© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2401.10099 · Journal of Physics A Mathematical and Theoretical

Time-optimal state transfer for an open qubit

Lev Vyacheslavovich Lokutsievskiy, Alexander Nikolaevich Pechen, Mikhail Il'ich Zelikin

Published Jun 21, 2024• 4 citations

Abstract

Abstract Finding minimal time and establishing the structure of the corresponding optimal controls which can transfer a given initial state of a quantum system into a given target state is a key problem of quantum control. In this work, this problem is solved for a basic component of various quantum technology processes—a qubit interacting with the environment and experiencing an arbitrary time-dependent coherent driving. We rigorously derive both upper and lower estimates for the minimal steering time. Surprisingly, we discover that the optimal controls have a very special form—they consist of two impulses, at the beginning and at the end of the control period, which can be assisted by a smooth time-dependent control in between. Moreover, an important for practical applications explicit almost optimal state transfer protocol is provided which only consists of four impulses and gives an almost optimal time of motion. The results can be directly applied to a variety of experimental situations for estimation of the ultimate limits of state control for quantum technologies.

DOI: 10.1088/1751-8121/ad5396

Related papers

Time-optimal single-scalar control on a qubit of unitary dynamics

Chungwei Lin, Qi Ding, Petros T. Boufounos +4 more

Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength ${u}_{\text{max}}$. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small ${u}_{\text{max}}$, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.

arXiv:2504.01785•Physical Review A•Apr 2, 2025•2 citations

Time optimal quantum control of two-qubit systems

Bin Li, Zuhuan Yu, Fei Shaoming +1 more

We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms. We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state. As applications the minimal time to implement Controlled-NOT gate, SWAP gate and Controlled-U gate is calculated in detail. The experimental realizations of these quantum gates are explicitly presented.

•Aug 17, 2016•11 citations

Introduction to quantum control: From basic concepts to applications in quantum technologies

Christiane P. Koch

Quantum control refers to our ability to manipulate quantum systems. This tutorial-style chapter focuses on the use of classical electromagnetic fields to steer the system dynamics. In this approach, the quantum nature of the control stems solely from the underlying dynamics, through the exploitation of destructive and constructive interference to reach the control target. We first discuss two basic control principles -- coherent control which uses manipulation in frequency or time to design these interferences, and adiabatic following where access to the control target is enabled by tracking the time-dependent ground state. For complex control targets and system dynamics that exceed the scope of these basic principles, optimal control theory provides a powerful suite of tools to design the necessary protocols. A key consideration for the successful application of optimal control theory is a proper choice of the optimization functional. All concepts are illustrated using recent work from my research group, with a focus on controlling atoms and superconducting qubits. The chapter concludes with an outlook on integrating coherent control with engineered dissipation and a discussion of open questions in the field.

quant-pharXiv:2512.04990•Dec 5, 2025•0 citations

Driving at the Quantum Speed Limit: Optimal Control of a Two-Level System

Gerhard C. Hegerfeldt

A remarkably simple result is derived for the minimal time Tmin required to drive a general initial state to a final target state by a Landau-Zener-type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is bounded by a constant c. Then, for large c, the optimal driving is of type bang-off-bang and for increasing c one recovers the unconstrained result. However, for smaller c the optimal driving can suddenly switch to bang-bang type. We discuss the notion of quantum speed limit time.

arXiv:1305.6403•Physical Review Letters•Dec 27, 2013•232 citations

Time-optimal quantum control of nonlinear two-level systems

Xi Chen, Yue Ban, Gerhard C. Hegerfeldt

Nonlinear two-level Landau-Zener type equations for systems with relevance for Bose-Einstein condensates and nonlinear optics are considered and the minimal time ${T}_{\mathrm{min}}$ to drive an initial state to a given target state is investigated. Surprisingly, the nonlinearity may be canceled by a time-optimal unconstrained driving and ${T}_{\mathrm{min}}$ becomes independent of the nonlinearity. For constrained and unconstrained driving explicit expressions are derived for ${T}_{\mathrm{min}}$, the optimal driving, and the protocol.

Physical Review A•Aug 19, 2016•28 citations

Time-Optimal Transfer of the Quantum State in Long Qubit Arrays

Andrei A. Stepanenko, Kseniia S. Chernova, Maxim A. Gorlach

Recent technological advances have allowed the fabrication of large arrays of coupled qubits, serving as prototypes for quantum processors. However, the optimal control of such systems remains notoriously challenging, limiting the potential of large-scale quantum systems. Here, we investigate a model problem of quantum state transfer in a large nearest-neighbor-coupled qubit array. We derive an optimal control strategy that simultaneously achieves maximal fidelity and minimal transfer time, reaching the quantum speed limit in a lattice with time-varying couplings.

arXiv:2501.11933•Physical Review Letters•Jun 10, 2025•8 citations

Abstract Finding minimal time and establishing the structure of the corresponding optimal controls which can transfer a given initial state of a quantum system into a given target state is a key problem of quantum control. In this work, this problem is solved for a basic component of various quantum technology processes—a qubit interacting with the environment and experiencing an arbitrary time-dependent coherent driving. We rigorously derive both upper and lower estimates for the minimal steering time. Surprisingly, we discover that the optimal controls have a very special form—they consist of two impulses, at the beginning and at the end of the control period, which can be assisted by a smooth time-dependent control in between. Moreover, an important for practical applications explicit almost optimal state transfer protocol is provided which only consists of four impulses and gives an almost optimal time of motion. The results can be directly applied to a variety of experimental situations for estimation of the ultimate limits of state control for quantum technologies.

Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength

{u}_{\text{max}}

. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small

{u}_{\text{max}}

, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.

We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms. We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state. As applications the minimal time to implement Controlled-NOT gate, SWAP gate and Controlled-U gate is calculated in detail. The experimental realizations of these quantum gates are explicitly presented.

Quantum control refers to our ability to manipulate quantum systems. This tutorial-style chapter focuses on the use of classical electromagnetic fields to steer the system dynamics. In this approach, the quantum nature of the control stems solely from the underlying dynamics, through the exploitation of destructive and constructive interference to reach the control target. We first discuss two basic control principles -- coherent control which uses manipulation in frequency or time to design these interferences, and adiabatic following where access to the control target is enabled by tracking the time-dependent ground state. For complex control targets and system dynamics that exceed the scope of these basic principles, optimal control theory provides a powerful suite of tools to design the necessary protocols. A key consideration for the successful application of optimal control theory is a proper choice of the optimization functional. All concepts are illustrated using recent work from my research group, with a focus on controlling atoms and superconducting qubits. The chapter concludes with an outlook on integrating coherent control with engineered dissipation and a discussion of open questions in the field.

A remarkably simple result is derived for the minimal time Tmin required to drive a general initial state to a final target state by a Landau-Zener-type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is bounded by a constant c. Then, for large c, the optimal driving is of type bang-off-bang and for increasing c one recovers the unconstrained result. However, for smaller c the optimal driving can suddenly switch to bang-bang type. We discuss the notion of quantum speed limit time.

Nonlinear two-level Landau-Zener type equations for systems with relevance for Bose-Einstein condensates and nonlinear optics are considered and the minimal time

{T}_{\mathrm{min}}

to drive an initial state to a given target state is investigated. Surprisingly, the nonlinearity may be canceled by a time-optimal unconstrained driving and

{T}_{\mathrm{min}}

becomes independent of the nonlinearity. For constrained and unconstrained driving explicit expressions are derived for

{T}_{\mathrm{min}}

, the optimal driving, and the protocol.

Recent technological advances have allowed the fabrication of large arrays of coupled qubits, serving as prototypes for quantum processors. However, the optimal control of such systems remains notoriously challenging, limiting the potential of large-scale quantum systems. Here, we investigate a model problem of quantum state transfer in a large nearest-neighbor-coupled qubit array. We derive an optimal control strategy that simultaneously achieves maximal fidelity and minimal transfer time, reaching the quantum speed limit in a lattice with time-varying couplings.