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arXiv:2412.17789 · Physical Review A

Power-optimized amplitude modulation for robust trapped-ion entangling gates: A study of gate-timing errors

Luke Ellert-Beck, Wenchao Ge

Published Jun 17, 2025• 1 citation
arXivHTMLPDF

Abstract

Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the M\o{}lmer-S\o{}rensen protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM, using a Fourier series expansion to maintain the generality of the laser amplitude's functional form. We then apply this general AM method to gate-timing errors by imposing conditions on these Fourier coefficients, producing trade-offs between the laser power and fidelity at a fixed gate time. The conditions derived here are linear and can be used, in principle, to achieve arbitrarily high orders of insensitivity to gate-timing errors. Numerical optimization is then employed to identify the minimum-power pulse satisfying these constraints. Our central result is that the leading-order dependence on gate-timing errors is improved from $O(\mathrm{\ensuremath{\Delta}}{t}^{2})$ to $O(\mathrm{\ensuremath{\Delta}}{t}^{6})$ with the addition of one linear constraint on the Fourier coefficients and to $O(\mathrm{\ensuremath{\Delta}}{t}^{10})$ with two linear constraints without a significant increase in the average laser power. The increase approaches zero as more Fourier coefficients are included. In further studies, this protocol can be applied to other error sources and used in conjunction with other error-mitigation techniques to improve two-qubit gates.

DOI
10.1103/physreva.111.062422

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Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the M\o{}lmer-S\o{}rensen protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM, using a Fourier series expansion to maintain the generality of the laser amplitude's functional form. We then apply this general AM method to gate-timing errors by imposing conditions on these Fourier coefficients, producing trade-offs between the laser power and fidelity at a fixed gate time. The conditions derived here are linear and can be used, in principle, to achieve arbitrarily high orders of insensitivity to gate-timing errors. Numerical optimization is then employed to identify the minimum-power pulse satisfying these constraints. Our central result is that the leading-order dependence on gate-timing errors is improved from O(\ensuremathΔt2)O(\mathrm{\ensuremath{\Delta}}{t}^{2})O(\ensuremathΔt2) to O(\ensuremathΔt6)O(\mathrm{\ensuremath{\Delta}}{t}^{6})O(\ensuremathΔt6) with the addition of one linear constraint on the Fourier coefficients and to O(\ensuremathΔt10)O(\mathrm{\ensuremath{\Delta}}{t}^{10})O(\ensuremathΔt10) with two linear constraints without a significant increase in the average laser power. The increase approaches zero as more Fourier coefficients are included. In further studies, this protocol can be applied to other error sources and used in conjunction with other error-mitigation techniques to improve two-qubit gates.
Almost every quantum circuit is built with two-qubit gates in the current stage, which are crucial to the quantum computing in any platform. The entangling gates based on Mølmer-Sørensen schemes are widely exploited in the trapped-ion system, with the utilization of the collective motional modes of ions and two laser-controlled internal states, which are served as qubits. The key to realize high-fidelity and robust gates is the minimization of the entanglement between the qubits and the motional modes under various sources of errors after the gate operation. In this work, we propose an efficient numerical method to search high-quality solutions for phase-modulated pulses. Instead of directly optimizing a cost function, which contains the fidelity and the robustness of the gates, we convert the problem to the combination of linear algebra and the solution to quadratic equations. Once a solution with the gate fidelity of 1 is found, the laser power can be further reduced while searching on the manifold where the fidelity remains 1. Our method largely overcomes the problem of the convergence and is shown to be effective up to 60 ions, which suffices the need of the gate design in current trapped-ion experiments.
Two-qubit gates in trapped-ion quantum computers are generated by applying spin-dependent forces that temporarily entangle the internal state of the ion with its motion. Laser pulses are carefully designed to generate a maximally entangling gate between the ions while minimizing any residual entanglement between the motion and the ion. The quality of the gates suffers when the actual experimental parameters differ from the ideal case. Here, we improve the robustness of frequency-modulated M\o{}lmer-S\o{}rensen gates to motional mode-frequency offsets by optimizing the average performance over a range of systematic errors using batch optimization. We then compare this method with frequency-modulated gates optimized for ideal parameters that include an analytic robustness condition. Numerical simulations show good performance up to 12 ions, and the method is experimentally demonstrated on a two-ion chain.
Trapped ion hardware has made significant progress recently and is now one of the leading platforms for quantum computing. To construct two-qubit gates in trapped ions, experimental manipulation approaches for ion chains are becoming increasingly prevalent. Given the restricted control technology, how implementing high-fidelity quantum gate operations is crucial. Many works in current pulse design optimization focus on ion-phonon and effective ion-ion coupling while ignoring the higher-order expansion impacts of these two terms brought on by experiment defects. This paper proposed a novel robust quantum control optimization method in trapped ions. By introducing the higher-order terms caused by the error into the optimization cost function, we generated an extremely robust Molmer-Sorensen gate with infidelity below 10−310^{-3}10−3 under drift noise range ±10\pm 10±10 kHz and time noise range ±0.02\pm 0.02±0.02. Our work reveals the vital role of higher-order coupling terms in trapped ion pulse control optimization, especially the higher ion-ion coupling order, and provides a robust optimization scheme for realizing more efficient entangled states in trapped ion platforms.
Abstract Trapped ion hardware has made significant progress recently and is now one of the leading platforms for quantum computing. To construct two-qubit gates in trapped ions, experimental manipulation approaches for ion chains are becoming increasingly prevalent. Given the restricted control technology, how implementing high-fidelity quantum gate operations is crucial. Many works in current pulse design optimization focus on ion–phonon and effective ion–ion couplings while ignoring the first-order derivative terms expansion impacts of these two terms brought on by experiment defects. This paper proposes a novel robust quantum control optimization method in trapped ions. By introducing the first-order derivative terms caused by the error into the optimization cost function, we generate an extremely robust Mølmer–Sørensen gate with infidelity below 10−3 under a drift noise range of ±10 kHz, the relative robustness achieves a tolerance of ±5%, compared to the 200-kHz frequency spacing between phonon modes, and for time noise drift, the tolerance reached to 2%. Our work reveals the vital role of the first-order derivative terms of coupling in trapped ion pulse control optimization, especially the first-order derivative terms of ion–ion coupling. It provides a robust optimization scheme for realizing more efficient entangled states in trapped ion platforms.
As quantum computing scales up, circuits relying on sequential two-qubit gates face severe challenges from excessive depth and decoherence. In trapped-ion systems, global entangling gates offer a highly parallel alternative but are hindered by complex numerical optimization. We propose an efficient numerical framework for piecewise-constant amplitude-modulated global gates. Based on the Mølmer-Sørensen protocol, we utilize null-space projection for linear constraints of spin-motion coupling and tangent-space gradient descent to minimize laser power, while achieving high-order robustness against frequency drifts. At the application level, we identify a structural correspondence between hardware-native global interactions and Pauli strings in Hamiltonian simulation and propose an efficient compilation scheme. By reducing common physical models to fundamental X X interactions and formulating interaction graph partitioning as a Set Cover problem, we leverage global gates to significantly optimize the compilation process. Explicit pulse-level benchmarks further show that, for the tested weighted interaction graphs, the reduction in entangling-gate count can translate into a substantial reduction in total entangling duration. These results demonstrate that integrating hardware-native operations with tailored compilation strategies can substantially reduce circuit overhead, providing a practical roadmap for the codesign of near-term quantum simulators.
To implement a scalable and fault-tolerant programmable quantum computing architecture, the realization of high-fidelity, robust, and efficient quantum entangling gates in a multiqubit system is an essential requirement. In this paper, we consider the optimization of the segmented amplitude-modulated entangling gates realized by applying spin-dependent forces to a chain of ions trapped in a hybrid quadratic and quartic potential. We propose a geometrical hypersurface-tangent method to optimize the laser power and the pulse gradient, carefully considering the experimental feasibility in realizing the effective Rabi frequency and its change between adjacent pulse segments. In addition, our method allows us to optimize the solution for the case of a few segments by constructing an approximated null space through trading a negligible amount of gate fidelity. Finally, we show that the present scheme can provide a unified framework to improve the robustness to an arbitrary order against random static drifts of the motional frequencies, the gate duration errors, the laser detuning drifts, and their simultaneous drifts.