Optimal control of two-level quantum systems

We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing an energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices of dimension two, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. This includes, in particular, a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed     

两能级封闭量子系统任意量子态的最优制备

在分析单量子位的Bloch球面表示的基础上,结合量子门实现量子态幺正演化的量子态调控机制,提出一种针对两能级封闭量子系统任意量子态的最优制备策略．该策略首先建立两能级量子系统及其控制场的模型;然后借助李群李代数．由经典最优控制的思想和约化动力学来获得最优控制,从而达到两能级封闭量子系统任意量子态的最优制备．理论分析与仿真实验表明了该策略的优越性．     

Real-time optimization of energy consumption under adaptive cruise control for connected HEVs

This paper presents a real-time energy optimization algorithm for a hybrid electric vehicle (HEV) that operates with adaptive cruise control (ACC). Real-time energy optimization is an essential issue such that the HEV powertrain system is as efficient as possible. With connected vehicle technique, ACC system shows considerable potential of high energy efficiency. Combining a classical ACC algorithm, a two-level cooperative control scheme is constructed to realize real-time power distribution for the host HEV that operates in a vehicle platoon. The proposed control strategy actually provides a solution for an optimal control problem with multi objectives in terms of string stable of vehicle platoon and energy consumption minimization of the individual following vehicle. The string stability and the real-time optimization performance of the cooperative control system are confirmed by simulations with respect to several operating scenarios.     

Problem of optimal control in the system of semiconductor quantum points

For the operation of spin information swapping in the system of two vertically superposed layers of tunnel-coupled semiconductor quantum points, a problem of optimization of the time function of control voltage pulse was formulated. The corresponding problem of optimal control with terminal constraints was stated and solved.     

基于相干控制的二能级量子系统退相干抑制

对于二能级开放量子系统,研究了利用相干控制抑制退相干效应的问题.首先讨论了二能级开放量子系统在相干控制下的建模问题,将退相干抑制归结为与环境噪声解耦的控制问题.然后,引入开环控制抑制退相干,并证明该控制可使系统状态中的部分分量与环境噪声渐近解耦.最后引入反馈控制,使得系统状态的相应分量可以与环境精确解耦,同时能够避免测量引入的量子噪声的影响.     

Moment quantization of inhomogeneous spin ensembles

Robust excitation of a large spin ensemble is a long-standing problem in the field of quantum information science and engineering and presents a grand challenge in quantum control. A formal theoretical treatment of this task is to formulate it as an ensemble control problem defined on an infinite-dimensional space. In this paper, we present a distinct perspective to understand and control quantum ensemble systems. Instead of directly analyzing spin ensemble systems defined on a Hilbert space, we transform them to a space where the systems have reduced dimensions with distinctive network structures through the introduction of moment representations. In particular, we illustrate the idea of moment quantization for a spin ensemble and illuminate how this technique leads to a dynamically equivalent control system of moments. This equivalence enables the control of spin ensembles through the control of their moment systems, which in turn creates a new control analysis and design paradigm for quantum ensemble systems based on the use of truncated moment systems.     

The optimal control problem on SO(4) and its applications toquantum control

We consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU(4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4×4 proper orthogonal matrices, SO(4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one     

Multilevel Training of Binary Morphological Operators

The design of binary morphological operators that are translation-invariant and locally defined by a finite neighborhood window corresponds to the problem of designing Boolean functions. As in any supervised classification problem, morphological operators designed from training sample also suffer from overfitting. Large neighborhood tends to lead to performance degradation of the designed operator. This work proposes a multi-level design approach to deal with the issue of designing large neighborhood based operators. The main idea is inspired from stacked generalization (a multi-level classifier design approach) and consists in, at each training level, combining the outcomes of the previous level operators. The final operator is a multi-level operator that ultimately depends on a larger neighborhood than of the individual operators that have been combined. Experimental results show that two-level operators obtained by combining operators designed on subwindows of a large window consistently outperforms the single-level operators designed on the full window. They also show that iterating two-level operators is an effective multi-level approach to obtain better results.     

Optimal Population Transfers in a Quantum System for Large Transfer Time

Transferring a quantum system to a final state with given populations is an important problem with applications to quantum chemistry and atomic physics. In this paper, we consider such transfers that minimize L² the norm of the control. This problem is challenging, both analytically and numerically. With the exception of the simplest cases, there is no general understanding of the nature of optimal controls and trajectories. We find that, by examining the limit of large transfer times, we can uncover such general properties. In particular, for transfer times large with respect to the time scale of the free dynamics of the quantum system, the optimal control is a sum of terms, each being a Bohr frequency sinusoid modulated by a slow amplitude, i.e., a profile that changes considerably only on the scale of the transfer time. Moreover, we show that the optimal trajectory follows a ldquomeanrdquo evolution modulated by the fast free dynamics of the system. The calculation of the ldquomeanrdquo optimal trajectory and the slow control profiles is done via an ldquoaveragedrdquo two-point boundary value problem that we derive and which is much easier to solve than the one expressing the necessary conditions for optimality of the original optimal transfer problem.     

A decomposition technique for an optimal control problem with "PQDZ" cost and bounded controls

In this note we present a solution method for a discrete-time linear optimal control problem where the controls are bounded and both the states and controls have asymmetric costs with dead zones containing the nominal values. This model generalizes the well-known optimal control model for a linear system with symmetric (quadratic) objective functional. We transform the problem into an equivalent large quadratic programming problem with equality and inequality, constraints and decompose it intoNsimpler subproblems, each with very easily computed optimal solutions. Using a two-level approach suggested by Lasdon and Schoeffler [11] the optimal solution to the overall problem is obtained.     

On one class of optimal control problems for quantum systems

We present a new modification of the global control improvement method based on a known Krotov’s method for optimal control in quantum systems from a certain class. The algorithm is implemented for high-dimensional systems as a parallel program. We give computations for the control in a quantum dynamical system that represents a well-known model of communicating the quantum state in spin chains.     

Decoherence effects in the quantum qubit flip game using Markovian approximation

We are considering a quantum version of the penny flip game, whose implementation is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. We focus our attention on the phase damping, amplitude damping and amplitude raising channels. Our results show that the Pauli strategy is no longer a Nash equilibrium under decoherence. We attempt to optimize the players’ control pulses in the aforementioned setup to allow them to achieve higher probability of winning the game compared with the Pauli strategy.     

Decentralized,stochastic control for a static LQ problem

An original problem statement for the optimal control laws design is presented. A large scale system composed of M linear static subsystems with an interaction and quadratic performance index is considered. A two-level hierarchical control structure is assumed, in which a coordinator and local controllers have access to different information. The so-called elastic constraint (Gessing 1985) is used for coordination. For the problem the possibility of partial decomposition of calculations and a decentralization of the control, as well as an analytical form of the optimal laws are obtained. The influence of the particular subsystem on the control quality is investigated.     

Fast quantum codes based on Pauli block jacket matrices

Jacket matrices motivated by the center weight Hadamard matrices have played an important role in signal processing, communications, image compression, cryptography, etc. In this paper, we suggest a design approach for the Pauli block jacket matrix achieved by substituting some Pauli matrices for all elements of common matrices. Since, the well-known Pauli matrices have been widely utilized for quantum information processing, the large-order Pauli block jacket matrix that contains commutative row operations are investigated in detail. After that some special Abelian groups are elegantly generated from any independent rows of the yielded Pauli block jacket matrix. Finally, we show how the Pauli block jacket matrix can simplify the coding theory of quantum error-correction. The quantum codes we provide do not require the dual-containing constraint necessary for the standard quantum error-correction codes, thus allowing us to construct quantum codes of the large codeword length. The proposed codes can be constructed structurally by using the stabilizer formalism of Abelian groups whose generators are selected from the row operations of the Pauli block jacket matrix, and hence have advantages of being fast constructed with the asymptotically good behaviors.     

基于双线性模型的动态系统优化和参数估计集成方法

针对双线性模型与实际系统之间的差异，提出一种基于双线性模型求解非线性动态系统最优控制的迭代算法。该算法通过重复求解修正的基于双线性模型的优化控制问题和参数估计问题，获得实际系统的最优解。同时提出求解修正的基于双线性模型的优化控制问题的一种新的分解方法，克服了非线性和双线性两点边值问题求解的困难。仿真例子表明该算法的有效性和实用性。     

基于恒定磁场的电子自旋量子比特系统任意量子态的最优制备

借助单量子位的Bloch球面表示,结合量子门实现量子态幺正演化的量子态调控机制,以恒定磁场为控制场,引入开关控制思想,提出了一种针对电子自旋量子系统任意量了态的最优制备策略.建立了量子系统及控制场的模型,并借助李群李代数,由经典最优控制的思想,获得任意量子态的最优制备.理论分析与仿真实验说明了该策略的优越性.     

Dynamical behavior of quantum correlations between two qubits coupled to an external environment

We investigate the dynamics of quantum correlations of a two-qubit system coupled to an external environment. We have considered both cases: a spin environment and a bosonic environment. In all cases, we have chosen the Bell-diagonal state as the initial state and computed the evolution of quantum correlations in terms of entanglement, quantum discord and trace distance geometric quantum discord. Special attention is paid to the singular quantum phenomena, such as entanglement sudden death, sudden transition and double sudden transitions from classical to quantum decoherence, which all depend on the initial state and the parameters related to the system and the environment. We find the trace distance geometric quantum discord has a good robustness in resisting the spin and bosonic environmental noise.     

Optimal maintenance policy incorporating system level and unit level for mechanical systems

The study works on a multi-level maintenance policy combining system level and unit level under soft and hard failure modes. The system experiences system-level preventive maintenance (SLPM) when the conditional reliability of entire system exceeds SLPM threshold, and also undergoes a two-level maintenance for each single unit, which is initiated when a single unit exceeds its preventive maintenance (PM) threshold, and the other is performed simultaneously the moment when any unit is going for maintenance. The units experience both periodic inspections and aperiodic inspections provided by failures of hard-type units. To model the practical situations, two types of economic dependence have been taken into account, which are set-up cost dependence and maintenance expertise dependence due to the same technology and tool/equipment can be utilised. The optimisation problem is formulated and solved in a semi-Markov decision process framework. The objective is to find the optimal system-level threshold and unit-level thresholds by minimising the long-run expected average cost per unit time. A formula for the mean residual life is derived for the proposed multi-level maintenance policy. The method is illustrated by a real case study of feed subsystem from a boring machine, and a comparison with other policies demonstrates the effectiveness of our approach.     

Active optimal vibration control for a lumped system

A dynamic absorber with active torque generating DC motors has been studied as an actively suppressing vibration system. A typical two-level spring-lumped mass system with slider undergoing internal excitation vibrations has been considered. Both the main mass and the secondary absorber mass have DC servo motors planned to suppress the amplitude of vibration. State variable techniques are used to formulate the complete system and three optimal control schemes are used to control such a system. First, the discrete time optimal disturbance rejection control for a system with known disturbance is considered, in the second approach a digital optimal control for a linear quadratic regulator problem is studied and in the last approach a discrete optimal control with selected system characteristic matrix is investigated. The response of the system obtained from these optimal control approaches are compared with the response without control action first. The performance of these optimal control approaches are also compared and the availability of implementations of them are evaluated.     

Control of Quantum Langevin Equations

The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply the recently discovered closed form of the unitarity conditions for stochastic evolutions driven by the square of white noise [9] to solve this problem in the case of quadratic cost functionals (cf. (5.5) below). The optimal control is explicitly given in terms of the solution of an operator Riccati equation. Under general conditions on the system Hamiltonian part of the stochastic evolution and on the system observable to be controlled, this equation admits solutions with the required properties and they can be explicitly described.