Pointwise and dynamic programming control synthesis for finite-level open quantum memory systems

Abstract: This paper is concerned with finite-level quantum memory systems for retaining initial dynamic variables in the presence of external quantum noise. The system variables have an algebraic structure, similar to that of the Pauli matrices, and their Heisenberg picture evolution is governed by a quasilinear quantum stochastic differential equation. The latter involves a Hamiltonian whose parameters depend affinely on a classical control signal in the form of a deterministic function of time. The memory performance is quantified by a mean-square deviation of quantum system variables of interest from their initial conditions. We relate this functional to a matrix-valued state of an auxiliary classical control-affine dynamical system. This leads to a pointwise control design where the control signal minimises the time-derivative of the mean-square deviation with an additional quadratic penalty on the control. In an alternative finite-horizon setting with a terminal-integral cost functional, we apply dynamic programming and obtain a quadratically nonlinear Hamilton-Jacobi-Bellman equation, for which a solution is outlined in the form of a recursively computed asymptotic expansion.

• The paper introduces a framework that reduces quantum performance functionals to classical control-affine systems for real-time memory stabilization.
• It employs pointwise greedy control with quadratic regularization alongside a dynamic programming approach using the Hamilton-Jacobi-Bellman formulation.
• Implications include robust fault-tolerant quantum memory design and tractable algorithms for high-fidelity quantum information processing.

Pointwise and Dynamic Programming Control for Finite-Level Open Quantum Memory Systems

Overview

This paper introduces a structured approach to the control synthesis of finite-level open quantum memory systems subjected to external quantum noise and deterministic time-dependent control, focusing on the minimization of the mean-square deviation of system variables from their initial conditions. The framework centers around a class of systems whose operator algebra mirrors that of the Pauli matrices, and whose open-system dynamics are dictated by quasilinear quantum stochastic differential equations (QSDEs). The central contribution is the explicit reduction of quantum performance functionals to the field of classical control-affine dynamical systems, enabling the deployment of pointwise and dynamic programming optimal control techniques—including a quadratic regularization for admissibility. Both analytical and algorithmic aspects for practical synthesis are detailed, and the implications for fault-tolerant quantum memory design are discussed.

System Model and Algebraic Structure

The study targets systems with $n$ self-adjoint dynamic variables $X_1,\ldots,X_n$ satisfying an operator-algebra similar to the Pauli algebra, leading to polynomial reduction into affine forms in $X$ through structural constants $\alpha$ and $\beta$. These variables evolve under QSDEs driven by a combination of deterministic control (linear Hamiltonian with affine parameter dependence), quantum vacuum field disturbances (modeled via Hudson-Parthasarathy calculus), and linear coupling to external bosonic fields. The system's algebraic closure and canonical commutation relations enable efficient computation of all multi-point moments and thus facilitate the tractable formulation of statistical control objectives.

Mean-Square Deviation as Performance Metric

Quantum memory is quantified through the mean-square deviation $\Delta(t)$ of selected observables (defined as $\varphi(t) = F X(t)$ for a selection matrix $F$) from their initial state. The paper rigorously derives the evolution equations for both the mean and the second-moment matrices, leveraging the algebraic structure to obtain closed equations for all required expectations. Crucially, the authors relate the quantum mean-square deviation to the state of an auxiliary matrix-valued, classical, control-affine dynamical system:

$\dot{z}(t) = A(t) z(t) + c,$

where $A(t)$ is affinely dependent on the control input and $X_1,\ldots,X_n$0 encodes both the mean and cross-moment states.

Pointwise (Greedy) Control Synthesis

A first mode of control synthesis minimizes, at every instant, the time-derivative of the mean-square deviation augmented with a quadratic (energy-like) regularization on the control. The cost function at each time is, thus,

$X_1,\ldots,X_n$1

Owing to the affine dependence on the control variable, the optimal instant control input is given explicitly in feedback form:

$X_1,\ldots,X_n$2

where $X_1,\ldots,X_n$3 is the control gradient derived from the reduction of the quantum objective to the classical system. Substitution yields a quadratically nonlinear closed-loop ODE for $X_1,\ldots,X_n$4. The regulation effect of $X_1,\ldots,X_n$5 is critical: a small $X_1,\ldots,X_n$6 allows aggressive reduction of $X_1,\ldots,X_n$7, but increases nonlinearity, complicating global behavior analysis.

Dynamic Programming and Hamilton-Jacobi-Bellman Approach

For finite-horizon cost functions (terminal deviation plus integral quadratic control cost), the control problem is recast as classical dynamic programming, with the value function $X_1,\ldots,X_n$8 governed by a Hamilton-Jacobi-Bellman equation of the form:

$X_1,\ldots,X_n$9

with a specified terminal cost. The optimal feedback controller is derived analytically in terms of the Frechet derivative of the value function. The nonlinearity (quadratic in $X$0) of the HJBE precludes closed-form solutions; thus, the authors propose a recursively defined asymptotic expansion using a small-penalty parameter approach. The first terms can be solved via characteristic methods, and higher-order corrections account for the impact of the quadratic regularizer.

Theoretical and Practical Implications

The reduction of a quantum control problem with noncommutative dynamics and noise to an explicit classical control-affine system is a salient feature that provides both analytical and numerical tractability. The explicit control laws derived allow for real-time feedback synthesis suitable for deterministic (classical) electronic controllers interacting with quantum hardware, bypassing the need for continuous quantum measurement-based feedback and enabling robust retention of encoded quantum information against decoherence.

Algorithmically, the pointwise control law offers practical implementability for online memory stabilization, while the dynamic programming-based approach, with its recursive expansion, provides a framework for near-optimal control over finite horizons with computationally manageable complexity for moderate system dimension.

Future Outlook

Anticipated extensions include: integrating time-varying or stochastic control parameters, accommodating measurement-based feedback protocols, and generalizing the algebraic structure for multi-qudit systems and oscillator networks. The explicit classical embedding of the quantum control problem also suggests routes for rigorous analysis of robustness, stochastic stability, and large-deviation performance in noisy, high-dimensional quantum memories—fundamental for scalable quantum information processing.

Conclusion

This work provides a mathematically rigorous and computationally viable framework for synthesizing deterministic control laws for finite-level quantum memory systems subject to noise. By representing quantum performance functionals via auxiliary classical control-affine systems, both pointwise and dynamic programming-based optimization are achieved, with explicit synthesis methods and recursive solutions to the nonlinear control problems encountered. These techniques lay the foundation for robust, high-fidelity quantum memory implementations in future quantum technologies.