Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1603.01545 · IEEE Transactions on Automatic Control

Minimum-Time Transitions Between Thermal Equilibrium States of the Quantum Parametric Oscillator

Dionisis Stefanatos

Published Mar 18, 2017• 28 citations
arXivHTMLPDF

Abstract

In this note, we use geometric optimal control to completely solve the problem of minimum-time transitions between thermal equilibrium states of the quantum parametric oscillator, which finds applications in various physical contexts. We discover a new kind of optimal solutions, absent from all the previous treatments of the problem.
DOI
10.1109/tac.2017.2684083

Related papers

Minimum-Time Transitions between Thermal and Fixed Average Energy States of the Quantum Parametric Oscillator

Dionisis Stefanatos

In this article we use geometric optimal control to completely solve the problem of minimum-time transitions between thermal equilibrium and fixed average energy states of the quantum parametric oscillator, a system which has been extensively used to model quantum heat engines and refrigerators. We subsequently use the obtained results to find the minimum driving time for a quantum refrigerator and the quantum finite-time availability of the parametric oscillator, i.e., the potential work which can be extracted from this system by a very short finite-time process.

arXiv:1608.02665•SIAM Journal on Control and Optimization•Jan 1, 2017•13 citations

Exponential bound in the quest for absolute zero

Dionisis Stefanatos

In most studies for the quantification of the third law of thermodynamics, the minimum temperature which can be achieved with a long but finite-time process scales as a negative power of the process duration. In this article, we use our recent complete solution for the optimal control problem of the quantum parametric oscillator to show that the minimum temperature which can be obtained in this system scales exponentially with the available time. The present work is expected to motivate further research in the active quest for absolute zero.

arXiv:1706.07483•Physical review. E•Oct 3, 2017•8 citations

Time optimal simultaneous control of two level quantum systems

Francesca Albertini, Domenico D’Alessandro

In this paper, we solve the problem of simultaneously driving in minimum time to arbitrary final conditions, N two level quantum systems subject to independent controls. The solution of this problem is obtained via an explicit description of the reachable set of the associated control system on SU(2). The treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on Lie groups.

arXiv:1510.07091•Oct 24, 2015•1 citation

Optimal control of a collection of parametric oscillators

K. H. Hoffmann, Bjarne Andresen, Peter Salamon

The problem of effectively adiabatic control of a collection of classical harmonic oscillators sharing the same time-dependent frequency is analyzed. The phase differences between the oscillators remain fixed during the process. This fact leads us to adopt the coordinates: energy, Lagrangian, and correlation, which have proved useful in a quantum description and which have the advantage of treating both the classical and quantum problem in one unified framework. A representation theorem showing that two classical oscillators can represent an arbitrary collection of classical or quantum oscillators is proved. An invariant, the Casimir companion, consisting of a combination of our coordinates, is the key to determining the minimum reachable energy. We present a condition for two states to be connectable using one-jump controls and enumerate all possible switchings for one-jump effectively adiabatic controls connecting any initial state to any reachable final state. Examples are discussed. One important consequence is that an initially microcanonical ensemble of oscillators will be transformed into another microcanonical ensemble by effectively adiabatic control. Likewise, a canonical ensemble becomes another canonical ensemble.

Physical Review E•Jun 5, 2013•22 citations

Quantum Optimal Control of a Lambda System in the Density Matrix Formulation

Julia Cen, Domenico D'Alessandro

In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.

quant-pharXiv:2511.06097•Nov 11, 2025•0 citations

Performance Measure for Optimal Quantum Control

Alexandre Coutinho Lisboa, José Roberto Castilho Piqueira

The problem concerning the minimum time for an initial state to evolve up to a target state plays an important role in the Classic Optimal Control theory. In the quantum context, as quantum states are so sensitive to environmental influences, the problem is more complex, but its formulation and solution are decisive to implement quantum information processing systems. As is well known, the decoherence phenomenon is unavoidable and the time-energy uncertainty relation must be used to study quantum dynamics. Here, the time-energy uncertainty relations are revisited, being fundamental to propose performance measures based on minimum time evolution. A minimum time performance measure is definedfor quantum control problems. Then, some practical examples are considered and the minimum time performance measure is applied providing results that are supposed to be useful for researchers to pursue strategies to optimize the control of the states of a quantum system.

arXiv:1504.03926•Apr 15, 2015•1 citation
In this article we use geometric optimal control to completely solve the problem of minimum-time transitions between thermal equilibrium and fixed average energy states of the quantum parametric oscillator, a system which has been extensively used to model quantum heat engines and refrigerators. We subsequently use the obtained results to find the minimum driving time for a quantum refrigerator and the quantum finite-time availability of the parametric oscillator, i.e., the potential work which can be extracted from this system by a very short finite-time process.
In most studies for the quantification of the third law of thermodynamics, the minimum temperature which can be achieved with a long but finite-time process scales as a negative power of the process duration. In this article, we use our recent complete solution for the optimal control problem of the quantum parametric oscillator to show that the minimum temperature which can be obtained in this system scales exponentially with the available time. The present work is expected to motivate further research in the active quest for absolute zero.
In this paper, we solve the problem of simultaneously driving in minimum time to arbitrary final conditions, N two level quantum systems subject to independent controls. The solution of this problem is obtained via an explicit description of the reachable set of the associated control system on SU(2). The treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on Lie groups.
The problem of effectively adiabatic control of a collection of classical harmonic oscillators sharing the same time-dependent frequency is analyzed. The phase differences between the oscillators remain fixed during the process. This fact leads us to adopt the coordinates: energy, Lagrangian, and correlation, which have proved useful in a quantum description and which have the advantage of treating both the classical and quantum problem in one unified framework. A representation theorem showing that two classical oscillators can represent an arbitrary collection of classical or quantum oscillators is proved. An invariant, the Casimir companion, consisting of a combination of our coordinates, is the key to determining the minimum reachable energy. We present a condition for two states to be connectable using one-jump controls and enumerate all possible switchings for one-jump effectively adiabatic controls connecting any initial state to any reachable final state. Examples are discussed. One important consequence is that an initially microcanonical ensemble of oscillators will be transformed into another microcanonical ensemble by effectively adiabatic control. Likewise, a canonical ensemble becomes another canonical ensemble.
In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.
The problem concerning the minimum time for an initial state to evolve up to a target state plays an important role in the Classic Optimal Control theory. In the quantum context, as quantum states are so sensitive to environmental influences, the problem is more complex, but its formulation and solution are decisive to implement quantum information processing systems. As is well known, the decoherence phenomenon is unavoidable and the time-energy uncertainty relation must be used to study quantum dynamics. Here, the time-energy uncertainty relations are revisited, being fundamental to propose performance measures based on minimum time evolution. A minimum time performance measure is definedfor quantum control problems. Then, some practical examples are considered and the minimum time performance measure is applied providing results that are supposed to be useful for researchers to pursue strategies to optimize the control of the states of a quantum system.