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arXiv:2409.12287 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Constructing Noise-Robust Quantum Gates via Pontryagin's Maximum Principle

Joshua Hanson, Dennis G. Lucarelli

Published Sep 15, 2024• 2 citations
arXivHTMLPDF

Abstract

Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of un-desired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.

DOI
10.1109/qce60285.2024.00165

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Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of un-desired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.
The presence of decoherence in quantum computers necessitates the suppression of noise. Dynamically corrected gates via specially designed control pulses offer a path forward, but hardware-specific experimental constraints can cause complications. Existing methods to obtain smooth pulses are either restricted to two-level systems, require an optimization over noise realizations, or limited to piecewise-continuous pulse sequences. In this paper, we present the first general method for obtaining truly smooth pulses that minimizes sensitivity to noise, eliminating the need for sampling over noise realizations and making assumptions regarding the underlying statistics of the experimental noise. We parametrize the Hamiltonian using a neural network, which allows the use of a large number of optimization parameters to adequately explore the functional control space. We demonstrate the capability of our approach by finding smooth shapes, which suppress the effects of noise within the logical subspace as well as leakage out of that subspace.
The main goal of this work is to make use of sub-Riemannian geometry and Pontryagins Maximum Principle to investigate optimal control for computing one- and two-qubit quantum gates, as well as to analyze how the presence of uncontrollable interactions, such as noise and cross-talk operators, influence on the energetic cost of performing such unitary operations. Initially, we study the case of a single noisy qubit. We aim to identify sub-Riemannian geodesics that connect the identity to a unitary operation, sufficiently close to the ideal case, despite the presence of decoherence noise. Next, we expand the study to the case of two qubits, initially only under a time-constant cross-talk interaction, and then, under both: cross-talk interaction and decoherence noise. For all case studies, the main challenge lies in determining the unknown parameters in the geodesic equation for the desired unitary operator. We present a novel method, based on random sampling, for determining such parameters and how they can lead to high-fidelity quantum gates which are synthesized through smooth external control fields.
We introduce a method for designing smooth single-qubit control pulses that implement a desired gate while suppressing the effect of unknown static error sources to first order. Unlike dynamically corrected gate constructions that require prior knowledge of the noise model, the present approach is agnostic to the detailed form of the target-bath interaction. The method parametrizes the control propagator through an auxiliary matrix expansion over orthogonal basis functions and enforces decoupling through algebraic orthogonality and equal-norm constraints on the expansion coefficients. These conditions guarantee that the leading Magnus contribution of an arbitrary static interaction reduces to a term proportional to the identity on the target system, thereby cancelling first-order error effects independently of the microscopic origin of the noise. We further show that the same construction suppresses, to first order, mediated couplings between simultaneously controlled qubits when their interaction occurs through intermediate environmental degrees of freedom, yielding effective second-order decoupling of the induced inter-qubit interaction. By using a discrete cosine transform parametrization, the pulse-synthesis problem is cast into a numerically stable constrained optimization with a minimal number of free parameters. Numerical examples for RzR_zRz​ rotations and random single-qubit unitaries demonstrate smooth control fields that realize the target gates while remaining robust against arbitrary static single-qubit noise and mediated multi-qubit couplings. These results provide a hardware-friendly route toward noise-agnostic dynamically corrected single-qubit gates.
Implementing high-fidelity quantum control and suppressing the unwanted environmental noise has been one of the essential challenges in developing quantum information technologies. In the past, driving pulse sequences based on Dirac delta functions or square wave functions, such as Hahn spin echo or CPMG, have been developed to dynamically correcting the noise effects. However, implementing these ideal pulses with high fidelity is a challenging task in real experiments. In this thesis, we provide a new and simple method to explore the entire solution space of driving pulse shapes that suppress environmental noise in the evolution of the system. In this method, any single-qubit phase gate that is first-order robust against quasi-static transversal noise corresponds to a closed curve on a two-dimensional plane, and more general first-order robust single-qubit gates correspond to closed three-dimensional space curves. Second-order robust gates correspond to closed curves having the property that their projection onto any two-dimensional planes shall enclose a zero net area. The driving pulse shapes that implement the gates can be determined by the curvature, torsion, and the length of the curve. By utilizing the framework it is possible to obtain globally optimal solutions in pulse shaping in respect of experimental constraints by mapping them into geometrical optimization problems. One such problem we solved is to prove that the fastest possible single-qubit phase gates that are second-order noise-resistant shall be implemented using sign-flipping square functions. Since square waves are not experimentally feasible, we provide a method to smooth these pulses with minimal loss in gate speed while maintaining the robustness, based on the geometrical framework. This framework can also be useful in diagnosing the noise-cancellation properties of pulse shapes generated from numerical methods such as GRAPE. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation.
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.
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.