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arXiv:2103.16015 · Quantum Science and Technology

Dynamically corrected gates from geometric space curves

Edwin Barnes, F. A. Calderon-Vargas, Wenzheng Dong, Bikun Li, Junkai Zeng, Fei Zhuang

Published Dec 17, 2021• 38 citations
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Abstract

Abstract 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.

DOI
10.1088/2058-9565/ac4421

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Abstract 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.
Noise-induced gate errors remain one of the main obstacles to realizing a broad range of quantum information technologies. Dynamical error suppression using carefully designed control schemes is critical for overcoming this challenge. Such schemes must be able to correct against multiple noise sources simultaneously afflicting a qubit to reach error-correction thresholds. Here we present a general framework for designing control fields that simultaneously suppress both noise in the fields themselves as well as transverse dephasing noise. Using the recently developed space curve quantum control formalism, in which robust quantum evolution is mapped to closed geometric curves in a multidimensional Euclidean space, we derive the minimal conditions necessary to guarantee the simultaneous cancellation of both types of noise to leading order. In particular, we find that the cancellation of control field noise requires the derivative of the space curve to have zero-area projections, which is a much more subtle property compared to the closed-curve condition needed to suppress transverse dephasing. We present several techniques for solving both these conditions simultaneously and provide explicit examples of error-resistant control fields. Our work also sheds light on the relation between holonomic evolution and the suppression of control field errors.
Abstract The noisy nature of quantum hardware necessitates the implementation of high-fidelity quantum gates in a noise-insensitive manner. While there exist many powerful methods for designing dynamically corrected gates (DCGs), they often use a single cost function to simultaneously achieve a target gate and suppress noise. This can lead to unnecessary tradeoffs that lower gate fidelities and complicate the discovery of globally-optimal solutions. Here, we present a method for single-qubit DCGs called Bézier Ansatz for Robust Quantum (BARQ) control to address these challenges. Rather than numerically optimizing the controls directly, BARQ instead makes use of the Space Curve Quantum Control formalism in which the quantum evolution is mapped to a geometric space curve. In the formulation used by BARQ, the boundary conditions of the space curve dictate the target gate while its shape determines the gate’s noise sensitivity. We eliminate the aforementioned tradeoffs by employing a control-point parameterization that allows the target gate to be fixed upfront and use numerical optimization only for noise-robustness. BARQ introduces a global perspective into the control landscape and provides ample freedom to design experimentally friendly and robust control pulses. The pulse design is facilitated through the developed software package qurveros .
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 Schr\"odinger 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 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.
The ability to perform gates in multiqubit systems that are robust to noise is of crucial importance for the advancement of quantum information technologies.However, finding control pulses that cancel noise while performing a gate is made difficult by the intractability of the time-dependent Schrödinger equation, especially in multiqubit systems.Here, we show that this issue can be sidestepped by using a formalism in which the cumulative error during a gate is represented geometrically as a curve in a multidimensional Euclidean space.Cancelation of noise errors to leading order corresponds to closure of the curve, a condition that can be satisfied without solving the Schrödinger equation.We develop and uncover general properties of this geometric formalism, and derive a recursion relation that maps control fields to curvatures for Hamiltonians of arbitrary dimension.We demonstrate the utility of the formalism by employing it to design pulses that simultaneously correct against both noise errors and crosstalk for two qubits coupled by an Ising interaction.We give examples both of a single-qubit rotation and a two-qubit maximally entangling gate.The results obtained in this example are relevant to both superconducting transmon qubits and semiconductor quantum-dot spin qubits.We propose this geometric formalism as a general technique for pulse-induced error suppression in quantum computing gate operations.
Quantum information processing faces a significant hurdle: noise. Different noise sources induce varying errors in quantum operations depending on the underlying dynamics. To gain a deeper understanding of these error mechanisms, we introduce the concept of quantum error evolution diagrams (QEEDs). QEEDs establish a dual correspondence between driven noisy quantum dynamics and geometric space curves, providing quantitative geometric metrics to assess the severity of the noises. This framework enables the design of universal robust quantum gates that correct errors induced by generic noise. Additionally, we present a protocol for constructing a universal set of single- and two-qubit robust quantum gates with simple and smooth control pulses of arbitrary length and fidelities exceeding 99.99% across a wide range of noise strengths. Our work provides new insights into the geometric nature of noisy quantum dynamics and paves the way for developing strategies to dynamically correct quantum errors.
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 phase gates. 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 phase gates while canceling 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.