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arXiv:2602.01688

Optimal Control to Minimize Dissipation and Fluctuations in Open Quantum Systems Beyond Slow and Rapid Regimes

Yuki Kurokawa, Yoshihiko Hasegawa

quant-phSubmitted Feb 2, 2026• Announced Feb 3, 2026
v1 · Feb 2, 2026v2 · Apr 7, 2026
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Abstract

Optimal control is a central problem in quantum thermodynamics. While control theories in the rapid-driving and slow-driving limits have been developed, to the best of our knowledge there is no general optimization method applicable to intermediate timescales. We introduce an optimal-control framework to minimize dissipated work and work variance, defined via the two-point measurement scheme, in open quantum systems governed by time-dependent Lindblad master equations. By introducing an auxiliary operator, we convert the history-dependent work variance into a time-local integral, enabling efficient gradient-based optimization beyond slow or rapid driving regimes. Applying our method, we find that in the coherent spin-boson model the optimized protocol can switch discontinuously between distinct locally optimal solutions as the relative weight between dissipation and fluctuations is varied. Moreover, for a single-level quantum dot coupled to a fermionic reservoir, the optimized fluctuation-minimizing protocol develops a qualitatively different multi-step structure that is not captured by approaches based on slow- or rapid-driving limits.

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Optimal control is a central problem in quantum thermodynamics. While control theories in the rapid-driving and slow-driving limits have been developed, to the best of our knowledge there is no general optimization method applicable to intermediate timescales. We introduce an optimal-control framework to minimize dissipated work and work variance, defined via the two-point measurement scheme, in open quantum systems governed by time-dependent Lindblad master equations. By introducing an auxiliary operator, we convert the history-dependent work variance into a time-local integral, enabling efficient gradient-based optimization beyond slow or rapid driving regimes. Applying our method, we find that in the coherent spin-boson model the optimized protocol can switch discontinuously between distinct locally optimal solutions as the relative weight between dissipation and fluctuations is varied. Moreover, for a single-level quantum dot coupled to a fermionic reservoir, the optimized fluctuation-minimizing protocol develops a qualitatively different multi-step structure that is not captured by approaches based on slow- or rapid-driving limits.
Abstract To achieve efficient and reliable control of microscopic systems one should look for driving protocols that mitigate both the average dissipation and stochastic fluctuations in work. This is especially important in fast driving regimes in which the system is driven far out of equilibrium, potentially creating large amounts of unwanted entropy production. Here we characterise these optimal protocols in rapidly driven classical and quantum systems and prove that they consist of two discontinuous jumps in the full set of control variables. These jumps can be tuned to interpolate between processes with either minimal dissipation or minimal fluctuations, and in some situations allow for simultaneous minimisation. We illustrate our general results with rapidly driven closed quantum systems, classical bit erasure and a dissipative Ising chain driven close to a quantum phase transition.
We apply advanced methods of control theory to open quantum systems and we determine finite-time processes which are optimal with respect to thermodynamic performances. General properties and necessary conditions characterizing optimal drivings are derived, obtaining bang-bang-type solutions corresponding to control strategies switching between adiabatic and isothermal transformations. A direct application of these results is the maximization of the work produced by a generic quantum heat engine, where we show that the maximum power is directly linked to a particular conserved quantity naturally emerging from the control problem. Finally we apply our general approach to the specific case of a two-level system, which can be put in contact with two different baths at fixed temperatures, identifying the processes that minimize heat dissipation. Moreover, we explicitly solve the optimization problem for a cyclic two-level heat engine driven beyond the linear-response regime, determining the corresponding optimal cycle, the maximum power, and the efficiency at maximum power.
Optimal control in open quantum systems is considered, with a focus on a two-level logical qubit system whose time evolution is governed by Markovian dynamics described within the Lindblad (GKSL) formalism. The objective is to minimize the net heat transferred to the environment during the transition while explicitly accounting for the work performed by the external control field. In this context, we formulate a variational optimal control framework for minimizing dissipated heat under fidelity constraints, which establishes existence under standard assumptions of bounded controls and lower semi-continuity of at least one bounded control that minimizes the environmental heat dissipation functional while maintaining the final-state fidelity above a designated threshold. The analysis combines the direct method of the calculus of variations with the Pontryagin Maximum Principle. The direct method in the calculus of variations (in L ∞ spaces) guarantees existence by demonstrating the compactness of admissible controls and the lower semi-continuity of the cost functional. Second, the Pontryagin Maximum Principle for open quantum systems establishes essential optimality conditions, resulting in coupled state–costate equations and switching functions that define optimal controls. We establish a physical foundation for the optimization problem by defining internal energy, heat flow, and work within the Lindblad framework, resulting in a thermodynamically coherent expression for dissipated energy. This formulation allows for both analytical understanding together with a practical numerical approximation based on the MDR-GRAPE algorithm. To demonstrate numerical consistency, we present MDR-GRAPE, a modified gradient-based algorithm that extends GRAPE to dissipative environments, integrating positivity-preserving techniques for density matrices. Numerical experiments demonstrate reproducible convergence across the five independent initializations considered in this study, providing empirical evidence of consistent convergence toward feasible low-dissipation solutions.
We investigate the optimal control problem for a non-Markovian open, dissipative quantum system. Optimal control using the Pontryagin maximum principle is specifically derived. The influences of ohmic reservoir with Lorentz-Drude regularization are numerically studied in a two-level system under the following three conditions: ${\ensuremath{\omega}}_{0}⪡{\ensuremath{\omega}}_{c}$, ${\ensuremath{\omega}}_{0}\ensuremath{\approx}{\ensuremath{\omega}}_{c}$, or ${\ensuremath{\omega}}_{0}⪢{\ensuremath{\omega}}_{c}$, where ${\ensuremath{\omega}}_{0}$ is the characteristic frequency of the quantum system of interest, and ${\ensuremath{\omega}}_{c}$ the cutoff frequency of the ohmic reservoir. The optimal control process shows its remarkable influences on the decoherence dynamics. The temperature is a key factor in the decoherence dynamics. We analyze the optimal decoherence control in high temperature, intermediate temperature, and low temperature reservoirs, respectively. It implies that designing some engineered reservoirs with the controlled coupling and state of the environment can slow down the decoherence rate and delay the decoherence time. Moreover, we compare the non-Markovian optimal decoherence control with the Markovian one and find that with non-Markovian the engineered artificial reservoirs are better than with the Markovian approximation in controlling the open, dissipative quantum system's decoherence.
We investigate the optimal control problem in non-Markovian open, dissipative quantum system. Optimal control using Pontryagin maximum principle of this problem has been specifically derived. The influences of Ohmic reservoir with Lorentz-Drude regularization are numerically studied in a two-level system under the following three conditions w0Lt wc, w0ap wcor w0Gt wcrespectively. The optimal control process shows its remarkable influences to the decoherence dynamics. The temperature is a key factor in the decoherence dynamics. We analysis the optimal decoherence control in high temperature, intermediate temperature, and low temperature reservoirs respectively. It implies that designing some engineered reservoirs with the controlled coupling and state of the environment can slow down the decoherence rate and delay the decoherence time. Moreover, we compared the non-Markovian optimal decoherence control with the Markovian one, we find that with non-Markovian the engineered artificial reservoirs are better than with the Markovian approximate in controlling the open, dissipative quantum systempsilas decoherence.
The second law of thermodynamics sets a lower bound on the work required to drive a system between thermal equilibrium states, with equality attained in the quasistatic limit. For finite-time processes, work in excess of this bound is inevitably dissipated, motivating the search for driving protocols that minimize this excess work. While classical stochastic systems have been extensively explored, quantum analyses remain limited and often rely on Markovian master equations valid only in the weak-coupling regime. Here, we study minimal work protocols for representative two-level systems coupled to a harmonic-oscillator bath using a numerically exact method. Inspired by known optimal solutions for Brownian oscillators, we introduce an impulse ansatz that incorporates possible boundary impulses and test it across a wide range of bath parameters. We find that impulse-like features remain nearly optimal in the quantum, non-Markovian regime at short times. We also identify cases in which the widely used Markovian master equation fails even at weak coupling, highlighting the necessity of going beyond standard approximations in the quantum regime for finite-time thermodynamic optimization.