an exact replacement for the quantum master equation. Our ersatz will describe a special process of exchange with the environment.

Resolving mean-field solutions of dissipative phase transitions using permutational symmetry

Phase transitions in dissipative quantum systems have been investigated using various analytical approaches, particularly in the mean-field (MF) limit. However, analytical results often depend on specific methodologies. For instance, the Keldysh formalism shows that the dissipative transverse Ising (DTI) model exhibits a discontinuous transition at the upper critical dimension, dc=3, whereas the fluctuationless MF approach predicts a continuous transition in infinite dimensions (d∞). These two solutions cannot be reconciled, because the MF solutions above dc should be the same. Thus numerical check is required. However, numerical studies in large systems may not be feasible owing to the exponential increase in computational complexity as O(22N) with system size N. Here, we notice that because spins can be regarded as being fully connected at d∞, the spin indices can be permutation invariant, and then the number of quantum states can be considerably contracted as O(N3). The Lindblad equation is transformed into the dynamic equation based on the contracted states. Applying the Runge–Kutta algorithm to the dynamic equation, we obtain all critical exponents including the dynamic exponent as z≈0.5. Moreover, using the property that the DTI model has Z2 symmetry and thus the hyperscaling relation has the form 2β+γ=ν(dc+z), we obtain the relation dc+z=4 in the MF limit. Hence, dc≈3.5 and thus the discontinuous transition in d=3 cannot be treated as an MF solution. We conclude that the permutation invariance at d∞ can be used effectively for checking the validity of an analytic MF solution in quantum phase transitions.

Engineering an imaginary Stark ladder in a dissipative lattice: Passive PT symmetry, K symmetry, and localized damping

We study an imaginary stark ladder model and propose a realization of the model in a dissipative chain with linearly increasing site-dependent dissipation strength. Due to the existence of a $K$-symmetry and passive $\mathcal{PT}$ symmetry, the model exhibits quite different feature from its Hermitian counterpart. With the increase of dissipation strength, the system first undergoes a passive $\mathcal{PT}$-symmetry breaking transition, with the shifted eigenvalues changing from real to complex, and then a $K$-symmetry restoring transition, characterized by the emergence of pure imaginary spectrum with equal spacing. Accordingly, the eigenstates change from $\mathcal{PT}$-unbroken extended states to the $\mathcal{PT}$-broken states, and finally to stark localized states. In the framework of the quantum open system governed by Lindblad equation with linearly increasing site-dependent dissipation, we unveil that the dynamical evolution of single particle correlation function is governed by the Hamiltonian of the imaginary stark ladder model. By studying the dynamical evolution of the density distribution under various initial states, we demonstrate that the damping dynamics displays distinct behaviors in different regions. A localized damping is observed in the strong dissipation limit.

Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics

Density Matrix Exponentiation is a technique for simulating Hamiltonian dynamics when the Hamiltonian to be simulated is available as a quantum state. In this paper, we present a natural analogue to this technique, for simulating Markovian dynamics governed by the well known Lindblad master equation. For this purpose, we first propose an input model in which a Lindblad operator [Formula: see text] is encoded into a quantum state [Formula: see text]. Then, given access to [Formula: see text] copies of the state [Formula: see text], the task is to simulate the corresponding Markovian dynamics for time [Formula: see text]. We propose a quantum algorithm for this task, called Wave Matrix Lindbladization, and we also investigate its sample complexity. We show that our algorithm uses [Formula: see text] samples of [Formula: see text] to achieve the target dynamics, with an approximation error of [Formula: see text].

Random Lindblad Operators Obeying Detailed Balance

We introduce different ensembles of random Lindblad operators [Formula: see text], which satisfy quantum detailed balance condition with respect to given stationary state [Formula: see text] of size [Formula: see text], and investigate their spectral properties. Such operators are known as ‘Davies generators’ and their eigenvalues are real; however, their spectral densities depend on [Formula: see text]. We propose different structured ensembles of random matrices, which allow us to tackle the problem analytically in the extreme cases of Davies generators corresponding to random [Formula: see text] with a nondegenerate spectrum or the maximally mixed stationary state, [Formula: see text]. Interestingly, in the latter case the density can be reasonably well approximated by integrating out the imaginary component of the spectral density characteristic to the ensemble of random unconstrained Lindblad operators. The case of asymptotic states with partially degenerated spectra is also addressed. Finally, we demonstrate that similar universal properties hold for the detailed balance-obeying Kolmogorov generators obtained by applying superdecoherence to an ensemble of random Davies generators. In this way we construct an ensemble of random classical generators with imposed detailed balance condition.

Exceptional Points and Exponential Sensitivity for Periodically Driven Lindblad Equations

In this contribution to the memorial issue of Göran Lindblad, we investigate the periodically driven Lindblad equation for a two-level system. We analyze the system using both adiabatic diagonalization and numerical simulations of the time-evolution, as well as Floquet theory. Adiabatic diagonalization reveals the presence of exceptional points in the system, which depend on the system parameters. We show how the presence of these exceptional points affects the system evolution, leading to a rapid dephasing at these points and a staircase-like loss of coherence. This phenomenon can be experimentally observed by measuring, for example, the population inversion. We also observe that the presence of exceptional points seems to be related to which underlying Lie algebra the system supports. In the Floquet analysis, we map the time-dependent Liouvillian to a non-Hermitian Floquet Hamiltonian and analyze its spectrum. For weak decay rates, we find a Wannier-Stark ladder spectrum accompanied by corresponding Stark-localized eigenstates. For larger decay rates, the ladders begin to dissolve, and new, less localized states emerge. Additionally, their eigenvalues are exponentially sensitive to perturbations, similar to the skin effect found in certain non-Hermitian Hamiltonians.

Simulating Open Quantum System Dynamics on NISQ Computers with Generalized Quantum Master Equations.

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.

Exact solution for the Lindbladian dynamics for the open XX spin chain with boundary dissipation

We obtain exact formulas for the time-dependence of a few physical observables for the open XX spin chain with Lindbladian dynamics. Our analysis is based on the fact that the Lindblad equation for an arbitrary open quadratic system of NN fermions is explicitly solved in terms of diagonalization of a 4N\times4N4N×4N matrix called structure matrix by following the scheme of the third quantization. We mainly focus on the time-dependence of magnetization and spin current. As a short-time behavior at a given site, we observe the plateau regime except near the center of the chain. Basic features of this are explained by the light-cone structure created by propagations of boundary effects from the initial time, but we can explain their more detailed properties analytically using our exact formulas. On the other hand, after the plateau regime, the magnetization and spin current exhibit a slow decay to the steady state values described by the Liouvillian gap. We analytically establish its O(N^{-3})O(N−3) scaling and also determine its coefficient.

Site recurrence for continuous-time open quantum walks on the line

In recent years, several properties and recurrence criteria of discrete-time open quantum walks (OQWs) have been presented. Recently, Pellegrini introduced continuous-time open quantum walks (CTOQWs) as continuous-time natural limits of discrete-time OQWs. In this work, we study semifinite CTOQWs and some of their basic properties concerning statistics, such as transition probabilities and site recurrence. The notion of SJK-recurrence for CTOQWs is introduced, and it is shown to be equivalent to the traditional concept of recurrence. This statistic arises from the definition of $\delta$-skeleton of CTOQWs, which is a dynamic that allows us to obtain a discrete-time OQW in terms of a CTOQW. We present a complete criterion for site recurrence in the case of CTOQW induced by a coin of finite dimension with a set of vertices $\mathbb{Z}$ such that its auxiliary Lindblad operator has a single stationary state. Finally, we present a similar criterion that completes the case in which the internal degree of freedom of each site is of dimension 2.