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

Physical Review A

Optimal control of a single qubit by direct inversion

Markus Wenin, W. Pötz

Published Aug 23, 2006• 39 citations
Publisher

Abstract

Optimal control of a driven single dissipative qubit is formulated as an inverse problem. We show that direct inversion is possible which allows an analytic construction of optimal control fields. Exact inversion is shown to be possible for dissipative qubits which can be described by a Lindblad equation. It is shown that optimal solutions are not unique. For a qubit with weak coupling to phonons we choose, among the set of exact solutions for the dissipationless qubit, one which minimizes the dissipative contribution in the kinetic equations. Examples are given for state trapping and $Z$-gate operation. Using analytic expressions for optimal control fields, favorable domains for dynamic stabilization in the Bloch sphere are identified. In the case of approximate inversion, the identified approximate solution may be used as a starting point for further optimization following standard methods.

DOI
10.1103/physreva.74.022319

Related papers

State-independent control theory for weakly dissipative quantum systems

Markus Wenin, W. Pötz

We present a general state-independent optimal control strategy for weakly dissipative quantum systems which allows the identification of Hamilton operators which approximately perform a specified unitary operation and minimize the adverse effects of the dissipative environment. Compared to traditional approaches, it avoids the need for Lagrange multipliers and the repeated solving of the full kinetic equations of the quantum subsystem. This direct method is outlined for a single qubit raealization, such as the spin of an excess electron in a quantum dot or a Josephson qubit. We show that for this system formulated within the Lindblad equation, optimal solutions for arbitrary unitary single-qubit operations may be found analytically provided there is full control over the system Hamiltonian. Absolute bounds for controllability are derived analytically. Numerical implementations of this approach confirm these results and allow an efficient determination of optimal solutions for various dissipators, starting from various initial guesses, both for limited and full control over the Hamiltonian of the quantum system.

Physical Review A•Jul 29, 2008•19 citations

State-independent optimal control of dissipative qubits

Markus Wenin, W. Pötz

A general state-independent optimal control strategy for weakly dissipative quantum systems is presented. It allows the identification of Hamilton operators which approximately perform a preselected unitary operation and minimize the adverse effects from a dissipative environment. This direct method is demonstrated at the example of a single qubit, as realized, for example, by the spin of an excess electron in a quantum dot or a Josephson flux qubit, with Lindblad dissipator. We show that optimal solutions for arbitrary unitary single-qubit operations may be found both analytically and numerically.

Applied Physics Letters•Mar 10, 2008•7 citations

Optimal control of one-qubit gates

K. M. Fonseca-Romero, G Useche Laverde, Federico Ardila

We consider the problem of carrying an initial Bloch vector to a final Bloch vector in a specified amount of time under the action of three control fields (a vector control field). We show that this control problem is solvable and therefore it is possible to optimize the control. We choose the physically motivated criteria of minimum energy expended in the control, minimum magnitude of the rate of change of the control and a combination of both. We find exact analytical solutions, determine the fields for a general one-qubit gate, and use the Y gate as an example. We argue that in the case of less than three controls, only the physical intuition does not provide a straight reasonable solution, and solve the problem in the case of a unique control minimizing the energy consumption.

arXiv:quant-ph/0203100•Journal of Physics A Mathematical and General•Jan 8, 2003•3 citations

Optimal Control of a Markovian Qubit with Unitary Control

Emanuel Malvetti

We study a single Markovian qubit governed by a Lindblad master equation and subject to fast unitary control. Using reduced control systems and optimal control theory we determine (i) controls for cooling and heating such systems in a time-optimal way as well as (ii) the set of stabilizable states in the Bloch ball. No restrictions on the Lindblad equation are assumed, and several known results, for instance for the Bloch equations, are recovered. Furthermore we introduce integral systems, for which the solutions take a particularly nice form. These integral systems include all systems with real Lindblad terms as well as all coolable systems. The method allows for intuitive visualizations and is mostly analytical, making use of only basic numerical methods.

arXiv:2404.09279•Apr 14, 2024•1 citation

Quantum optimal control theory and dynamic coupling in the spin-boson model

Hamza Jirari, W. Pötz

A Markovian master equation describing the evolution of open quantum systems in the presence of a time-dependent external field is derived within the Bloch-Redfield formalism. It leads to a system-bath interaction which depends on the control field. Optimal control theory is used to select control fields which allow accelerated or decelerated system relaxation, or suppression of relaxation (dissipation) altogether, depending on the dynamics we impose on the quantum system. The control-dissipation correlation and the nonperturbative treatment of the control field are essential for reaching this goal. The optimal control problem is formulated within Pontryagin's minimum principle and the resulting optimal differential system is solved numerically. As an application, we study the dynamics of a spin-boson model in the strong coupling regime under the influence of an external control field. We show how trapping the system in unstable quantum states and transfer of population can be achieved by optimized control of the dissipative quantum system. We also used optimal control theory to find the driving field that generates the quantum $Z$ gate. In several cases studied, we find that the selected optimal field which reduces the purity loss significantly is a multicomponent low-frequency field including higher harmonics, all of which lie below the phonon cutoff frequency. Finally, in the undriven case we present an analytic result for the Lamb shift at zero temperature.

arXiv:cond-mat/0602497•Physical Review A•Aug 7, 2006•76 citations

Optimal control of dissipative quantum dynamics

Farhang Haddadfarshi

In this thesis, we develop a perturbative approximation for the solution of Lindblad master equations with time-dependent generators that satisfies the fundamental property of complete positivity, as essential for quantum simulations and optimal control of open quantum systems. By probing our method to several explicit examples we show that ensuring this property not only improves the accuracy of the perturbative approximation substantially, but also it permits to read off the effective dissipative processes . Subsequently, we design optimal entangling quantum gates for trapped ions mediated by a bus mode which is subject to decoherence. We show that suitably designed polychromatic control pulses, help to suppress the qubit-phonon entanglement substantially while maintaining the mediated interaction. This leads to a considerable reduction in the gate infidelity, in particular, for multi-qubit gates this yields a significant improvement in the gate performance.

•Feb 1, 2017•0 citations
Optimal control of a driven single dissipative qubit is formulated as an inverse problem. We show that direct inversion is possible which allows an analytic construction of optimal control fields. Exact inversion is shown to be possible for dissipative qubits which can be described by a Lindblad equation. It is shown that optimal solutions are not unique. For a qubit with weak coupling to phonons we choose, among the set of exact solutions for the dissipationless qubit, one which minimizes the dissipative contribution in the kinetic equations. Examples are given for state trapping and ZZZ-gate operation. Using analytic expressions for optimal control fields, favorable domains for dynamic stabilization in the Bloch sphere are identified. In the case of approximate inversion, the identified approximate solution may be used as a starting point for further optimization following standard methods.
We present a general state-independent optimal control strategy for weakly dissipative quantum systems which allows the identification of Hamilton operators which approximately perform a specified unitary operation and minimize the adverse effects of the dissipative environment. Compared to traditional approaches, it avoids the need for Lagrange multipliers and the repeated solving of the full kinetic equations of the quantum subsystem. This direct method is outlined for a single qubit raealization, such as the spin of an excess electron in a quantum dot or a Josephson qubit. We show that for this system formulated within the Lindblad equation, optimal solutions for arbitrary unitary single-qubit operations may be found analytically provided there is full control over the system Hamiltonian. Absolute bounds for controllability are derived analytically. Numerical implementations of this approach confirm these results and allow an efficient determination of optimal solutions for various dissipators, starting from various initial guesses, both for limited and full control over the Hamiltonian of the quantum system.
A general state-independent optimal control strategy for weakly dissipative quantum systems is presented. It allows the identification of Hamilton operators which approximately perform a preselected unitary operation and minimize the adverse effects from a dissipative environment. This direct method is demonstrated at the example of a single qubit, as realized, for example, by the spin of an excess electron in a quantum dot or a Josephson flux qubit, with Lindblad dissipator. We show that optimal solutions for arbitrary unitary single-qubit operations may be found both analytically and numerically.
We consider the problem of carrying an initial Bloch vector to a final Bloch vector in a specified amount of time under the action of three control fields (a vector control field). We show that this control problem is solvable and therefore it is possible to optimize the control. We choose the physically motivated criteria of minimum energy expended in the control, minimum magnitude of the rate of change of the control and a combination of both. We find exact analytical solutions, determine the fields for a general one-qubit gate, and use the Y gate as an example. We argue that in the case of less than three controls, only the physical intuition does not provide a straight reasonable solution, and solve the problem in the case of a unique control minimizing the energy consumption.
We study a single Markovian qubit governed by a Lindblad master equation and subject to fast unitary control. Using reduced control systems and optimal control theory we determine (i) controls for cooling and heating such systems in a time-optimal way as well as (ii) the set of stabilizable states in the Bloch ball. No restrictions on the Lindblad equation are assumed, and several known results, for instance for the Bloch equations, are recovered. Furthermore we introduce integral systems, for which the solutions take a particularly nice form. These integral systems include all systems with real Lindblad terms as well as all coolable systems. The method allows for intuitive visualizations and is mostly analytical, making use of only basic numerical methods.
A Markovian master equation describing the evolution of open quantum systems in the presence of a time-dependent external field is derived within the Bloch-Redfield formalism. It leads to a system-bath interaction which depends on the control field. Optimal control theory is used to select control fields which allow accelerated or decelerated system relaxation, or suppression of relaxation (dissipation) altogether, depending on the dynamics we impose on the quantum system. The control-dissipation correlation and the nonperturbative treatment of the control field are essential for reaching this goal. The optimal control problem is formulated within Pontryagin's minimum principle and the resulting optimal differential system is solved numerically. As an application, we study the dynamics of a spin-boson model in the strong coupling regime under the influence of an external control field. We show how trapping the system in unstable quantum states and transfer of population can be achieved by optimized control of the dissipative quantum system. We also used optimal control theory to find the driving field that generates the quantum ZZZ gate. In several cases studied, we find that the selected optimal field which reduces the purity loss significantly is a multicomponent low-frequency field including higher harmonics, all of which lie below the phonon cutoff frequency. Finally, in the undriven case we present an analytic result for the Lamb shift at zero temperature.
In this thesis, we develop a perturbative approximation for the solution of Lindblad master equations with time-dependent generators that satisfies the fundamental property of complete positivity, as essential for quantum simulations and optimal control of open quantum systems. By probing our method to several explicit examples we show that ensuring this property not only improves the accuracy of the perturbative approximation substantially, but also it permits to read off the effective dissipative processes . Subsequently, we design optimal entangling quantum gates for trapped ions mediated by a bus mode which is subject to decoherence. We show that suitably designed polychromatic control pulses, help to suppress the qubit-phonon entanglement substantially while maintaining the mediated interaction. This leads to a considerable reduction in the gate infidelity, in particular, for multi-qubit gates this yields a significant improvement in the gate performance.