Quantum Lyapunov Control Research Articles

Feedback-based quantum algorithm inspired by counterdiabatic driving

In recent quantum algorithmic developments, a feedback-based approach has shown promise for preparing quantum many-body system ground states and solving combinatorial optimization problems. This method utilizes quantum Lyapunov control to iteratively construct quantum circuits. Here, we propose a substantial enhancement by implementing a protocol that uses ideas from quantum Lyapunov control and the counterdiabatic driving protocol, a key concept from quantum adiabaticity. Our approach introduces an additional control field inspired by counterdiabatic driving. We apply our algorithm to prepare ground states in one-dimensional quantum Ising spin chains. Comprehensive simulations demonstrate a remarkable acceleration in population transfer to low-energy states within a significantly reduced time frame compared to conventional feedback-based quantum algorithms. This acceleration translates to a reduced quantum circuit depth, a critical metric for potential quantum computer implementation. We validate our algorithm on the IBM cloud computer, highlighting its efficacy in expediting quantum computations for many-body systems and combinatorial optimization problems. Published by the American Physical Society 2024

Feedback-based quantum algorithms for ground state preparation

The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024

Closed-loop designed open-loop control of quantum systems: An error analysis

Quantum Lyapunov control, an important class of quantum control methods, aims at generating converging dynamics guided by Lyapunov-based theoretical tools. However, unlike the case of classical systems, disturbance caused by quantum measurement hinders direct and exact realization of the theoretical feedback dynamics designed with Lyapunov theory. Regarding this issue, the idea of closed-loop designed open-loop control has been mentioned in literature, which means to design the closed-loop dynamics theoretically, simulate the closed-loop system, generate control pulses based on simulation and apply them to the real plant in an open-loop fashion. Based on bilinear quantum control model, we analyze in this article the error, i.e., difference between the theoretical and real systems’ time-evolved states, incurred by the procedures of closed-loop designed open-loop control. It is proved that the error at an arbitrary time converges to a unitary transformation of initialization error as the number of simulation grids between 0 and that time tends to infinity. Moreover, it is found that once the simulation accuracy reaches a certain level, adopting more accurate (thus possibly more expensive) numerical simulation methods does not efficiently improve convergence. We also present an upper bound on the error norm and an example to illustrate our results.

Quantum Control and Its Application: A Brief Introduction

The development of quantum information science has brought great changes to the fields of computer and communication, and deepened people’s understanding of basic physical problems. The core problems of overcoming decoherence, preparing and manipulating quantum states are quantum control problems in essence, so the research on quantum control will attract more and more attention and be important in the development of quantum information. This paper studies several applications of quantum Lyapunov control and quantum feedback control in quantum information processing by using Schrodinger equation, and some principal equations as tools for finite-dimensional systems. In this paper, a brief introduction of quantum control and its applications is presented to review the recent development and its prospect.

Dissipative preparation of multipartite Greenberger-Horne-Zeilinger states of Rydberg atoms* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11774047 and 12047525).

The multipartite Greenberger–Horne–Zeilinger (GHZ) states play an important role in large-scale quantum information processing. We utilize the polychromatic driving fields and the engineered spontaneous emissions of Rydberg states to dissipatively drive three- and four-partite neutral atom systems into the steady GHZ states, at the presence of the next-nearest neighbor interaction of excited Rydberg states. Furthermore, the introduction of quantum Lyapunov control can help us optimize the dissipative dynamics of the system so as to shorten the convergence time of the target state, improve the robustness against the spontaneous radiations of the excited Rydberg states, and release the limiting condition for the strengths of the polychromatic driving fields. Under the feasible experimental conditions, the fidelities of three- and four-partite GHZ states can be stabilized at 99.24 % and 98.76 %, respectively.

Lyapunov Control of High-Dimensional Closed Quantum Systems Based on Particle Swarm Optimization

For high-dimensional closed quantum systems, this paper proposes a novel quantum Lyapunov control scheme based on the particle swarm optimization algorithm and achieves a high-probability population transfer of the system to a non-isolated target eigenstate in the LaSalle invariant set under usual smooth Lyapunov control laws. Via a quadratic Lyapunov function with unknown parameters, a control law with the unknown parameters is designed; based on the LaSalle invariance principle and the energy-level connectivity graph, the stability of the system is analyzed; by using the particle swarm optimization algorithm, a set of optimal parameters is obtained to achieve the control goal. In particular, we propose a path planning method based on the energy-level connectivity graph to determine the initial values of the unknown parameters, which is such that the optimization algorithm can efficiently and conveniently find a set of optimal solutions of the unknown parameters. Numerical simulation experiments on a five-level quantum system and a three-qubit system demonstrate that the proposed Lyapunov control scheme based on the particle swarm optimization algorithm has a good control effect.

Quantum Lyapunov control with machine learning

Quantum state engineering is a central task in Lyapunov-based quantum control. Given different initial states, better performance may be achieved if the control parameters, such as the Lyapunov function, are individually optimized for each initial state, however, at the expense of computing resources. To tackle this issue, we propose an initial-state-adaptive Lyapunov control strategy with machine learning. Specifically, artificial neural networks are used to learn the relationship between the optimal control parameters and initial states through supervised learning with samples. Two designs are presented where the feedforward neural network and the general regression neural network are used to select control schemes and design Lyapunov functions, respectively. We demonstrate the performance of the designs with a three-level quantum system for an eigenstate control problem. Since the sample generation and the training of neural networks are carried out in advance, the initial-state-adaptive Lyapunov control can be implemented for new initial states without much increase of computational resources.

Influence of localization transition on dynamical properties for an extended Aubry–André–Harper model

We show the localization transition and its influence on two dynamical processes of an extended Aubry–André–Harper model with incommensurate on-site and hopping potentials. After specifying an extended Aubry–André–Harper model, we check the localization transition for all the eigenstates and eigenenergy band splitting behavior versus a system parameter. To check the influence of localization transition on dynamical processes, firstly, the slowly pumping of edge states are examined. The adiabatic pumping is suppressed in the localized region. Then by quantum Lyapunov control method in terms of different control Hamiltonians, we prepare an edge-localized state in the nonlocal region. The control effect is suppressed in the localized region compared to that in the nonlocal region. In the dynamical processes, the system acts as conductor for the excitation in the nonlocal region and insulator in the localized region. Then we adopt the occupation imbalance between even and odd sites and entropy to indicate the localization transition further. Finally, the experimental schemes based on cold atoms trapped in quasiperiodic optical lattice and coupled optical waveguide arrays are suggested.

Edge state preparation in a one-dimensional lattice by quantum Lyapunov control

Quantum Lyapunov control uses a feedback control methodology to determine control fields applied to control quantum systems in an open-loop way. In this work, we employ two Lyapunov control schemes to prepare an edge state for a fermionic chain consisting of cold atoms loaded in an optical lattice. Such a chain can be described by the Harper model. Corresponding to the two schemes, two types of quantum Lyapunov functions are considered. The results show that both the schemes are effective at preparing the edge state within a wide range of parameters. We found that the edge state can be prepared with high fidelity even if there are moderate fluctuations of on-site or hopping potentials. Both control schemes can be extended to similar chains (3m + d, d = 2) of different lengths. Since a regular amplitude control field is easier to apply in practice, an amplitude-modulated control field is used to replace the unmodulated one. Such control approaches provide tools to explore the edge states of one-dimensional topological materials.

Lyapunov control on quantum systems

We review the scheme of quantum Lyapunov control and its applications into quantum systems. After a brief review on the general method of quantum Lyapunov control in closed and open quantum systems, we apply it into controlling quantum states and quantum operations. The control of a spin-1/2 quantum system, driving an open quantum system into its decoherence free subspace (DFS), constructing single qubit and two-qubit logic gates are taken to illustrate the scheme. The optimalization of the Lyapunov control is also reviewed in this article.

Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity

The convergence of closed quantum systems in the degenerate cases to the arbitrary desired target state by using the quantum Lyapunov control based on the average value of an imaginary mechanical quantity is studied in this paper. On the basis of the existing methods which can only ensure the pure states and single-control Hamiltonian systems converge toward a set, we propose a control laws design to make the multi-control Hamiltonian systems, which can also converge from the arbitrary initial state to the arbitrary target state of the quantum systems whose internal Hamiltonian are not strongly regular or/and control Hamiltonians are not full connected. The degenerate cases' problems solved in this paper widely exist in the practical quantum systems, so it has the great significance in quantum systems control. People can make use of those conditions obtained to design a convergent controller for the quantum control system, which can instruct the experimental scientists to obtain high successful probability to the actual quantum systems control. This research work establishes a completed quantum Lyapunov control theory in closed quantum systems. The convergence of the control system is proved. How to make the convergence conditions to be satisfied is proved or analyzed. Finally, numerical simulations for a three level system in the degenrate case transfering from an initial eigenstate to a target superposition state are studied.

Characteristics Analysis and State Transfer for Non-Markovian Open Quantum Systems

The weak-coupled two-level open quantum system described by non-Markovian time-convolution-less master equation is investigated in this paper. The system characteristics with respect to the decay rate β(t), coherence factor Coh and purity p in terms of the cut-off frequency ωc, coupling constant α and transition frequency ω0 are investigated, respectively. The appropriate parameters used in system simulation experiments are determined by analyzing results of different values of parameters for the effects of system performance. The control laws used to transfer the system states are designed on the basis of the Lyapunov stability theorem. Numerical simulation experiments are implemented under the Matlab environment. The features of the free evolution trajectory of the non-Markovian systems and the state transfer from a pure state to a desired pure state under the action of the proposed control laws are studied, respectively. By comparing the experimental results, the effectiveness of the proposed quantum Lyapunov control method applied to the state transfer in non-Markovian open quantum systems is verified. Meanwhile, the influences of different control parameters and cut-off frequencies on the system performance are analyzed.

Optimal Lyapunov-based quantum control for quantum systems

Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.

Time-delay effects and simplified control fields in quantum Lyapunov control

Lyapunov-based quantum control has the advantage that it is free from the measurement-induced decoherence and it includes the instantaneous information of the system in the control. The Lyapunov control is often confronted with time delay in the control fields and difficulty in practical implementations of the control. In this paper, we study the effect of time delay on the Lyapunov control and explore the possibility of replacing the control field with a pulse train or a bang–bang signal. The efficiency of the Lyapunov control is also presented through examining the convergence time of the system. These results suggest that the Lyapunov control is robust against time delay, easy to realize and effective for high-dimensional quantum systems.