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arXiv:2210.02372

Frequency-robust Mølmer-Sørensen gates via balanced contributions of multiple motional modes

Brandon Ruzic, Matthew N. H. Chow, Ashlyn D. Burch, Daniel Lobser, Melissa Revelle, Wilson, Joshua M., Christopher G. Yale, Susan Clark

Published Oct 5, 2022• 0 citations
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Abstract

In this work, we design and implement frequency-robust Molmer-Sorensen gates on a linear chain of trapped ions, using Gaussian amplitude modulation and a constant laser frequency. We select this frequency to balance the entanglement accumulation of all motional modes during the gate to produce a strong robustness to frequency error, even for long ion chains. We demonstrate this technique on a three-ion chain, achieving $<\,1\%$ reduction from peak fidelity over a $20\,$kHz range of frequency offset, and we analyze the performance of this gate design through numerical simulations on chains of two to 33 ions.

DOI
10.48550/arxiv.2210.02372

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In this work, we design and implement frequency-robust Molmer-Sorensen gates on a linear chain of trapped ions, using Gaussian amplitude modulation and a constant laser frequency. We select this frequency to balance the entanglement accumulation of all motional modes during the gate to produce a strong robustness to frequency error, even for long ion chains. We demonstrate this technique on a three-ion chain, achieving < 1%<\,1\%<1% reduction from peak fidelity over a 20 20\,20kHz range of frequency offset, and we analyze the performance of this gate design through numerical simulations on chains of two to 33 ions.
Near-term quantum algorithms require high-performance entangling gates in a scalable quantum processor, and M\o{}lmer-S\o{}rensen (MS) gates on trapped ions have achieved some of the highest entanglement fidelities to date. However, technical noise that induces small changes in the motional frequencies of the ion chain is a dominant error source in these gates that limits scalability. Here, we report an improved MS-gate design that mitigates the impact of motional-frequency changes by using a class of laser waveforms with three salient features: a spectrally compact Fourier transform, a constant frequency that balances contributions to the MS-gate rotation angle from all motional modes, and a simple parameterization. Together, these features suppress residual coherent displacement of the motional modes, generate robustness to rotation-angle error, and provide low technical complexity and computational overhead. As a proof of concept, we implement this gate design on a four-ion chain, achieving a $<1\mathrm{%}$ fidelity reduction over a $\ifmmode\pm\else\textpm\fi{}10$ kHz range of frequency offset from the calibrated detuning. We also analyze the applicability of our gate design to longer ion chains through numerical simulations of our gate design on chains of up to 33 ions. This robustness to motional-frequency changes represents an order-of-magnitude improvement over the more complicated waveform-shaping techniques that currently dominate the robust-gate literature, and our simulations predict that a high level of robustness can be maintained in these systems for longer ion chains. In addition, the robustness and simplicity of our approach promises to accelerate the development of quantum architectures that invoke long algorithmic runtimes and frequent calibration.
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.
High-fidelity two-qubit gates in quantum computers are often hampered by fluctuating experimental parameters. The effects of time-varying parameter fluctuations lead to coherent noise on the qubits, which can be suppressed by designing control signals with appropriate filter functions. Here, we develop filter functions for M\o{}lmer-S\o{}rensen gates of trapped-ion quantum computers that accurately predict the change in gate error due to small parameter fluctuations at any frequency. We then design the filter functions of frequency-modulated laser pulses, and compare this method with pulses that are robust to static offsets of the motional-mode frequencies. Experimentally, we measure the noise spectrum of the motional modes and use it for designing the filter functions, which improves the gate fidelity from 99.23(7)% to 99.55(7)% in a five-ion chain.
We analyze the impact of correlated laser frequency noise on Mølmer-Sørensen gates in qubit registers based on trapped-ion chains. Using perturbation theory, we calculate gate fidelities in the presence of noise with arbitrary power spectral density for different chain lengths and ion positions in the chain. With our approach, we account for simultaneous excitation of multiple phonon modes during gate operation. We find out that the impact of medium-frequency laser noise depends considerably on the positions of the ions in the chain. In contrast, low-frequency noise has similar effect for different chain lengths and ion positions.
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.
The Mølmer-Sørensen (MS) gate is a two-qubit controlled-phase gate in ion traps that is highly valued due to its ability to preserve the motional state of the ions. However, its fidelity is obstructed by errors affecting the motion of the ions as well as the rotation of the qubits. In this work, we propose an amplitude-modulated composite MS gate which features high fidelity robust to gate timing, detuning and coupling errors and is also tolerant of a.c. Stark shifts and drifting detuning errors.