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arXiv:2605.04681

Finite steps optimise dissipation in stochastically controlled quantum systems

Theodore McKeever, Harry J. D. Miller, Ahsan Nazir

quant-phcond-mat.stat-mechSubmitted May 6, 2026• Announced May 7, 2026• 0 citations
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Abstract

Motivated by the need for precise, energy-efficient, and experimentally realistic quantum control protocols, we investigate the thermodynamic cost of performing quantum step-equilibration processes under the influence of classical stochastic control fields. Whereas purely deterministic protocols exhibit dissipation that scales inversely with the number of steps, we show that weak Gaussian noise in the control variables induces dissipative contributions that grow linearly with the number of steps. Consequently, we derive the finite optimal number of steps and minimal achievable average dissipated work and its variance using the quantum thermodynamic length. These results are demonstrated using two paradigmatic examples: a Landau-Zener sweep of a qubit strongly coupled to a thermal bath and the erasure of a transverse-field Ising model.

Comments
18 pages, 10 figures

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Motivated by the need for precise, energy-efficient, and experimentally realistic quantum control protocols, we investigate the thermodynamic cost of performing quantum step-equilibration processes under the influence of classical stochastic control fields. Whereas purely deterministic protocols exhibit dissipation that scales inversely with the number of steps, we show that weak Gaussian noise in the control variables induces dissipative contributions that grow linearly with the number of steps. Consequently, we derive the finite optimal number of steps and minimal achievable average dissipated work and its variance using the quantum thermodynamic length. These results are demonstrated using two paradigmatic examples: a Landau-Zener sweep of a qubit strongly coupled to a thermal bath and the erasure of a transverse-field Ising model.
Constructing optimal thermodynamic processes in quantum systems relies on managing the balance between the average excess work and its stochastic fluctuations. Recently it has been shown that two different quantum generalisations of thermodynamic length can be utilised to determine protocols with either minimal excess work or minimal work variance. These lengths measure the distance between points on a manifold of control parameters, and optimal protocols are achieved by following the relevant geodesic paths given some fixed boundary conditions. Here we explore this problem in the context of Gaussian quantum states that are weakly coupled to an environment and derive general expressions for these two forms of thermodynamic length. We then use this to compute optimal thermodynamic protocols for various examples of externally driven Gaussian systems with multiple control parameters.
Constructing optimal thermodynamic processes in quantum systems relies on managing the balance between the average excess work and its stochastic fluctuations. Recently it has been shown that two different quantum generalizations of thermodynamic length can be utilized to determine protocols with either minimal excess work or minimal work variance. These lengths measure the distance between points on a manifold of control parameters, and optimal protocols are achieved by following the relevant geodesic paths given some fixed boundary conditions. Here we explore this problem in the context of Gaussian quantum states that are weakly coupled to an environment and derive general expressions for these two forms of thermodynamic length. We then use this to compute optimal thermodynamic protocols for various examples of externally driven Gaussian systems with multiple control parameters.
Optimal (reversible) processes in thermodynamics can be modelled as step-by-step processes, where the system is successively thermalized with respect to different Hamiltonians by an external thermal bath. However, in practice interactions between system and thermal bath will take finite time, and precise control of their interaction is usually out of reach. Motivated by this observation, we consider finite-time and uncontrolled operations between system and bath, which result in thermalizations that are only partial in each step. We show that optimal processes can still be achieved for any non-trivial partial thermalizations at the price of increasing the number of operations, and characterise the corresponding tradeoff. We focus on work extraction protocols and show our results in two different frameworks: A collision model and a model where the Hamiltonian of the working system is controlled over time and the system can be brought into contact with a heat bath. Our results show that optimal processes are robust to noise and imperfections in small quantum systems, and can be achieved by a large set of interactions between system and bath.
Optimal (reversible) processes in thermodynamics can be modelled as step-by-step processes, where the system is successively thermalized with respect to different Hamiltonians by an external thermal bath. However, in practice interactions between system and thermal bath will take finite time, and precise control of their interaction is usually out of reach. Motivated by this observation, we consider finite-time and uncontrolled operations between system and bath, which result in thermalizations that are only partial in each step. We show that optimal processes can still be achieved for any non-trivial partial thermalizations at the price of increasing the number of operations, and characterise the corresponding tradeoff. We focus on work extraction protocols and show our results in two different frameworks: A collision model and a model where the Hamiltonian of the working system is controlled over time and the system can be brought into contact with a heat bath. Our results show that optimal processes are robust to noise and imperfections in small quantum systems, and can be achieved by a large set of interactions between system and bath.
Quantum systems are typically controlled by classical devices that necessarily dissipate energy to the surrounding environment. By the fluctuation-dissipation theorem, this implies noise in the control field produced. We investigate the relationship between the resulting errors and the entropy produced by the macroscopic controller, working in the context of a concrete minimal model. In the most experimentally relevant regime of fast control and high fidelity, we find a tradeoff between dissipation and control errors. These findings are exemplified by perturbative calculations of the gate fidelity for simple gate implementations and of the fidelity of state transfer via shortcuts to adiabaticity in the Landau-Zener model. Our results quantitatively demonstrate that suppressing errors in quantum control typically comes at the cost of macroscopic dissipation, which is needed to tame fluctuations in the classical controller.
Landauer's limit on heat dissipation during information erasure is critical as devices shrink, requiring optimal pure-state preparation to minimize errors. However, Nernst's third law states this demands infinite resources in energy, time, or control complexity. We address the challenge of cooling quantum systems with finite resources. Using Markovian collision models, we explore resource trade-offs and present efficient cooling protocols (that are optimal for qubits) for coherent and incoherent control. Leveraging thermodynamic length, we derive bounds on heat dissipation for swap-based strategies and discuss the limitations of preparing pure states efficiently.