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:1506.02701 · Physical Review A

Minimum time generation of SU(2) transformations with asymmetric bounds on the controls

Raffaele Romano

Published Nov 11, 2015• 1 citation
arXivHTMLPDF

Abstract

We study how to generate in minimum time special unitary transformations for a two-level quantum system under the assumptions that (i) the system is subject to a constant drift, (ii) its dynamics can be affected by three independent, bounded controls, and (iii) the bounds on the controls are asymmetric, that is, the constraint on the control in the direction of the drift is independent of that on the controls in the orthogonal plane. Using techniques recently developed for the analysis of $\text{SU}(2)$ transformations, we fully characterize the reachable sets of the system and the optimal control strategies for any possible target transformation.

DOI
10.1103/physreva.92.052314

Related papers

Geometric analysis of minimum-time trajectories for a two-level quantum system

Raffaele Romano

We consider the problem of controlling in minimum time a two-level quantum system which can be subject to a drift. The control is assumed to be bounded in magnitude and to affect two or three independent generators of the dynamics. We describe the time optimal trajectories in SU(2), the Lie group of possible evolutions for the system, by means of a particularly simple parametrization of the group. A key ingredient of our analysis is the introduction of the optimal front line. This tool allows us to fully characterize the time evolution of the reachable sets and to derive the worst-case operators and the corresponding times. The analysis is performed in any regime---controlled dynamics stronger than, of the same magnitude as, or weaker than the drift term---and gives a method to synthesize quantum logic operations on a two-level system in minimum time.

arXiv:1410.4906•Physical Review A•Dec 1, 2014•19 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

Minimum time control of a pair of two-level quantum systems with opposite drifts*

Raffaele Romano, Domenico D’Alessandro

Abstract In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin maximum principle, and provide a complete characterization of singular and nonsingular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.

arXiv:1504.07219•Journal of Physics A Mathematical and Theoretical•Jul 22, 2016•8 citations

Minimum time optimal synthesis for two level quantum systems

Francesca Albertini, Domenico D’Alessandro

For the time optimal problem of an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU(2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.

Journal of Mathematical Physics•Jan 1, 2015•31 citations

The roles of drift and control field constraints upon quantum control speed limits

Christian Arenz, Benjamin Russell, Daniel Klaus Burgarth +1 more

In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.

arXiv:1704.06289•New Journal of Physics•Oct 23, 2017•35 citations

Time-optimal synthesis of SU(2) transformations for a spin-1/2 system

A. David. Boozer

We consider a quantum control problem involving a spin-$1/2$ particle in a magnetic field. The magnitude of the field is held constant, and the direction of the field, which is constrained to lie in the $x--y$ plane, serves as a control parameter that can be varied to govern the evolution of the system. We analytically solve for the time dependence of the control parameter that will synthesize a given target SU(2) transformation in the least possible amount of time, and we show that the time-optimal solutions have a simple geometric interpretation in terms of the fiber bundle structure of SU(2). We also generalize our time-optimal solutions to a control problem that includes a constant bias field along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{z}}$ axis and to the case of inhomogeneous control, in which a single control parameter governs the evolution of an ensemble of spin-$1/2$ systems.

arXiv:1109.5412•Physical Review A•Jan 19, 2012•53 citations
We study how to generate in minimum time special unitary transformations for a two-level quantum system under the assumptions that (i) the system is subject to a constant drift, (ii) its dynamics can be affected by three independent, bounded controls, and (iii) the bounds on the controls are asymmetric, that is, the constraint on the control in the direction of the drift is independent of that on the controls in the orthogonal plane. Using techniques recently developed for the analysis of SU(2)\text{SU}(2)SU(2) transformations, we fully characterize the reachable sets of the system and the optimal control strategies for any possible target transformation.
We consider the problem of controlling in minimum time a two-level quantum system which can be subject to a drift. The control is assumed to be bounded in magnitude and to affect two or three independent generators of the dynamics. We describe the time optimal trajectories in SU(2), the Lie group of possible evolutions for the system, by means of a particularly simple parametrization of the group. A key ingredient of our analysis is the introduction of the optimal front line. This tool allows us to fully characterize the time evolution of the reachable sets and to derive the worst-case operators and the corresponding times. The analysis is performed in any regime---controlled dynamics stronger than, of the same magnitude as, or weaker than the drift term---and gives a method to synthesize quantum logic operations on a two-level system in minimum time.
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.
Abstract In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin maximum principle, and provide a complete characterization of singular and nonsingular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.
For the time optimal problem of an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU(2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.
In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.
We consider a quantum control problem involving a spin-1/21/21/2 particle in a magnetic field. The magnitude of the field is held constant, and the direction of the field, which is constrained to lie in the x−−yx--yx−−y plane, serves as a control parameter that can be varied to govern the evolution of the system. We analytically solve for the time dependence of the control parameter that will synthesize a given target SU(2) transformation in the least possible amount of time, and we show that the time-optimal solutions have a simple geometric interpretation in terms of the fiber bundle structure of SU(2). We also generalize our time-optimal solutions to a control problem that includes a constant bias field along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{z}}$ axis and to the case of inhomogeneous control, in which a single control parameter governs the evolution of an ensemble of spin-1/21/21/2 systems.