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arXiv:2204.02876 · Physical Review Applied

Designing Globally Time-Optimal Entangling Gates Using Geometric Space Curves

Ho Lun Tang, Kyle Connelly, Ada Warren, Fei Zhuang, Sophia E. Economou, Edwin Barnes

Published Apr 28, 2023• 12 citations
arXivHTMLPDF

Abstract

High-fidelity entangling gates are essential for quantum computation. Currently, most approaches to designing such gates are based either on simple, analytical pulse waveforms or on ones obtained from numerical optimization techniques. In both cases, it is typically not possible to obtain a global understanding of the space of waveforms that generate a target gate operation, which can make it challenging to identify globally time-optimal waveforms that respect amplitude and bandwidth constraints. Here, we show that in the case of weakly coupled qubits, it is possible to find all pulses that implement a target entangling gate in near-minimal time. We do this by mapping quantum evolution onto geometric space curves. We derive the minimal conditions these curves must satisfy in order to guarantee a gate with a desired entangling power is implemented. Pulse waveforms are extracted from the curvatures of these curves. We illustrate our method by designing fast, cnot-equivalent entangling gates for silicon quantum dot spin qubits with fidelities exceeding 99 %. We show that fidelities can be further improved while maintaining low bandwidth requirements by using geometrically derived pulses as initial guesses in numerical optimization routines.

DOI
10.1103/physrevapplied.19.044094

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High-fidelity entangling gates are essential for quantum computation. Currently, most approaches to designing such gates are based either on simple, analytical pulse waveforms or on ones obtained from numerical optimization techniques. In both cases, it is typically not possible to obtain a global understanding of the space of waveforms that generate a target gate operation, which can make it challenging to identify globally time-optimal waveforms that respect amplitude and bandwidth constraints. Here, we show that in the case of weakly coupled qubits, it is possible to find all pulses that implement a target entangling gate in near-minimal time. We do this by mapping quantum evolution onto geometric space curves. We derive the minimal conditions these curves must satisfy in order to guarantee a gate with a desired entangling power is implemented. Pulse waveforms are extracted from the curvatures of these curves. We illustrate our method by designing fast, cnot-equivalent entangling gates for silicon quantum dot spin qubits with fidelities exceeding 99 %. We show that fidelities can be further improved while maintaining low bandwidth requirements by using geometrically derived pulses as initial guesses in numerical optimization routines.
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 .
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.
The speed of quantum evolution is limited under finite energy resources. While most quantum speed limits (QSLs) are formulated in terms of quantum states, they can be extended to the evolution operator itself, and thus impose fundamental limits on how quickly logical gate operations can be implemented on a quantum computer. Here, we derive a general, tight QSL that holds for any unitary evolution under the constraint that the spectral width of the Hamiltonian is bounded. We apply this result to obtain QSLs for several standard quantum gates, including Hadamard, CNOT, and Toffoli gates, finding that the QSL can vary significantly across different gates, including ones with the same entangling power. These findings can be understood geometrically using the Space Curve Quantum Control formalism, which maps unitary evolution to space curves in Euclidean space. In this formalism, the problem of finding QSLs is recast as the problem of finding minimal-length curves obeying a curvature bound. We find that time-optimal gates map to helices of varying dimensions, and that QSLs can be understood from the perspective of a bottleneck principle in which the operator that evolves the slowest governs the minimal gate time.
Implementing high-fidelity two-qubit gates in single-electron spin qubits in silicon double quantum dots is still a major challenge. In this work, we employ analytical methods to design control pulses that generate high-fidelity entangling gates for quantum computers based on this platform. Using realistic parameters, we present simple control pulses that generate CNOT, CPHASE, and CZ gates with average fidelities greater than 99.99% and gate times as short as 45 ns. Moreover, using the local invariants of the system's evolution operator, we show that a simple square pulse generates a CNOT gate in less than 27 ns and with a fidelity greater than 99.99%. Last, we use the same analytical methods to generate two-qubit gates locally equivalent to CNOT\sqrt{CNOT}CNOT​ and CZ\sqrt{CZ}CZ​ that are used to implement simple two-piece pulse sequences that produce high-fidelity CNOT and CZ gates in the presence of low-frequency noise.
Implementing high-fidelity two-qubit gates in single-electron spin qubits in silicon double quantum dots is still a major challenge. In this work we employ analytical methods to design control pulses that generate high-fidelity entangling gates for quantum computers based on this platform. Using realistic parameters and initially assuming a noise-free environment, we present simple control pulses that generate cnot, cphase, and cz gates with average fidelities greater than 99.99% and gate times as short as 45 ns. Moreover, using the local invariants of the system's evolution operator, we show that a simple square pulse generates a cnot gate in less than 27 ns and with a fidelity greater than 99.99%. Last, we use the same analytical methods to generate two-qubit gates locally equivalent to CNOT\sqrt{\mathrm{CNOT}}CNOT​ and CZ\sqrt{\mathrm{CZ}}CZ​ that are used to implement simple two-piece pulse sequences that produce high-fidelity cnot and cz gates in the presence of low-frequency noise.
Quantum optimal control methods are widely used to design experimental control pulses such as laser amplitudes, phases, or detunings, that implement a target unitary evolution. In practice, what makes a pulse "good" depends not only on its fidelity, but also on the experimental setting and the relevant hardware constraints. Here, we introduce geometric quantum control with kernel optimisation (GECKO), a model-agnostic method for improving control pulses after a high-fidelity solution has been found. GECKO uses the Riemannian geometry of the special unitary group to identify directions in pulse space that leave the implemented unitary unchanged to first order, allowing one to traverse level sets of the control landscape while optimising a chosen differentiable pulse-quality function. We demonstrate GECKO on a transverse-field Ising Hamiltonian implementing CZ and CNOT gates, optimising pulse properties including spectral filtering, smoothness, robustness to parameter deviations, and pulse duration. In all cases, GECKO finds substantially improved pulse solutions.