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Modeling and Control of Quantum Systems: An Introduction

Claudio Altafini, Francesco Ticozzi

arXiv:1210.7127v1quant-ph

TL;DR

Quantum control requires models and synthesis tools that account for probabilistic dynamics, dissipation, and irreversible measurement effects. This paper presents finite-dimensional closed- and open-system models, analyzes controllability and stability, and surveys open- and closed-loop design methods. It provides a self-contained tutorial and roadmap while identifying application-specific adaptation and large-scale, non-Markovian settings as continuing challenges.

  • Problem

    Quantum control needs theory tailored to probabilistic systems, dissipation, and irreversible measurements, beyond methods inherited from classical open-loop control.

  • Method

    The paper develops models for closed and open quantum systems, analyzes controllability and stability, and surveys available open- and closed-loop control-synthesis methods.

  • Results

    The paper presents main controllability and stability results, including Lie-algebraic controllability for the Schrödinger system and uniqueness as the criterion for global asymptotic stability of a steady state.

  • Takeaways & Limitations

    The review provides interested readers with a roadmap for further studies in quantum-control modeling and synthesis.

Abstract

from arXiv · show

The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.

I. INTRODUCTION

This work introduces finite-dimensional quantum theory and surveys modeling, controllability, stability, and control-synthesis methods for quantum systems. It is designed as a concise tutorial and roadmap for control-engineering readers, while balancing breadth against detailed treatment of individual experiments.

  • Motivation: Quantum control is motivated by increasing ability to manipulate atomic-scale matter and light and by quantum information processing's demand for precise information control.The paper highlights storage, manipulation, and retrieval of information as requiring control schemes tailored to quantum mechanics.
  • Scope and limitations: The paper deliberately avoids detailed treatment of particular physical examples and requires readers to verify and adapt theoretical results for their own settings.Its continuous-time focus also excludes many discrete-time results, while simplifying the connection with classical control theory.
  • Scope and organization: The tutorial introduces quantum systems through states, observables, measurement probabilities, and expectations before presenting dynamical models and control-oriented analyses.Its first part covers static quantum tools and models for isolated and open systems; the second addresses controllability and synthesis methods.
  • Basic quantum representation: Finite-dimensional systems are represented on complex Hilbert spaces, with state vectors for known pure states and density operators for ensembles or classical uncertainty.Density operators are positive semidefinite, unit-trace matrices, and the same density operator can correspond to different ensembles.
  • Basic quantum representation: Observables are Hermitian operators whose eigenvalues are measurement outcomes, while their projectors determine outcome probabilities and conditional states.Expectation values can be computed using the Hilbert-Schmidt inner product of the observable and density operator.

D. Probabilities and expectations

Quantum probabilities are computed from observable projectors and the system state, with conditioning producing state updates after measurements. Non-selective measurements generally alter the state, distinguishing quantum from classical probability.

  • Measurement probabilities: Measurement outcomes are observable eigenvalues, and their probabilities are computed from the corresponding projectors and the state.The same conditioning framework applies to pure and mixed states by linearity.
  • State conditioning: After an outcome is recorded, the state is replaced by the state conditioned on that measurement result, while global phase does not affect predictions.The phase equivalence applies to state vectors related by multiplication by e^iφ.
  • Expectations: Expectation values of observables are computed as the Hilbert-Schmidt inner product of the observable and density operator.This follows from the trace representation of quantum expectations.
  • Quantum versus classical probability: A non-selective measurement generally produces an average state different from the pre-measurement state, unlike the corresponding classical situation.The distinction follows when measurement outcomes are unavailable and the post-measurement states are averaged.
  • Quantum versus classical probability: When the relevant operators commute with a fixed non-degenerate observable, quantum probabilities and expectations reduce to classical rules on an abstract sample space.Spectra act as random variables or indicator functions in the commutative representation.

E. Two-level systems and Bloch representation.

A qubit is represented using the Pauli basis, yielding a Bloch-vector description in which valid states form the Bloch ball. The section also introduces entangled states of composite systems.

  • Qubits: A qubit is a two-dimensional quantum system whose pure state vector lies on S3 and whose reference basis is |0⟩, |1⟩.It represents the quantum-mechanical unit of information in analogy with the classical bit.
  • Bloch representation: The Pauli matrices provide a convenient basis for two-level observables and express the density operator through its expectation values.The identity component ensures unit trace, while the x, y, and z components form the Bloch vector.
  • Bloch representation: The Bloch vectors form a convex subset of an affine hyperplane, with the induced Hilbert-Schmidt geometry corresponding to Euclidean geometry.The Frobenius norm of the density operator induces the standard Euclidean norm in the representation.
  • Bloch representation: The Bloch ball contains all two-level density operators, with pure states on its surface and the ground and excited states at opposite z-axis points.The completely mixed state is also identified within this geometric representation.
  • Entanglement: Composite systems can possess entangled states that cannot be represented as factorized pure states or classical mixtures of factorized density operators.For separated systems, such pure states can be intrinsically nonlocal.

III. DYNAMICAL MODELS OF CLOSED QUANTUM SYSTEMS

Closed quantum dynamics are modeled by the Schrödinger equation and controlled through Hamiltonian couplings. Equivalent unitary and density-operator descriptions support Lie-algebraic controllability analysis and identify invariant state spectra.

  • State-vector dynamics: The state vector of a closed quantum system evolves according to an autonomous linear ordinary differential equation governed by the system Hamiltonian.The paper uses units in which the Planck constant is fixed to 1.
  • Hamiltonian control: Hamiltonian control couples the system to tunable electromagnetic fields, with first-order interaction terms producing control amplitudes and a bilinear system on a sphere.The resulting controls preserve unitary evolution and are therefore called unitary controls in this presentation.
  • Unitary propagator: The state-vector dynamics can be lifted to a right-invariant matrix differential equation on SU(N), whose time-varying solution is represented by a time-ordered exponential.The paper identifies this expansion as Dyson's series and relates it to a Volterra series in control theory.
  • Density-operator dynamics: Density operators evolve according to the quantum Liouville-von Neumann equation corresponding to the unitary propagator.This provides the mixed-state formulation of the same closed-system Hamiltonian dynamics.
  • Geometric structure: Closed-system density dynamics are isospectral, so the eigenvalues remain constant and the state space is foliated into SU(N) conjugation orbits.These compact connected leaves are complex flag manifolds whose dimensions depend on eigenvalue multiplicities.

D. Schr¨odinger’s and Heisenberg’s dual pictures of dynamics

Closed-system predictions can be described either by evolving the state or by evolving observables under the dual Heisenberg picture. For two-level systems, this duality yields the Bloch equation as an infinitesimal-rotation model.

  • In the Schrödinger picture, the state evolves under conjugation by the unitary propagator while observables remain time-invariant.
  • In the Heisenberg picture, observables evolve by conjugation with the adjoint propagator, producing identical measurement predictions.
  • For two-level systems, decomposing the Hamiltonian and state with Pauli matrices converts the Liouville equation into a vector ODE.
  • The Bloch equation represents infinitesimal rotations of the Bloch vector, so its norm remains constant and trajectories stay on a sphere.

F. N-level system: explicit parametrizations

N-level quantum dynamics can be vectorized through coherence-vector or superoperator representations, although these become substantially more complex than the two-level case. Open-system models reduce environment-coupled unitary dynamics to Markovian master equations whose stability is characterized by invariant sets and steady states.

  • F. N-level system: explicit parametrizations: An N-level density operator has N^2−1 degrees of freedom and can be represented using coherence vectors or Liouville-space superoperators.
  • F. N-level system: explicit parametrizations: These N-level parametrizations are substantially more complex to describe explicitly than the two-level representation.
  • F. N-level system: explicit parametrizations: Open-system dynamics arise by coupling the system to an environment and reducing the joint unitary evolution through a partial trace.
  • F. N-level system: explicit parametrizations: Under suitable memory assumptions, the reduced dynamics can be modeled as a Markovian quantum dynamical semigroup with completely positive composition-preserving maps.
  • F. N-level system: explicit parametrizations: The master-equation generator combines Hamiltonian and dissipative terms, and its non-unitary dynamics generally produce non-imaginary-axis eigenvalues.
  • F. N-level system: explicit parametrizations: A steady state is globally asymptotically stable if and only if it is unique; analogous subspace stability depends on the absence of invariant support outside the target.

D. Two-level MME in Bloch vector representation

For two-level systems, Lindblad dissipation becomes an affine Bloch-vector dynamics whose geometry distinguishes unital from non-unital noise. Examples include dephasing, which preserves an axis, and decay, which selects a stable state.

  • D. Two-level MME in Bloch vector representation: The dissipative part of a two-level master equation is generally affine in Bloch-vector coordinates, with unital dynamics reducing to a linear field.
  • D. Two-level MME in Bloch vector representation: For invertible unital dissipation, the completely mixed state is the unique fixed point and is globally asymptotically stable.
  • D. Two-level MME in Bloch vector representation: Continuous-time dephasing squeezes the Bloch sphere along the x and y axes while leaving the z axis invariant.
  • D. Two-level MME in Bloch vector representation: For a decaying two-level atom, the excited-state survival probability is e^−γt under the stated Markovian model.
  • D. Two-level MME in Bloch vector representation: Quantum stochastic processes for electromagnetic fields use non-commutative noise operators on Fock space, with calculus rules depending on the input field state.

2) Joint evolution and Heisenberg dynamics:

The system and field evolve jointly through a unitary quantum stochastic model, while measurement conditioning produces a quantum filter or stochastic master equation. For homodyne detection, trajectories converge in probability to measurement-operator eigenstates.

  • 2) Joint evolution and Heisenberg dynamics:: The joint system-field evolution is unitary and is modeled by a quantum stochastic differential equation with bounded interaction operators.
  • 2) Joint evolution and Heisenberg dynamics:: Heisenberg system operators and output fields are obtained by conjugating initial operators with the joint propagator.
  • 3) Conditional expectation, homodyne detection measurement and filtering equations:: Self-nondemolition observations commute across times and with system observables, enabling conditional expectations and quantum filtering equations.
  • 3) Conditional expectation, homodyne detection measurement and filtering equations:: The stochastic master equation combines Lindblad drift with a measurement-induced diffusion term driven by innovation noise.
  • 3) Conditional expectation, homodyne detection measurement and filtering equations:: Under homodyne detection of a two-level system, the conditional Bloch dynamics form a nonlinear diffusion with equilibrium states given by the eigenstates of σz.
  • 3) Conditional expectation, homodyne detection measurement and filtering equations:: All trajectories converge in probability to those equilibria, reproducing the probabilistic state-selection effect of a projective measurement.

V. CONTROLLABILITY OF QUANTUM SYSTEMS

Controllability of quantum systems is characterized through reachable sets and Lie-algebraic conditions, with generic and graph-based criteria for closed and open models.

  • For the Schrödinger control system, controllability is equivalent to Lie{−iH0, −iH1} = su(N).
  • The Lie-algebraic condition is sufficient but not necessary for pure-state controllability, because a proper subalgebra can still act transitively on S2N−1.
  • Controllability is generic: almost all pairs H0 and H1 satisfy the required Lie-algebraic condition.
  • Almost any pair of unitary gates is universal, allowing arbitrary compositions to reach any point of SU(N).
  • For a two-level system, H0 = ∆σz and H1 = σx generate su(2) through their commutator, ensuring controllability.
  • For diagonal H0, connectivity of the control Hamiltonian's graph is necessary, while strong regularity of H0 plus graph connectivity is sufficient for controllability.

B. Controllability of the MME

For controlled open quantum systems, dissipation enlarges the dynamics beyond unitary Lie-group motion, constraining short-time and finite-time controllability while enabling distinct long-time characterizations.

  • Controllability of the MME: The MME dynamics may act through linear or affine Lie algebras, depending on whether the dissipator is unital.For non-unital dissipation, the algebra has a semidirect-sum structure; for unital dissipation, the affine part is absent.
  • Controllability of the MME: The MME is accessible in the stated general Lie-algebra cases but is never small-time controllable or controllable in finite time.The theorem also states that unital dissipation prevents controllability in the density-operator space.
  • Controllability of the MME: Non-unitary dissipative directions cannot be reversed by unitary controls, producing Lie-semigroup reachable sets rather than locally reversible motion.This irreversibility is the stated reason for the absence of small-time controllability.
  • Controllability of the MME: For accessible unital systems, reachable sets strictly expand with time and the state spectrum becomes progressively more majorized.For t2 > t1 > 0, the reachable-set inclusion is strict and Φ(ρ(t2)) ≺ Φ(ρ(t1)).
  • Controllability of the MME: With unital dissipation, a unique steady state is globally asymptotically stable, and in the two-level example it is the completely mixed state.The purity norm acts as a quadratic Lyapunov function centered at that state in the controlled two-level example.
  • Controllability of the MME: For a two-level atom with decay, the uncontrolled MME has a unique globally asymptotically stable fixed point, while constant coherent controls can shift it to a mixed state.A global ordering of reachable sets cannot generally be established in this non-unital case.

C. Average Hamiltonian Methods and Dynamical Decoupling

Average Hamiltonian and dynamical-decoupling methods use pulsed controls to reshape quantum dynamics, particularly to suppress environmental interactions. The paper also surveys measurement-based and coherent feedback, including stabilization results and constructive feedback design.

  • C. Average Hamiltonian Methods and Dynamical Decoupling: Pulsed-control protocols based on average Hamiltonians suppress undesired noise and engineer quantum dynamics in closed and open systems.These protocols were inspired by NMR pulse sequences and applied across solid-state and optical settings.
  • C. Average Hamiltonian Methods and Dynamical Decoupling: Periodic, piecewise-constant control propagators generated by unitary sequences enable dynamical-decoupling analysis through the Magnus expansion.The control propagator is cyclic, with U_c(nT_c) = I, and is modeled using a finite set of unitary operators.
  • C. Average Hamiltonian Methods and Dynamical Decoupling: As K →∞ with fixed T, higher-order Magnus terms become negligible and the evolution is approximated by the average-Hamiltonian propagator U(T) ≈e^−i H̄(0)T.For group-based sequences, the averaged interaction is obtained from the corresponding group projection.
  • Feedback control: Feedback control divides into measurement-based schemes using classical output signals and coherent schemes that retain quantum information within a quantum controller network.Coherent feedback uses dynamically entangled system-controller networks and feeds back an input field to the system.
  • Feedback control: Wiseman–Milburn feedback produces a valid Lindblad generator whose noise operator depends jointly on the measurement and control actions.The resulting Markovian dynamics represent average closed-loop behavior over noise trajectories.
  • Feedback control: For any measurement operator L, a feedback Hamiltonian F and compensation H_c can make an arbitrary desired pure state ρ_d globally asymptotically stable when the stated condition holds.The theorem’s proof provides a constructive algorithm for designing the required feedback and correction Hamiltonians.

B. Filtering-based feedback

Filtering-based feedback uses a real-time state estimate from the measurement record to design state-feedback laws for stochastic quantum dynamics. The surveyed methods combine numerical Lyapunov analysis with patched or hybrid controls, while general systematic design remains limited.

  • B. Filtering-based feedback: Filtering-based feedback integrates the stochastic master equation in real time to update a state estimate used in the control law.This extends simple output feedback by conditioning the controller on the measurement record and estimated state.
  • B. Filtering-based feedback: Limited detection efficiency is represented by 0 < η ≤1, and the resulting state dynamics are stochastic master equations with fixed Hamiltonians H_0 and H_1.The state-space topology makes feedback design difficult and prevents direct use of standard methods.
  • B. Filtering-based feedback: An affine Bloch-vector control is chosen to vanish at the target state but not at the antipodal stationary state.Its dependence on z_t and x_t follows geometric considerations about the target and free dynamics.
  • B. Filtering-based feedback: A polynomial Lyapunov function can be numerically constructed so that its generator is a negative sum of squares, supporting a proof of control effectiveness.The argument combines invariant-set analysis for uncontrolled diffusions with LaSalle-type results.
  • B. Filtering-based feedback: No systematic and efficient stabilizing-control design is available for arbitrary stochastic master equations.A patched control law solves the problem for a class of N-level spin systems with H_1 = F_y and L = F_z.
  • B. Filtering-based feedback: A hybrid feedback law using threshold regions and a constant control globally stabilizes the stochastic dynamics around ρ_d, with E(ρ_t) →ρ_d as t →∞.The design uses a stochastic Lyapunov law near the target and destabilizes the remaining state space with constant control.

C. Quantum optical networks

Quantum optical networks model local field interactions and compose subsystems into interconnected input-output systems. Their coherent controllers support quantum feedback design, although physical realizability and nonconvex optimization remain central challenges.

  • C. Quantum optical networks: Local-in-space-and-time interactions with optical components permit quantum input-output signals and interconnection of network elements.This supports designing desired input-output behavior from cavities, beam-splitters, mirrors, and related components.
  • C. Quantum optical networks: Series connection and concatenation represent interconnected systems as composite systems with parameters G = (S, L, H).Series connection feeds one system’s output field into another, while concatenation combines systems in parallel.
  • C. Quantum optical networks: For coherent states of a quantum harmonic oscillator with scalar S and L, the state remains coherent and the input-output relation is time-invariant.The relation is dB_out = S dB_in + Ldt.
  • C. Quantum optical networks: Linear quantum networks are modeled by state and output equations involving noncommutative operators, input increments, output increments, and matrices A, B, C, and D.The model is given by dx(t) = Ax(t)dt + Bdw(t) and dy(t) = Cx(t)dt + Ddw(t).
  • C. Quantum optical networks: Coherent-controller optimization is non-convex because physical constraints complicate otherwise linear interconnected dynamics.A key design requirement is ensuring that numerically optimized controllers are physically realizable.
  • C. Quantum optical networks: Recent numerical results indicate that a coherent controller can outperform a linear classical controller in the quantum LQG problem.This suggests a potential advantage for fully quantum controllers within the studied setting.
  • C. Quantum optical networks: Feedback control of non-Markovian models remains relatively unexplored from a control-theoretic viewpoint, while large-scale quantum systems pose further theoretical challenges.The outlook identifies non-Markovian feedback and high-dimensional or networked systems as areas for further research.

APPENDIX

The appendix introduces finite-dimensional quantum notation and basic operator concepts, then reviews harmonic-oscillator representations, coherent states, and their evolution. These definitions establish the mathematical language used throughout the paper.

  • APPENDIX: Finite-dimensional quantum systems are represented on complex Hilbert spaces H_N ≃ C^N, with observables as Hermitian operators.The appendix restricts the basic presentation to finite-dimensional systems to avoid technical complications.
  • APPENDIX: Dirac notation represents states by kets and bras, operator actions by A|ψ⟩, and outer products by |ψ⟩⟨ϕ|.In matrix form, kets are column vectors, bras are row vectors, and adjoints are conjugate transposes.
  • APPENDIX: Tensor products describe joint quantum systems, while the partial trace reduces operators to subsystem descriptions.Observables on one subsystem are embedded using the identity operator on the other subsystem.
  • APPENDIX: For a harmonic oscillator, the annihilation operator lowers energy levels, the creation operator raises them, and [a, a†] = I.The number operator N := a†a is diagonal in the energy basis and measures the number of quanta.
  • APPENDIX: The oscillator Hamiltonian H = ω(N + 1/2 I) makes a and a† evolve as a(t) = e^−iωt a(0) and a†(t) = e^iωt a†(0).Thus, the ladder operators are energy eigen-operators for the evolution.
  • APPENDIX: Coherent states are annihilation-operator eigenstates parameterized by α, with both the expectation and variance of N equal to |α|^2.Under harmonic-oscillator evolution, a coherent state remains coherent while its phase oscillates.
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