non-Markovian Quantum Systems Research Articles

Unifying methods for optimal control in non-Markovian quantum systems via process tensors.

The large dimensionality of environments is the limiting factor in applying optimal control to open quantum systems beyond the Markovian approximation. Various methods exist to simulate non-Markovian systems, which effectively reduce the environment to a number of active degrees of freedom. Here, we show that several of these methods can be expressed in terms of a process tensor in the form of a matrix-product-operator, which serves as a unifying framework to show how they can be used in optimal control and to compare their performance. The matrix-product-operator form provides a general scheme for computing gradients using back propagation and allows the efficiency of the different methods to be compared via the bond dimensions of their respective process tensors.

Quantum approximate optimization algorithm in non-Markovian quantum systems

Although quantum approximate optimization algorithm (QAOA) has demonstrated its quantum supremacy, its performance on Noisy Intermediate-Scale Quantum (NISQ) devices would be influenced by complicated noises, e.g. quantum colored noises. To evaluate the performance of QAOA under these noises, this paper presents a framework for running QAOA on non-Markovian quantum systems which are represented by an augmented system model. In this model, a non-Markovian environment carrying quantum colored noises is modelled as an ancillary system driven by quantum white noises which is directly coupled to the corresponding principal system; i.e. the computational unit for the algorithm. With this model, we mathematically formulate QAOA as piecewise Hamiltonian control of the augmented system, where we also optimize the control depth to fit into the circuit depth of current quantum devices. For efficient simulation of QAOA in non-Markovian quantum systems, a boosted algorithm using quantum trajectory is further presented. Finally, we show that non-Markovianity can be utilized as a quantum resource to achieve a relatively good performance of QAOA, which is characterized by our proposed exploration rate.

Analysis of non-Markovian passive quantum linear systems’ response to single-photon input fields*

In this paper, we analyze the response of passive quantum linear systems in a non-Markovian environment to single photon input fields. Based on an augmented modelling method for non-Markovian quantum systems, analytic forms of output intensity and output covariance function in steady-state are derived. These results allow us to quantitatively analyze the influence of non-Markovian environment on system response under a framework of linear system theory.

Nonlinear Optimal Feedback Control of the Two-Level Open Non-Markovian Stochastic Quantum System

In this paper, a nonlinear optimal feedback control based on the nonlinear state estimator for a two-level open non-Markovian stochastic quantum system is proposed. The proposed nonlinear state estimator is designed by using the state-dependent differential Riccati equation and constructed to optimally estimate the state. The estimated state is updated by the output data of continuous weak measurement of the controlled quantum system and used to design the nonlinear state feedback controller to track the output of the reference model. The output state of reference model is chosen as the desired performance. The numerical simulation results verify the achievability of the proposed state feedback control strategy, and the capability to steer the state of system from any arbitrary initial state to the final reference target state with high state transfer success rate.

Inverse-system method for identification of damping-rate functions in non-Markovian quantum systems

Identification of complicated quantum environments lies in the core of quantum engineering, which systematically constructs an environment model with the aim of accurate control of quantum systems. In this paper, we present an inverse-system method to identify damping rate functions which describe non-Markovian environments in time-convolution-less master equations. To access information on the environment, we couple a finite-level quantum system to the environment and measure time traces of local observables of the system. By using sufficient measurement results, an algorithm is designed, which can simultaneously estimate multiple damping rate functions for different dissipative channels. Further, we show that identifiability for the damping rate functions corresponds to the invertibility of the system and a necessary condition for identifiability is also given. The effectiveness of our method is shown in examples of an atom and three-spin-chain non-Markovian systems.

Modeling for Non-Markovian Quantum Systems

This paper presents an augmented Markovian system model for non-Markovian quantum systems. In this augmented system model, ancillary systems are introduced to play the role of internal modes of the non-Markovian environment converting white noise to colored noise. Consequently, non-Markovian dynamics are represented as resulting from direct interaction of the principal system with the ancillary system. To demonstrate the utility of the proposed augmented system model, it is applied to design whitening quantum filters for non-Markovian quantum systems. Examples are presented to illustrate how whitening quantum filters can be utilized for estimating non-Markovian linear quantum systems and qubit systems. In particular, we showed that the augmented Markovian formulation can be used to theoretically model the environment for an observed non-Markovian behavior in a recent experiment on quantum dots [ Phys. Rev. Lett., 108:046807, 2012]

Identification of Non-Markovian Environments for Spin Chains

Correlations of an environment are crucial for the dynamics of non-Markovian quantum systems, which may not be known in advance. In this brief, we propose a gradient algorithm for identifying the correlations in terms of the time-varying damping rate functions in a time-convolution-less master equation for spin chains. By measuring time trace observables of the system, the identification procedure can be formulated as an optimization problem. The gradient algorithm is designed based on a calculation of the derivative of an objective function with respect to the damping rate functions, whose effectiveness is shown in a comparison with a differential approach for a two-qubit spin chain.

Unconventional single-photon blockade in non-Markovian systems

The photon blockade based on weak nonlinearities is called unconventional photon blockade. The studies of the unconventional photon blockade in recent decades are all limited to Markovian approximation. In this paper, we propose a scheme to achieve a strong photon antibunching in non-Markovian reservoir. An exact non-Markovian master equation for the optical cavity coupled with a degenerate optical parametric amplifier (DOPA) is derived. We show that this unconventional photon blockade originates from the destructive interference between two paths which are formed by the DOPA pump and a driving field. Unconventional single-photon blockades that exhibit a transition from the non-Markovian to Markovian regime by controlling environmental spectral width regardless of the weakness of DOPA gain and driving field are found. We analytically derive an optimal condition for strong antibunching, which is in good agreement with that obtained by the full numerical simulation. The presented results then are compared with that given by Markovian approximation in the weak-coupling limit. Our scheme might have paved an avenue towards the investigations of unconventional photon blockades in many other non-Markovian quantum systems in quantum optics.

Measurement of quantum correlation by Frobenius norm in non-Markovian system

In order to measure the quantum correlation of a bipartite state quickly, an easy method is to construct a test matrix through the commutations among the blocks of its density matrix. Then, the Frobenius norm of the test matrix can be used to measure the quantum correlation. In this paper, we apply the measurement by Frobenius norm ([Formula: see text] to the dynamics evolution of the non-Markovian quantum system and compare it with the typical quantum discord ([Formula: see text] proposed by Ollivier and Zurek. The research results show that [Formula: see text] can indeed measure the quantum correlation of a bipartite state as same as [Formula: see text]. Further studies find that there are still differences between the two measurements: in some regions, when [Formula: see text] is zero, [Formula: see text] is not zero. It indicates that [Formula: see text] is more detailed than [Formula: see text] to measure quantum correlation of a bipartite state.

Feedback Tracking Control of Non-Markovian Quantum Systems

In this paper, we present a feedback tracking control strategy to reject colored noise in linear non-Markovian quantum systems. This non-Markovian system is modeled as an augmented linear Markovian system, which includes a principal system to be controlled and multiancillary systems representing the effect of the internal modes of the non-Markovian environment. The colored noise generated by the multiancillary systems disturbs the principal system through a direct interaction. Based on this model, a whitening quantum filter can be obtained for estimating the dynamical variables of both the principal and multiancillary systems, which can be used for feedback control. To reject the colored noise, a linear quadratic Gaussian controller is designed to drive the dynamical variables of the principal system to track those of a reference system. In examples, we show that colored noise with both rational and nonrational spectra in linear non-Markovian quantum systems can be efficiently rejected using our tracking control strategy.

Orbit Tracking of Quantum State for Non-Markovian Quantum System Based on Model Reference Adaptive Control

Utilizing model reference adaptive control theory and Lyapunov stability theorem, we derive the adaptive law for the model reference adaptive system. Then we design the Lyapunov control law by double control functions and investigate the orbit tracking of quantum state for non-Markovian quantum system with phase relaxation and energy dissipative relaxation. The influence of Ohmic reservoir with Lorentz-Drude regularization is numerically studied for a two-level system. The simulations show that the controlled quantum system will track the target orbit with a small oscillation due to the non-Markovian environmental memory effect, which indicates the orbit tracking of non-Markovian quantum system is incomplete.

Rapid state transfer for a two-level non-Markovian quantum system

In quantum information processing, rapid control is a basic requirement for performance improvement because a realistic quantum system cannot be perfectly separated from its environment, which will cause relaxation or decoherence effect. Rapid control may make the control law more robust to uncertainties in the model or in the control process. From the time optimization standpoint, bang-bang control is an excellent method for engineering practice. However, quantum bang-bang control relies on quantum Zeno effect, quantum anti-Zeno effect will occur under low measurement frequency, which intensifies the decoherence and leads to high-frequency oscillation phenomenon. In order to realize rapid state transfer and avoid high-frequency oscillation with an infinitesimal period in bang-bang Lyapunov control, we design control fields to realize arbitrary state transfer for non-Markovian system with phase relaxation and energy dissipative relaxation by Lyapunov stability theory. The numerical simulations illustrate that arbitrary state (eigenstate, superposition state or mixed state) transfer and maintenance for non-Markovian system can be realized under the approximate bang-bang Lyapunov control.

Lyapunov-based state transfer and maintenance for non-Markovian quantum system

One main problem of quantum control in quantum systems is the manipulation of states including state transfer and state tracking. We design control fields to realize arbitrary state transfer for open quantum system by Lyapunov stability theory, and investigate the state transfer for non-Markovian system with phase relaxation and energy dissipative relaxation. The numerical simulations illustrate that arbitrary state (eigenstate, superposition state or mixed state) transfer and maintenance for non-Markovian system can be realized under Lyapunov control function by an external steady control field of proper amplitude, with a success rate of more than 98%. When the right control field and function are implemented, not only the decoherence is compensated completely but also the purity of quantum states is maintained in the process of state transfer.

Optimal control-based states transfer for non-Markovian quantum system

Utilizing the method of optimal control, we investigate the tactics of state transfer in the non-Markovian quantum system with phase relaxation and energy dissipative relaxation. The influence of Ohmic reservoir with Lorentz–Drude regularization is numerically studied. Owing to the decoherence and memory effects of non-Markovian channel, the purity of quantum state attenuates damply in the free evolution. The numerical simulations indicate that arbitrary state transfer for non-Markovian system can be realized under the optimal control function by a proper external control field with a success rate of more than 98 percent. When the right control field and function is implemented, not only the decoherence is compensated completely but also the purity of quantum states are maintained in the process of state transfer.

Preparation of Quantum Gates for Open Quantum Systems by Lyapunov Control Method

In this paper the control laws of preparing quantum gates are designed based on Lyapunov stability theorem for two level open quantum systems. We propose a novel Lyapunov function according to the matrix logarithm function, which has higher accuracy and faster convergence speed by comparing them with those of the Lyapunov function of distance. Based on the proposed function, we design two types of control laws to prepare quantum gates for different systems including Markovian quantum systems with phase damping and amplitude damping, non-Markovian quantum systems and closed quantum systems. Furthermore, the system robustness when the Hamiltonian contains uncertainty is further investigated. In order to verify the superiorities of proposed function and control method, NOT gates are prepared by the designed control laws for different systems in the numerical experiments, and the results are comparatively analyzed.

Realizing the dynamics of a non-Markovian quantum system by Markovian coupled oscillators: a Green’s function-based root locus approach

In this paper, we develop a Green's function-based root locus approach to realizing a Lorentzian-noise-disturbed non-Markovian quantum system by Markovian coupled oscillators in an extended Hilbert space. By using a Green's function-based root locus method, we design an ancillary oscillator for Markovian coupled oscillators to be a Lorentzian noise generator. Thus a principal oscillator coupled to the ancillary oscillator via a direct interaction can capture the dynamics of a Lorentzian-noise-disturbed non-Markovian quantum system. By matching the root locus in the frequency domain, conditions for the realization are obtained and a critical transition in the non-Markovian quantum system can also be observed in the Markovian coupled oscillators.