Implementing high-fidelity quantum control and reducing the effect of the coupling between a quantum system and its environment is a major challenge in developing quantum information technologies. Here, we show that there exists a geometrical structure hidden within the time-dependent Schrödinger equation that provides a simple way to view the entire solution space of pulses that suppress noise errors in a system's evolution. In this framework, any single-qubit gate that is robust against quasistatic noise to first order corresponds to a closed three-dimensional space curve, where the driving fields that implement the robust gate can be read off from the curvature and torsion of the space curve. Gates that are robust to second order are in one-to-one correspondence with closed curves whose projections onto three mutually orthogonal planes each enclose a vanishing net area. We use this formalism to derive new examples of dynamically corrected gates generated from smooth pulses. We also show how it can be employed to analyze the noise-cancellation properties of pulses generated from numerical algorithms such as GRAPE. A similar geometrical framework exists for quantum systems of arbitrary Hilbert space dimension. arXiv:1811.04864v2 [quant-ph]

In order to achieve the high-fidelity quantum control needed for a broad range of quantum information technologies, reducing the effects of noise and system inhomogeneities is an essential task. It is well known that a system can be decoupled from noise or made insensitive to inhomogeneous dephasing dynamically by using carefully designed pulse sequences based on square or delta-function waveforms such as Hahn spin echo or CPMG. However, such ideal pulses are often challenging to implement experimentally with high fidelity. Here, we uncover a new geometrical framework for visualizing all possible driving fields, which enables one to generate an unlimited number of smooth, experimentally feasible pulses that perform dynamical decoupling or dynamically corrected gates to arbitrarily high order. We demonstrate that this scheme can significantly enhance the fidelity of singlequbit operations in the presence of noise and when realistic limitations on pulse rise times and amplitudes are taken into account.In recent years, the prospect of enhanced technologies that exploit the principles of quantum mechanics has attracted great interest from many fields in physics and beyond. These efforts are geared toward several envisioned applications, including information processing [1-6], secure communications [7,8], and sensing [9][10][11], and enormous progress has been made in engineering and optimizing coherent quantum systems for these purposes. However, decoherence caused by the environment or other factors remains a primary impediment to realizing quantum technologies [12][13][14][15]; overcoming this challenge requires improvements not only in system engineering [16][17][18], but also in how such systems are controlled.It has been known since the early days of nuclear magnetic resonance that it is possible to design driving fields that suppress adverse effects caused by fluctuations in the system Hamiltonian or in the driving field itself. The simplest example is the Hahn spin echo [19], in which a fast (δ-function) π-pulse is applied halfway through the evolution of a precessing spin, guaranteeing that the spin returns to its initial state at the end of the evolution regardless of the precession rate. This has long been a standard technique to combat inhomogeneous dephasing -the loss of coherence due to variations in precession frequency in spin ensembles. Spin echo and related multipulse sequences [20][21][22][23][24][25][26][27] have also been widely employed to mitigate other types of decoherence such as environmental noise fluctuations [28][29][30]. Much work has been done to extend dynamical decoupling to not only preserve the state of the system, but to also cancel errors while performing operations on the system (dynamical gate correction) [31][32][33][34][35][36][37][38][39][40][41][42][43].Although these dynamical decoupling methods have been broadly successful, there are many systems, especially in the context of quantum information technologies, where they exhibit substantial drawbacks. This is because the hig...

Dynamically correcting for unwanted interactions between a quantum system and its environment is vital to achieving the high-fidelity quantum control necessary for a broad range of quantum information technologies. In recent work, we uncovered the complete solution space of all possible driving fields that suppress transverse quasistatic noise errors while performing single-qubit operations. This solution space lives within a simple geometrical framework that makes it possible to obtain globally optimal pulses subject to a set of experimental constraints by solving certain geometrical optimization problems. In this work, we solve such a geometrical optimization problem to find the fastest possible pulses that implement single-qubit gates while cancelling transverse quasistatic noise to second order. Because the time-optimized pulses are not smooth, we provide a method based on our geometrical approach to obtain experimentally feasible smooth pulses that approximate the time-optimal ones with minimal loss in gate speed. We show that in the presence of realistic constraints on pulse rise times, our smooth pulses significantly outperform sequences based on ideal pulse shapes, highlighting the benefits of building experimental waveform constraints directly into dynamically corrected gate designs.

Quantum information technologies demand highly accurate control over quantum systems. Achieving this requires control techniques that perform well despite the presence of decohering noise and other adverse effects. Here, we review a general technique for designing control fields that dynamically correct errors while performing operations using a close relationship between quantum evolution and geometric space curves. This approach provides access to the global solution space of control fields that accomplish a given task, facilitating the design of experimentally feasible gate operations for a wide variety of applications.