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arXiv:2008.01168 · PRX Quantum

Geometrical Formalism for Dynamically Corrected Gates in Multiqubit Systems

Donovan Buterakos, S. Das Sarma, Edwin Barnes

Published Mar 9, 2021• 39 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.2.010341

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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.
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.
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.
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.
In holonomic quantum computation, quantum gates are performed using driving protocols that trace out closed loops on the Bloch sphere, making them robust to certain pulse errors.However, dephasing noise that is transverse to the drive, which is significant in many qubit platforms, lies outside the family of correctable errors.Here, we present a general procedure that combines two types of geometry-holonomy loops on the Bloch sphere and geometric space curves in three dimensions-to design gates that simultaneously suppress pulse errors and transverse noise errors.We demonstrate this doubly geometric control technique by designing explicit examples of single-qubit and two-qubit dynamically corrected holonomic gates.
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.
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.