Optimal pulse design in quantum control: A unified computational method

Abstract

Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π∕2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.pseudospectral methods | ensemble control | Lie algebra | broadband excitation C ompelling applications for quantum control have received particular attention and have motivated seminal studies in wide-ranging areas from coherent spectroscopy and MRI to quantum optics. Designing and implementing time-varying excitations (rf pulses) to manipulate complex dynamics of a large quantum ensemble on the order of Avogadro's number is a long-standing problem and an indispensable step that enables every application of quantum control (1). For example, magnetic resonance applications often suffer from imperfections such as inhomogeneity in the static magnetic field (B 0 inhomogeneity) and in the applied rf field (rf inhomogeneity). In addition, there is dispersion in the Larmor frequency of spins due to chemical shifts. A good pulse design strategy must be robust to these effects, and such variations need to be considered in the modeling and pulse design stages in order to match theoretical predictions to experimental outcomes. As difficult experiments with more demanding performance specifications have emerged, the complexity of finding optimal pulse sequences has drastically increased. For example, as high-field NMR spectrometers are increasingly more accessible and required, broadband excitation pulses are needed to cover a wide 13 C chemical-shift range (e.g., up to 40 kHz). In addition, to design excitation and inversion pulses that are practical for a typical NMR spectrometer, methods must accommodate realistic maximum rf power and pulse duration while accomplishing the desired spin evolution. In the majority of cases, the length of an rf pulse is constrained by the fixed delays that dictate a certain coherence transfer. S...

Progress in Compensating Pulse Sequences for Quantum Computation

“…Furthermore, the inverse problem (solving for the control functions) is especially difficult except for in the most simple cases. For this reason, various numerical optimization methods have become popular, including gradient-accent techniques, optimal control methods [68,69,70], and simulated annealing [71,72]. Methods which use elements of optimal-control theory merit special attention; in recent years the GRAPE [73,74] and Krotov algorithms have been especially successful in pulse design, and has been applied to NMR [68], trapped ions [75], and ESR [76].…”

Consistent approximation of a nonlinear optimal control problem with uncertain parameters

“…4), Runge-Kutta (Kameswaran & Biegler, 2008;Schwartz & Polak, 1996), and Pseudospectral (Gong, Kang, & Ross, 2006;Kang, 2010;Ross & Karpenko, 2012). These computational optimal control methods have achieved great success in many areas of control applications (Bedrossian, Bhatt, Kang, & Ross, 2009;Bedrossian, Karpenko, & Bhatt, 2012;Chung, Polak, Royset, & Sastry, 2011;Li, Ruths, Yu, & Arthanari, 2011). In a standard nonlinear optimal control problem, the objective functional is of the Bolza type, which consists of an end cost as well as an integral over the time domain.…”

High-fidelity quantum gates in the presence of dispersion

“…With further enhancement of numerical and analytical techniques, there is no reason that higher fidelities and shorter times should be impossible. Much work has been done in this field however there is potential for improved techniques for designing robust control pulses, allowing for greater control over quantum systems [50]. What has been shown in this paper is the application of optimal control to robust pulse design.…”