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Quantum control theory and applications: A survey
Daoyi Dong, Ian R Petersen
TL;DR
Quantum control theory seeks systematic ways to manipulate quantum systems while addressing controllability, coherence, measurement, noise, and uncertainty. This survey organizes the field from a control-systems perspective, covering open-loop and closed-loop methods, their applications, and selected theoretical results. It concludes that the field has made substantial progress but remains limited by scope-specific controllability conditions and the relatively sparse development of incoherent-control approaches.
Problem
Quantum control must address whether desired quantum states can be reached despite quantum-specific behavior, measurement disturbance, environmental noise, and uncertainty.
Method
The paper surveys controllability, open-loop strategies, closed-loop design, quantum feedback, and applications from a control-systems perspective.
Results
The survey presents developments in controllability, open-loop control strategies, closed-loop design methods, and selected quantum-control applications.
Takeaways & Limitations
Quantum control combines coherent and incoherent resources with open-loop and closed-loop approaches to address tasks including controllability, feedback, robustness, and quantum-state manipulation.
Takeaways & Limitations
Only few results address quantum incoherent control, and the sufficient conditions stated for pure-state controllability are not necessary.
Abstract
from arXiv · showhide
This paper presents a survey on quantum control theory and applications from a control systems perspective. Some of the basic concepts and main developments (including open-loop control and closed-loop control) in quantum control theory are reviewed. In the area of open-loop quantum control, the paper surveys the notion of controllability for quantum systems and presents several control design strategies including optimal control, Lyapunov-based methodologies, variable structure control and quantum incoherent control. In the area of closed-loop quantum control, the paper reviews closed-loop learning control and several important issues related to quantum feedback control including quantum filtering, feedback stabilization, LQG control and robust quantum control.
9 January, 2011
The section identifies the paper’s central quantum-control topics, spanning controllability, control resources, and feedback.
- Quantum control and controllability are identified as core topics.
- The topic list distinguishes coherent control from incoherent control.
- Feedback control and robust control are included among the paper’s themes.
1 Introduction
The introduction frames quantum control as an established but still developing field and surveys its theoretical foundations, open-loop methods, closed-loop methods, and applications from a control-systems perspective.
- Quantum control aims to develop systematic methods for actively manipulating quantum systems despite quantum-specific features such as entanglement and coherence.
- Controllability asks whether a quantum system can be driven to a desired state and connects closely with universal quantum computation and atomic- or molecular-scale transformations.
- Open-loop control: The survey covers coherent control, which manipulates quantum states with semiclassical potentials while preserving coherence.
- Open-loop control: Because measurements can destroy quantum-state information, Lyapunov methods construct an artificial closed-loop controller and derive an open-loop law through simulation.
- Closed-loop control: Open-loop methods face limitations from environmental noise and uncertainty, motivating robust closed-loop control and related quantum-feedback areas.
- Scope: The paper surveys open-loop strategies, closed-loop learning control, quantum feedback, and selected applications to quantum information technology.
2 Prerequisites
The prerequisites introduce the quantum-mechanical concepts needed to formulate quantum control theory, including states, measurements, and control models.
- Quantum control systems have dynamics governed by the laws of quantum mechanics.
- The section briefly introduces quantum states as well as quantum measurements and quantum control models.
- These concepts provide the framework for developing quantum control theory.
2.1 Quantum states
Quantum states are represented differently for closed, open, and composite systems, with pure, mixed, separable, and entangled states distinguished through their mathematical descriptions.
- State representations: A closed quantum system can be represented by a unit vector |ψ⟩ in a complex Hilbert space, also called a wavefunction.
- Pure states: Pure states use unit-vector representations, while physically equivalent vectors may differ by a global phase.
- Qubits: Qubit states encode the classical values zero and one through the basis states |0⟩ and |1⟩.
- Mixed states: Quantum ensembles and open systems are represented by positive, trace-one density operators rather than unit vectors.
- Mixed states: A state is mixed when tr(ρ^2) < 1, whereas a pure state satisfies tr(ρ^2) = 1.
- Composite systems: Composite systems are modeled on tensor-product Hilbert spaces, with product states written using tensor products of subsystem states.
- Entanglement: A bipartite pure state is separable if it factors into subsystem states and entangled otherwise.
2.2 Quantum measurements
Quantum measurements differ from classical measurements because they affect the measured system. The section distinguishes instantaneous projective measurements from continuous measurements used for quantum feedback control.
- Projective measurement: A quantum observable is represented by a Hermitian operator, and projective measurement outcomes are its eigenvalues.For a pure state, outcome m occurs with probability p(m)=⟨ψ|P_m|ψ⟩; for a mixed state, p(m)=tr{P_mρ}.
- Projective measurement: After a projective measurement, the system state changes according to the projector associated with the observed outcome.For a pure state the post-measurement state is P_m|ψ⟩, while for a mixed state it is P_mρP_m divided by p(m).
- Projective measurement: Projective measurements are modeled as instantaneous when measurement strength is sufficiently large and the measurement timescale is much shorter than other task timescales.This approximation may not describe continuous monitoring.
- Continuous measurement: Continuous measurement theory is essential for quantum feedback control because feedback information must be extracted continuously to adjust system evolution.Continuous measurements have been experimentally realized for practical systems such as solid-state qubits.
- Continuous measurement: Under continuous monitoring, the measurement record and system evolution can be described using a stochastic master equation.Continuous measurements can be derived from projective measurements under appropriate assumptions.
2.3 Quantum control models
Quantum control models represent closed and open-system dynamics using bilinear, Markovian master-equation, stochastic master-equation, and linear quantum stochastic differential-equation formulations. These models support state-transfer and feedback descriptions while requiring physical-realizability constraints in the linear stochastic case.
- Model classes: The section introduces four quantum-control model classes: bilinear models, Markovian master equations, stochastic master equations, and linear quantum stochastic differential equations.They cover closed-system, open-system, measurement-driven, and linear quantum stochastic descriptions.
- Bilinear models: Closed-system states evolve according to the Schrödinger equation, while controlled evolution is determined by a Hamiltonian combining free and control-interaction terms.The Hamiltonian is H(t)=H0+Σ_k u_k(t)H_k, with real control functions and Hermitian interaction Hamiltonians.
- Bilinear models: A typical control objective is to choose a final time and admissible controls that drive an initial pure state to a predefined target state.The resulting Hamiltonian defines a unitary propagator for the state transition.
- Bilinear models: Finite-dimensional bilinear models can describe closed quantum systems such as molecular and spin systems, including spin-1/2 systems controlled by magnetic fields.They can be formulated through unitary transformations or coefficient trajectories when controllability holds.
- Open-system models: For open systems with Markovian dynamics, the quantum state can be described by a Markovian master equation rather than generally by a unitary transformation.The Markovian approximation neglects memory effects under a short environmental correlation-time assumption.
- Open-system models: Continuous measurements lead to stochastic master equations whose state is conditional because it incorporates information from the measurement record.Different measurement processes produce different forms of stochastic master equations.
- Linear quantum stochastic differential equations: Linear quantum stochastic differential equations model noncommutative system variables driven by input noises and producing measurement outputs.Their state and output equations use matrices A, B, C, and D, while initial variables satisfy specified commutation relations.
- Linear quantum stochastic differential equations: A linear quantum stochastic differential equation need not describe a physically meaningful system without additional physical-realizability constraints.This limitation applies even though the model can describe systems such as linear quantum optical systems.
3 Controllability and open-loop control of quantum systems
The survey frames controllability as the ability to reach desired quantum states or operators, then reviews open-loop strategies for constructing and improving such control.
- Controllability: Controllability is practically important because it connects to universal quantum computation and atomic- or molecular-scale transformations.
- Controllability: Operator controllability holds exactly when the dynamical Lie algebra equals u(N) or su(N), while pure-state controllability has additional Lie-algebraic criteria.
- Controllability: Quantum controllability includes pure-state, operator, and eigenstate notions, ordered from strongest operator controllability to weakest eigenstate controllability.
- Controllability: For finite-dimensional open systems with Markovian dynamics, coherent control alone cannot ensure controllability, whereas Kraus-map dynamics can provide complete kinematic state controllability.
- Optimal control: Lie-group decomposition gives constructive and exact controls but may be difficult to complete, motivating optimal control for practical target-state tasks.
- Optimal control: Time-optimal control seeks rapid task completion, including minimum-time quantum computation, to reduce relaxation effects.
- Lyapunov-based control design: Lyapunov-based design simulates an artificial closed loop to generate an open-loop control sequence because measurements can destroy the quantum state.
- Variable structure control: Variable structure control switches controller structures, enabling controllability for models that are not individually pure-state controllable.
4 Some results on closed-loop design methods
Closed-loop quantum control combines learning procedures, measurement-based feedback, filtering, stabilization, LQG synthesis, and robust-control methods. These approaches address complex tasks, noisy observations, disturbances, uncertainties, and physical-realizability constraints.
- 4.1 Closed-loop learning control: Closed-loop learning control iteratively designs, applies, observes, and updates laboratory controls using a learning algorithm.The objective is commonly formulated as optimizing a functional of quantum states, control inputs, and control time.
- 4.2 Quantum feedback control: Quantum feedback uses projective or continuous weak measurements, with classical, quantum, or hybrid controllers implementing Markovian, Bayesian, or coherent feedback.Measurement-based information acquisition is central, although quantum measurements disturb the measured system.
- 4.2.1 Identification and filtering/estimation: Quantum filtering extracts information from noisy continuous observations and supplies an essential component for feedback-control strategies.Filtering equations describe the evolution of estimated information for monitored quantum systems.
- 4.2.2 Feedback stabilization/control: Feedback controllers can stabilize quantum systems using measurement or estimation information; Markovian feedback is linear in the signal, whereas Bayesian feedback uses a general measurement-record function.Bayesian feedback uses more information but is harder to implement because it requires an estimation step.
- 4.2.2 Feedback stabilization/control: For the stated system, a suitable γ makes the feedback law globally stabilizing, with Eρ_t → ρ_f as t → ∞.The control law switches behavior according to the state's measured overlap with the target-state region.
- 4.2.3 LQG control / 4.2.4 Robust control: Quantum LQG control optimizes a quadratic cost for suitable linear quantum models, while robust H∞ synthesis uses Riccati solutions and physical-realizability constraints.The robust-control result gives necessity and sufficiency through stabilizing Riccati solutions satisfying the stated assumptions.
5 Conclusions and perspectives
The survey concludes that quantum control has progressed substantially but remains an emerging field with distinctive challenges arising from measurement, decoherence, entanglement, and uncertainty. It identifies quantum incoherent, feedback, robust, decoherence, and entanglement control as important directions for further development.
- Quantum control is still in its infancy despite substantial progress, and it differs fundamentally from classical control because measurement introduces essential uncertainties.Quantum entanglement and protecting quantum coherence are important control tasks without corresponding classical counterparts.
- Quantum incoherent control: Only few results address quantum incoherent control, motivating systematic and physically realizable coherent/incoherent hybrid controllers.Quantum measurement can serve as a useful control tool when coherent control alone is impractical for some tasks.
- Quantum feedback control: Existing quantum feedback-control results are usually restricted to special cases, leaving nonlinear, weak-measurement, and non-Markovian control problems open.The survey specifically raises questions about nonlinear quantum systems, linear controllers for nonlinear dynamics, and feedback under non-Markovian dynamics.
- Robust control of quantum systems: Robust control is essential for practical quantum technologies because quantum systems inevitably face disturbances, noise, and other uncertainties.Existing work focuses mainly on systems with linear Heisenberg-picture dynamics or restricted uncertainty descriptions.
- Decoherence control: Decoherence control remains central because environmental interactions, external controls, and measurement apparatus unavoidably introduce decoherence into many open quantum systems.Quantum error correction is identified as a typical decoherence-control approach.
- Entanglement control: Existing entanglement-control methods represent only a first step, motivating new approaches that account for entanglement’s unique nonclassical characteristics.The survey notes that both optimal-control and feedback-control approaches have been proposed for quantum entanglement.