Exploring the tradeoff between fidelity and time optimal control of quantum unitary transformations

Tibbetts, Katharine Moore; Brif, Constantin; Grace, Matthew D.; Donovan, Ashley; Hocker, David; Ho, Tak‐San; Wu, Re Bing; Rabitz, Herschel

doi:10.1103/physreva.86.062309

Cited by 71 publications

(109 citation statements)

References 73 publications

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“…Quantum optimal control theory (OCT) provides a basis to explore applied field tuning of quantum system dynamics [1,2,3,4,5]. Recent successes include achieving control of electrons in quantum dots through time-varying gate voltages [6], controlling electronic correlations in molecules [7], manipulating the dynamics of quantum many-body systems [8], cooling ultracold atomic gases [9], eliciting quantum revivals [10], and generating adiabatic time evolution [11].…”

Section: Introductionmentioning

confidence: 99%

Exploring the complexity of quantum control optimization trajectories

Nanduri

Shir

Donovan

et al. 2015

Phys. Chem. Chem. Phys.

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The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric R ≥ 1.0, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of R = 1 would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that R can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived, wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology 1 and structure provide a foundation to understand why optimal quantum control can be readily achieved.

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Section: Introductionmentioning

confidence: 99%

Exploring the complexity of quantum control optimization trajectories

Nanduri

Shir

Donovan

et al. 2015

Phys. Chem. Chem. Phys.

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“…Several OCT studies have shown that violating condition (3) by limiting the number of control variables [180] or the control period T [81,168,[170][171][172][173][174][175][176][177][178][179] can prevent the achievement of a globally optimal solution. In this section, we investigate the practical effects of imposing various types of constraints.…”

Section: Effects Of Severe Control Field Constraintsmentioning

confidence: 99%

“…In this paper, we focus on the subset of severe constraints, i.e., those that prevent achievement of the target objective by introducing local optima onto the control landscape. It has been shown, however, that even more mild constraints can have a significant effect on OCT optimizations, e.g., by increasing the search effort [163][164][165][166][167][168].…”

Section: Introductionmentioning

confidence: 99%

“…Special algorithms that facilitate successful optimization when the control field has significant spectral constraints have been introduced, for problems such as population transfer in a onedimensional asymmetric double well [169] and molecular alignment [166]. Other works have explored the effect of a specific constraint on OCT optimization; timeoptimal control, the problem of achieving a target objective in the minimum possible time, has received the greatest attention [81,168,[170][171][172][173][174][175][176][177][178][179], and constraints on the number of field components have also been investigated [180]. In this work, we perform extensive OCT simulations to evaluate constraints whose effects on the success of gradient optimization have not previously been examined, identifying values of each constrained parameter beyond which some or all of a set of searches fail to optimize.…”

Section: Introductionmentioning

confidence: 99%

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Searching for quantum optimal controls under severe constraints

Riviello¹,

Tibbetts²,

Brif³

et al. 2015

Phys. Rev. A

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The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

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“…1d were implemented with a control time and phase step duration identified as nearoptimal based on computer-numerical and laboratory exploration [29]. Figure 2 shows the average fidelity calculated for that set of maps when implemented with waveforms using a range of (T, δt).…”

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confidence: 99%

Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System

et al. 2015

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Quantum control in large dimensional Hilbert spaces is essential for realizing the power of quantum information processing. For closed quantum systems the relevant inputoutput maps are unitary transformations, and the fundamental challenge becomes how to implement these with high fidelity in the presence of experimental imperfections and decoherence. For two-level systems (qubits) most aspects of unitary control are well understood, but for systems with Hilbert space dimension d>2 (qudits), many questions remain regarding the optimal design of control Hamiltonians 1 and the feasibility of robust implementation 2,3 . Here we show that arbitrary, randomly chosen unitary transformations can be efficiently designed and implemented in a large dimensional Hilbert space (d=16) associated with the electronic ground state of atomic 133 Cs, 4 achieving fidelities above 0.98 as measured by randomized benchmarking 5 . Generalizing the concepts of inhomogeneous control 6 and dynamical decoupling 7 to d>2 systems, we further demonstrate that these qudit unitary maps can be made robust to both static and dynamic perturbations. Potential applications include improved fault-tolerance in universal quantum computation 8 , nonclassical state preparation for high-precision metrology 9 , implementation of quantum simulations 10 , and the study of fundamental physics related to open quantum systems and quantum chaos 11 .The goal of quantum control is to perform a desired transformation through dynamical evolution driven by a control Hamiltonian H C (t) . For example, one common objective is to evolve the system from a known initial state to a desired final state. If the control task is simple or special symmetries are present, it is sometimes possible to find a high-performing control Hamiltonian through intuition, or to construct one using group theoretic methods 12 . In this letter we explore the use of "optimal control" 1 to design control Hamiltonians for tasks of varying complexity, from state-to-state maps to unitary maps on the entire accessible Hilbert space. The basic procedure is well established: the Hamiltonian H C (t) is parameterized by a set of control variables, and a numerical search is performed to find values that optimize the fidelity with which the control objective is achieved. The application of optimal control to quantum systems originated in NMR 13 and physical chemistry 1 , and has since expanded to include, e. g., ultrafast physics 14 , cold atoms 15,16 , biological molecules 17 , spins in condensed matter 18 , and superconducting circuits 19 .We study the efficacy of numerical design and the performance of the resulting control Hamiltonians using a well developed testbed consisting of the electron and nuclear spins of individual 133 Cs atoms driven by radiofrequency (rf) and microwave (µw) magnetic fields (Fig. 1) 16 . Our experiments show that the optimal control strategy is adaptable to a wide range of control tasks, and that it can generate control Hamiltonians with excellent performance even in the prese...

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Exploring the tradeoff between fidelity and time optimal control of quantum unitary transformations

Cited by 71 publications

References 73 publications

Exploring the complexity of quantum control optimization trajectories

Exploring the complexity of quantum control optimization trajectories

Searching for quantum optimal controls under severe constraints

Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System

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