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Coherent quantum LQG control
H. I. Nurdin, M. R. James, I. R. Petersen
TL;DR
The paper asks how to design quantum LQG controllers when the controller may itself be quantum and must remain physically realizable. It formulates coherent feedback control, reformulates the constrained problem for numerical solution, and obtains fully quantum controller designs. In comparison with the classical schemes considered, one fully quantum controller achieves lower cost.
Problem
The central problem is quantum LQG controller design with a quantum controller and fully quantum plant output, subject to physical realizability constraints.
Method
The authors reformulate the constrained polynomial matrix problem as a rank-constrained LMI feasibility problem and solve it numerically using alternating projections.
Results
A fully quantum controller achieves J∞ = 4.1793 versus J∞ ≈ 4.4440 for the best classical linear controller with indirect measurement.
Takeaways & Limitations
The results indicate that coherent quantum controllers may have practical significance beyond being a theoretical curiosity.
Takeaways & Limitations
The quantum LQG problem is generally computationally hard, and the paper does not establish an exact or analytical solution to its synthesis problem.
Abstract
from arXiv · showhide
Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as "coherent feedback control.'' It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones.
1 Introduction
The paper formulates coherent quantum LQG control, allowing a quantum controller and fully quantum plant output, unlike prior classical-controller formulations. Because physical realizability adds constraints, the authors develop a numerical approach and demonstrate feasible fully quantum designs.
- Motivation: Quantum optical devices can, under appropriate assumptions, be modeled by linear quantum stochastic differential equations driven by quantum Wiener processes.Linear quantum optics is studied for communication systems, quantum computers, and physics.
- Contribution: The proposed quantum LQG formulation allows the controller itself to be another quantum system and is termed coherent feedback control.Earlier treatments used classical controllers driven by continuous measurements of the quantum plant output.
- Contribution: Physical realizability imposes additional constraints on controller coefficients, making the coherent LQG design problem more difficult than standard LQG.The paper characterizes the problem as constrained controller design rather than an unconstrained classical optimization.
- Numerical approach: The problem is converted into a rank constrained LMI problem using a nonlinear change of variables, then approached with an alternating projections algorithm.The method is demonstrated on stabilization of a quantum plant, where a solution was successfully obtained.
- Paper scope: The paper is organized around quantum linear stochastic models, physical realizability, quantum LQG formulation, numerical procedures, extensions, and design examples.The later sections include an extension of the methodology and discussion of classical and quantum controller designs.
2 General quantum linear stochastic models in quantum optics
This section specifies linear quantum stochastic models using system variables, quantum Wiener disturbances, commutation relations, and structured noise Ito matrices. Canonical and degenerate-canonical forms organize the quantum and classical components of these models.
- Model form: The models use linear QSDEs dx(t) = Ax(t)dt + Bdw(t) and dy(t) = Cx(t)dt + Ddw(t), with self-adjoint possibly non-commutative system variables.A, B, C, and D are real matrices of compatible dimensions.
- System variables: Initial system variables are Gaussian and satisfy commutation relations determined by a real antisymmetric matrix Θ.The commutator convention is [A, B] = AB − BA, and Θ may be canonical or degenerate canonical.
- Commutation structure: Canonical Θ has repeated J blocks, whereas degenerate canonical Θ includes a zero block alongside quantum conjugate-variable pairs.A system with one classical variable and two conjugate quantum variables has Θ = diag(0, J).
- Inputs: The input vector decomposes into a noise part and an adapted self-adjoint signal part representing variables passed through system connections.The signal part is assumed to commute with the noise increments and system variables.
- Noise: Noise commutation relations are determined by a non-negative Hermitian Ito matrix, whose antisymmetric component defines the noise commutator matrix.The noise can include classical components and pairs of conjugate quantum noises.
- Noise conventions: The paper adopts canonical noise conventions with even noise dimension and allows noncanonical noise to be enlarged into a canonical representation.The canonical form is F_ẇ = I + i diag(J, ..., J).
3 Physical realizability of linear QSDEs
Physical realizability distinguishes mathematically specified QSDEs from physically meaningful quantum systems. It requires preservation of canonical commutation relations and, for degenerate systems, embedding into a larger physically realizable system.
- Physical meaning: Arbitrary system matrices in a linear quantum stochastic model need not define a meaningful physical quantum system.Quantum mechanics restricts the allowable coefficients of the QSDE.
- Physical constraints: Closed quantum-system evolution is unitary, which requires preservation of canonical commutation relations for all times.This preservation restricts the matrices A, B, C, and D.
- Canonical systems: For fully quantum systems with canonical Θ, physical realizability means the QSDE represents an open quantum harmonic oscillator.The canonical case directly connects the model to a physically realizable quantum system.
- Degenerate systems: For degenerate canonical Θ, classical components are treated as members of fictitious conjugate pairs, and realizability is defined through embedding in a larger system.The augmented system has the special structure required for physical realizability.
- Characterization: Theorem 1 gives necessary and sufficient conditions for physical realizability and specifies the required feedthrough structure.For canonical Θ, the associated Hamiltonian and coupling matrices have explicit expressions.
4 Formulation of the quantum LQG problem
The quantum LQG problem minimizes an infinite-horizon quadratic cost over physically realizable quantum controllers. Unlike classical LQG, its controller constraints are nonlinear and non-convex, motivating a rank-constrained LMI reformulation and feasibility-based numerical strategy.
- Closed-loop model: The closed-loop plant-controller model combines plant variables, quantum Wiener disturbances, adapted control signals, and controller noise inputs.The controller has its own state ξ(t), noise channels, and output field driving the plant.
- Assumptions: The plant and controller are initially decoupled, with their initial cross-commutators equal to zero.The controller state has commutation structure ΘK and is independent of the plant initially.
- Performance index: For asymptotically stable closed-loop dynamics, the infinite-horizon cost is computed from the unique positive definite solution of a Lyapunov equation.The limiting covariance matrix P satisfies the Lyapunov equation and determines J∞ = Tr(CPCT).
- Synthesis problem: Quantum LQG synthesis minimizes J∞ over controller matrices subject to physical realizability constraints and a fixed ΘK determining the controller type.Canonical ΘK specifies a fully quantum controller, while other choices permit different controller structures.
- Constraints: The physical realizability constraint is a non-convex, non-linear equality constraint on the controller matrices.These constraints are additional to the usual performance objective and are required for the controller to represent a physical system.
- Computational strategy: Because an exact analytical solution is unknown and general-purpose optimization is difficult, the paper studies a relaxed cost-bound feasibility problem reformulated as a rank-constrained LMI.A solution to the original problem can in principle be approached by iterating over the bound γ.
5 Reformulation of the quantum LQG problem into a rank constrained LMI problem
The quantum LQG synthesis problem is transformed into a rank-constrained LMI formulation through nonlinear variable changes and matrix lifting. The reformulation can be made necessary and sufficient, although simplifying assumptions yield a less complex sufficient formulation.
- Problem transformation: The plant and controller equations are redefined so the controller’s quantum noise is included in a modified plant output, preserving the closed-loop equations.This places the problem in a form compatible with standard LQG reformulation techniques.
- Problem transformation: Auxiliary variables and a nonlinear change of variables rewrite the stability and cost-bound conditions as LMI constraints.The introduced variables include N, M, X, Y, and Q, with X, Y, and Q symmetric.
- Rank-constrained formulation: Theorem 4 characterizes solvability through LMIs together with polynomial matrix equalities linking lifted and controller variables.The constraints include ˘N = NΘK, NM T = I −YX, and C = CKM T.
- Rank-constrained formulation: Matrix lifting linearizes the polynomial constraints and produces an LMI system with a rank n constraint.The construction introduces 14 matrix lifting variables while retaining matrix structure rather than scalarizing the polynomial program.
- Controller recovery: A positive semidefinite matrix of rank at most n can be factorized to recover the transformed variables and reconstruct a physically realizable controller.The controller matrices are recovered using the reconstruction formulas involving M and N.
- Scope of the reformulation: Fixing M = I and N = I −YX makes the rank-constrained LMI problem only sufficient, whereas adding variables restores necessity at higher computational cost.The larger formulation may be useful for reducing complexity when the plant dimension exceeds 2.
6 Numerical solution of the rank constrained LMI problem
The resulting rank-constrained LMI problem is generally difficult because polynomial controller-design formulations are nonconvex and may be NP-hard. The paper therefore uses alternating projections, with heuristic initialization, to search numerically for feasible solutions.
- Computational challenge: Polynomial matrix programming problems are nonconvex, nonlinear, and generally difficult to solve, with some related control problems known to be NP-hard.This computational difficulty motivates the rank-constrained LMI approach.
- Computational challenge: Moment and sum-of-squares LMI relaxations can provide convergence guarantees under appropriate conditions, but the paper instead targets the rank-constrained formulation directly.Direct rank-constrained methods search for points satisfying both the LMIs and the rank constraint.
- Numerical procedure: The numerical procedure uses the LMIRank implementation of an alternating-projections algorithm augmented with a Newton step to potentially accelerate convergence.The solver is accessed through Yalmip and a freely available Matlab toolbox.
- Numerical procedure: Because convergence is not guaranteed from arbitrary starting points, the paper proposes initializing the solver from a solution of the relaxed LMIs.The initialization sets M = I and N = I −YX, then computes the lifted variables before forming the starting point.
7 An extension of the numerical procedure
The numerical procedure is extended by allowing the controller commutation matrix to vary, so the optimization can seek the controller type rather than fixing it in advance. This adds flexibility but increases problem complexity and may affect convergence.
- Free controller type: Allowing ΘK to be a free real skew-symmetric matrix lets the procedure seek an optimal classical, quantum, or mixed classical-quantum controller type.With ΘK fixed, the parameter determines the type of controller being sought.
- Similarity transformation: A solution of the extended problem can be transformed into a solution with canonical or degenerate canonical commutation structure through a similarity transformation.The LQG cost remains invariant under the corresponding controller-state similarity transformation.
- Extended formulation: The extension enlarges the lifted matrix and introduces additional variables and constraints for ΘK and the associated controller transformations.The lifted matrix is redefined to have dimension 25n × 25n under the stated simplifying assumptions.
- Limitations: The extended solver can complement the fixed-ΘK solver, but its additional free variables and constraints increase complexity and may affect alternating-projections convergence.An additional heuristic is needed to choose an initial ΘK.
- Limitations: The extended procedure does not supersede strict quantum-controller synthesis because freeing ΘK may produce a controller that is not quantum when a quantum controller is desired.The paper recommends retaining the strictly quantum procedure for such cases.
8 Quantum LQG control design examples
The paper demonstrates numerical quantum LQG controller designs and compares fully quantum control with classical controllers using direct or indirect measurements. The fully quantum controller achieves lower cost than the best tested classical linear controller with indirect measurement, while the realizations remain computationally demanding and experimentally unaddressed.
- 8.1 Quantum LQG controller design example I: A physically realizable fully quantum controller asymptotically stabilizes the plant and achieves an LQG cost J∞=5.7382.The controller was obtained numerically after solving the formulated design problem.
- 8.1 Quantum LQG controller design example I: The controller designs set BK2 effectively to zero because the associated controller-only quantum noise contributes to the LQG cost.For the first example, BK2 entries are of order 10^-10, and setting BK2=0 produces an identical numerical result.
- 8.2 Classical LQG controller designs: The classical direct-measurement controller achieves J∞=4.8468, compared with J∞=5.7382 for the first fully quantum controller.The classical controller uses a measured plant-output quadrature as its input.
- 8.2 Classical LQG controller designs: Indirect measurement gives its lowest classical cost near α=0.715, with J∞≈4.4440.The indirect measurement mixes the plant output with vacuum noise before homodyne detection of both output quadratures.
- 8.3 Quantum LQG controller design example II: The second fully quantum design achieves J∞=4.1793, outperforming the best tested classical linear controller with indirect measurement.The comparison is 4.1793 for the fully quantum controller versus approximately 4.4440 for indirect measurement.
9 Conclusions
The paper formulates quantum LQG control with potentially fully quantum controllers and addresses its computational difficulty using rank-constrained formulations and alternating projections. Numerical examples successfully produce fully quantum controllers, including one that improves performance over the classical schemes considered.
- The quantum LQG problem permits the controller itself to be another quantum system.
- The problem can be converted from polynomial matrix programming to a rank constrained LMI problem.
- An alternating projections procedure, including an extension for unspecified controller types, is proposed to solve the constrained problem.
- Two stabilization examples for a marginally stable quantum plant successfully compute fully quantum LQG controllers.
- A fully quantum LQG controller can achieve improved performance over the fully classical controller schemes considered.
- Standard LQG methodology does not guarantee physical realizability, and the quantum LQG problem is generally computationally hard.