Control of Quantum Systems

Shuang Cong University of Science and Technology of China

• About the Author xiii Preface xv1 Introduction 11.1 Quantum States 21.2 Quantum Systems Control Models 31.2.1 Schrödinger Equation 41.2.2 Liouville Equation 41.2.3 Markovian Master Equations 51.2.4 Non-Markovian Master Equations 51.3 Structures of Quantum Control Systems 61.4 Control Tasks and Objectives 81.5 System Characteristics Analyses 91.5.1 Controllability 91.5.2 Reachability 91.5.3 Observability 101.5.4 Stability 101.5.5 Convergence 101.5.6 Robustness 101.6 Performance of Control Systems 111.6.1 Probability 111.6.2 Fidelity 111.6.3 Purity 121.7 Quantum Systems Control 131.7.1 Description of Control Problems 131.7.2 Quantum Control Theory and Methods 131.8 Overview of the Book 16References 182 State Transfer and Analysis of Quantum Systems on the Bloch Sphere 212.1 Analysis of a Two-level Quantum System State 212.1.1 Pure State Expression on the Bloch Sphere 212.1.2 Mixed States in the Bloch Sphere 242.1.3 Control Trajectory on the Bloch Sphere 262.2 State Transfer of Quantum Systems on the Bloch Sphere 272.2.1 Control of a Single Spin-1/2 Particle 282.2.2 Situation with the Minimum Ωt of Control Fields 302.2.3 Situation with a Fixed Time T 312.2.4 Numerical Simulations and Results Analyses 33References 373 Control Methods of Closed Quantum Systems 393.1 Improved Optimal Control Strategies Applied in Quantum Systems 393.1.1 Optimal Control of Quantum Systems 403.1.2 Improved Quantum Optimal Control Method 423.1.3 Krotov-Based Method of Optimal Control 433.1.4 Numerical Simulation and Performance Analysis 453.2 Control Design of High-Dimensional Spin-1/2 Quantum Systems 483.2.1 Coherent Population Transfer Approaches 483.2.2 Relationships between the Hamiltonian of Spin-1/2 Quantum Systems under Control and the Sequence of Pulses 493.2.3 Design of the Control Sequence of Pulses 523.2.4 Simulation Experiments of Population Transfer 533.3 Comparison of Time Optimal Control for Two-Level Quantum Systems 573.3.1 Description of System Model 583.3.2 Geometric Control 593.3.3 Bang-Bang Control 613.3.4 Time Comparisons of Two Control Strategies 643.3.5 Numerical Simulation Experiments and Results Analyses 66References 714 Manipulation of Eigenstates – Based on Lyapunov Method 734.1 Principle of the Lyapunov Stability Theorem 744.2 Quantum Control Strategy Based on State Distance 754.2.1 Selection of the Lyapunov Function 764.2.2 Design of the Feedback Control Law 774.2.3 Analysis and Proof of the Stability 784.2.4 Application to a Spin-1/2 Particle System 804.3 Optimal Quantum Control Based on the Lyapunov Stability Theorem 814.3.1 Description of the System Model 824.3.2 Optimal Control Law Design and Property Analysis 844.3.3 Simulation Experiments and the Results Comparisons 864.4 Realization of the Quantum Hadamard Gate Based on the Lyapunov Method 884.4.1 Mathematical Model 894.4.2 Realization of the Quantum Hadamard Gate 904.4.3 Design of Control Fields 924.4.4 Numerical Simulations and Comparison Results Analyses 94References 965 Population Control Based on the Lyapunov Method 995.1 Population Control of Equilibrium State 995.1.1 Preliminary Notions 995.1.2 Control Laws Design 1005.1.3 Analysis of the Largest Invariant Set 1015.1.4 Considerations on the Determination of P 1045.1.5 Illustrative Example 1055.1.6 Appendix: Proof of Theorem 5.1 1075.2 Generalized Control of Quantum Systems in the Frame of Vector Treatment 1105.2.1 Design of Control Law 1105.2.2 Convergence Analysis 1135.2.3 Numerical Simulation on a Spin-1/2 System 1145.3 Population Control of Eigenstates 1175.3.1 System Model and Control Laws 1175.3.2 Largest Invariant Set of Control Systems 1185.3.3 Analysis of the Eigenstate Control 1185.3.4 Simulation Experiments 119References 1236 Quantum General State Control Based on Lyapunov Method 1256.1 Pure State Manipulation 1256.1.1 Design of Control Law and Discussion 1256.1.2 Control System Simulations and Results Analyses 1296.2 Optimal Control Strategy of the Superposition State 1316.2.1 Preliminary Knowledge 1326.2.2 Control Law Design 1336.2.3 Numerical Simulations 1346.3 Optimal Control of Mixed-State Quantum Systems 1356.3.1 Model of the System to be Controlled 1366.3.2 Control Law Design 1376.3.3 Numerical Simulations and Results Analyses 1426.4 Arbitrary Pure State to a Mixed-State Manipulation 1456.4.1 Transfer from an Arbitrary Pure State to an Eigenstate 1466.4.2 Transfer from an Eigenstate to a Mixed State by Interaction Control 1476.4.3 Control Design for a Mixed-State Transfer 1496.4.4 Numerical Simulation Experiments 151References 1547 Convergence Analysis Based on the Lyapunov Stability Theorem 1557.1 Population Control of Quantum States Based on Invariant Subsets with the Diagonal Lyapunov Function 1557.1.1 System Model and Control Design 1557.1.2 Correspondence between any Target Eigenstate and the Value of the Lyapunov Function 1567.1.3 Invariant Set of Control Systems 1577.1.4 Numerical Simulations 1617.1.5 Summary and Discussion 1647.2 A Convergent Control Strategy of Quantum Systems 1657.2.1 Problem Description 1657.2.2 Construction Method of the Observable Operator 1667.2.3 Proof of Convergence 1687.2.4 Route Extension Strategy 1737.2.5 Numerical Simulations 1747.3 Path Programming Control Strategy of Quantum State Transfer 1767.3.1 Control Law Design Based on the Lyapunov Method in the Interaction Picture 1777.3.2 Transition Path Programming Control Strategy 1787.3.3 Numerical Simulations and Results Analyses 182References 1868 Control Theory and Methods in Degenerate Cases 1878.1 Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Error 1878.1.1 Control Design 1888.1.2 Convergence Proof 1928.1.3 Relation between Two Lyapunov Functions 1938.1.4 Numerical Simulation and Result Analysis 1938.2 Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity 1958.2.1 Control Law Design and Convergence Proof 1958.2.2 Numerical Simulation and Result Analysis 1998.3 Implicit Lyapunov Control for the Quantum Liouville Equation 2008.3.1 Description of Problem 2018.3.2 Derivation of Control Laws 2028.3.3 Convergence Analysis 2058.3.4 Numerical Simulations 209References 2119 Manipulation Methods of the General State 2139.1 Quantum System Schmidt Decomposition and its Geometric Analysis 2139.1.1 Schmidt Decomposition of Quantum States 2149.1.2 Definition of Entanglement Degree Based on the Schmidt Decomposition 2159.1.3 Application of the Schmidt Decomposition 2169.2 Preparation of Entanglement States in a Two-Spin System 2209.2.1 Construction of the Two-Spin Systems Model in the Interaction Picture 2209.2.2 Design of the Control Field Based on the Lyapunov Method 2239.2.3 Proof of Convergence for the Bell States 2269.2.4 Numerical Simulations 2279.3 Purification of the Mixed State for Two-Dimensional Systems 2309.3.1 Purification by Means of a Probe 2309.3.2 Purification by Interaction Control 2329.3.3 Numerical Experiments and Results Comparisons 2339.3.4 Discussion 234References 23510 State Control of Open Quantum Systems 23710.1 State Transfer of Open Quantum Systems with a Single Control Field 23710.1.1 Dynamical Model of Open Quantum Systems 23710.1.2 Derivation of Optimal Control Law 23810.1.3 Control System Design 24110.1.4 Numerical Simulations and Results Analyses 24210.2 Purity and Coherence Compensation through the Interaction between Particles 24610.2.1 Method of Compensation for Purity and Coherence 24710.2.2 Analysis of System Evolution 25010.2.3 Numerical Simulations 25310.2.4 Discussion 255Appendix 10.A Proof of Equation 10.59 257References 25811 State Estimation, Measurement, and Control of Quantum Systems 26111.1 State Estimation Methods in Quantum Systems 26111.1.1 Background of State Estimation of Quantum Systems 26211.1.2 Quantum State Estimation Methods Based on the Measurement of Identical Copies 26211.1.3 Quantum State Reconstruction Methods Based on System Theory 26711.2 Entanglement Detection and Measurement of Quantum Systems 26811.2.1 Entanglement States 26911.2.2 Entanglement Witnesses 27111.2.3 Entanglement Measures 27311.2.4 Non-linear Separability Criteria 27711.3 Decoherence Control Based on Weak Measurement 27811.3.1 Construction of a Weak Measurement Operator 27911.3.2 Applicability of Weak Measurement 28011.3.3 Effects on States 282Appendix 11.A Proof of Normed Linear Space (A, ¡¬ • ¡¬) 286References 28712 State Preservation of Open Quantum Systems 29112.1 Coherence Preservation in a Λ-Type Three-Level Atom 29112.1.1 Models and Objectives 29212.1.2 Design of Control Field 29412.1.3 Analysis of Singularities Issues 29712.1.4 Numerical Simulations 29912.2 Purity Preservation of Quantum Systems by a Resonant Field 30112.2.1 Problem Description 30212.2.2 Purity Property Preservation 30312.2.3 Discussion 30612.3 Coherence Preservation in Markovian Open Quantum Systems 30712.3.1 Problem Formulation 30812.3.2 Design of Control Variables 31112.3.3 Numerical Simulations 31312.3.4 Discussion 315Appendix 12.A Derivation of HC 316References 31713 State Manipulation in Decoherence-Free Subspace 32113.1 State Transfer and Coherence Maintainance Based on DFS for a Four-Level Energy Open Quantum System 32113.1.1 Construction of DFS and the Desired Target State 32213.1.2 Design of the Lyapunov-Based Control Law for State Transfer 32513.1.3 Numerical Simulations 32613.2 State Transfer Based on a Decoherence-Free Target State for a Λ-Type N-Level Atomic System 32813.2.1 Construction of the Decoherence-Free Target State 32813.2.2 Design of the Lyapunov-Based Control Law for State Transfer 33113.2.3 Numerical Simulations and Results Analyses 33213.3 Control of Quantum States Based on the Lyapunov Method in Decoherence-Free Subspaces 33613.3.1 Problem Description 33613.3.2 Control Design in the Interaction Picture 33813.3.3 Construction of P and Convergence Analysis 33913.3.4 Numerical Simulation Examples and Discussion 345References 34814 Dynamic Decoupling Quantum Control Methods 35114.1 Phase Decoherence Suppression of an n-Level Atom in Ξ;-Configuration with Bang-Bang Controls 35114.1.1 Dynamical Decoupling Mechanism 35214.1.2 Design of the Bang–Bang Operations Group in Phase Decoherence 35514.1.3 Examples of Design 35714.2 Optimized Dynamical Decoupling in Ξ-Type n-Level Atom 36014.2.1 Periodic Dynamical Decoupling 36114.2.2 Uhrig Dynamical Decoupling 36114.2.3 Behaviors of Quantum Coherence under Various Dynamical Decoupling Schemes 36214.2.4 Examples 36514.2.5 Discussion 36614.3 An Optimized Dynamical Decoupling Strategy to Suppress Decoherence 36614.3.1 Universal Dynamical Decoupling for a Qubit 36714.3.2 An Optimized Dynamical Decoupling Scheme 36914.3.3 Simulation and Comparison 36914.3.4 Discussion 375References 37815 Trajectory Tracking of Quantum Systems 38115.1 Orbit Tracking of Quantum States Based on the Lyapunov Method 38215.1.1 Description of the System Model 38215.1.2 Design of Control Law 38415.1.3 Numerical Simulation Experiments and Results Analysis 38515.2 Orbit Tracking Control of Quantum Systems 38915.2.1 System Model and Control Law Design 39015.2.2 Numerical Simulation Experiments 39115.3 Adaptive Trajectory Tracking of Quantum Systems 39415.3.1 Description of the System Model 39615.3.2 Control System Design and Characteristic Analysis 39815.3.3 Numerical Simulation and Result Analysis 40015.4 Convergence of Orbit Tracking for Quantum Systems 40215.4.1 Description of the Control System Model 40315.4.2 Control Law Derivation 40415.4.3 Convergence Analysis 40415.4.4 Applications and Experimental Results Analyses 411References 416Index 419