We apply optimal control theory (OCT) to the design of refocusing pulses suitable for the CPMG sequence that are robust over a wide range of B(0) and B(1) offsets. We also introduce a model, based on recent progress in the analysis of unitary dynamics in the field of quantum information processing (QIP), that describes the multiple refocusing dynamics of the CPMG sequence as a dephasing Pauli channel. This model provides a compact characterization of the consequences and severity of residual pulse errors. We illustrate the methods by considering a specific example of designing and analyzing broadband OCT refocusing pulses of length 10t(180) that are constrained by the maximum instantaneous pulse power. We show that with this refocusing pulse, the CPMG sequence can refocus over 98% of magnetization for resonance offsets up to 3.2 times the maximum RF amplitude, even in the presence of ±10% RF inhomogeneity.

We describe how the transient behavior of a tuned and matched resonator circuit and a ringdown suppression pulse may be integrated into an optimal control theory (OCT) pulse-design algorithm to derive control sequences with limited ringdown that perform a desired quantum operation in the presence of resonator distortions of the ideal waveform. Inclusion of ringdown suppression in numerical pulse optimizations significantly reduces spectrometer deadtime when using high quality factor (high-Q) resonators, leading to increased signal-to-noise ratio (SNR) and sensitivity of inductive measurements. To demonstrate the method, we experimentally measure the free-induction decay of an inhomogeneously broadened solid-state free radical spin system at high Q. The measurement is enabled by using a numerically optimized bandwidth-limited OCT pulse, including ringdown suppression, robust to variations in static and microwave field strengths. We also discuss the applications of pulse design in high-Q resonators to universal control of anisotropic-hyperfine coupled electron-nuclear spin systems via electron-only modulation even when the bandwidth of the resonator is significantly smaller than the hyperfine coupling strength. These results demonstrate how limitations imposed by linear response theory may be vastly exceeded when using a sufficiently accurate system model to optimize pulses of high complexity.

High fidelity coherent control of quantum systems is critical to building quantum devices and quantum computers. We provide a general optimal control framework for designing control sequences that account for hardware control distortions while maintaining robustness to environmental noise. We demonstrate the utility of our algorithm by presenting examples of robust quantum gates optimized in the presence of nonlinear distortions. We show that nonlinear classical controllers do not necessarily incur additional computational cost to pulse optimization, enabling more powerful quantum devices. PACS numbers:The ability to coherently control the dynamics of quantum systems with high fidelity is a critical component of the development of modern quantum devices, including quantum computers [1], actuators [2,3], and sensors [4][5][6] that push beyond the capabilities of classical computation and metrology. In recent years, quantum computation has presented a compelling application for quantum control, as high-fidelity control is essential to implement quantum information processors that achieve fault-tolerance [7][8][9].The performance of numerically optimized quantum gates in laboratory applications strongly depends on the accuracy of the system model used to approximate the response of the experimental system to the applied control sequence. Here we develop a general framework whereby classical control hardware components are modelled explicitly, such that their effect on a quantum system can be computed and compensated for using numerical optimal control theory (OCT) [10] algorithms to optimize control sequences. Control sequences designed using OCT algorithms, such as the GRadient Ascent Pulse Engineering (GRAPE) [11] algorithm, can be made robust to a wide variety of inhomogenities, pulse errors and noise processes [12][13][14]. These methods are also easily extended [15][16][17][18] to other applications and may be integrated into other protocols [19]. Recently, it was demonstrated how a model of linear distortions of the control sequence, such as those arising from finite bandwidth of the classical control hardware, may also be integrated into OCT algorithms [20][21][22].Here, we improve and generalize those results beyond linear kernels to any operation that smoothly maps a list of control steps to a classical field seen by the quantum system. Importantly, our framework naturally allows for robustness against uncertainties and errors due to classical control hardware. We begin developing our method generally, without making assumptions about the device of interest, so that our results may be broadly applicable to a wide range of quantum devices. We briefly discuss how our theory is easily applied to any linear distortion, and then in more detail, demonstrate with numerics how nonlinearities in control hardware, such as those found in strongly-driven superconducting resonators used for pulsed electron spin resonance (PESR) [23][24][25], may be included in OCT algorithms.With this goal in mind, we briefly review th...