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arXiv:2112.05447 · Physical Review A

Analytical and experimental study of center-line miscalibrations in Mølmer-Sørensen gates

Fernando Martínez-García, Lukas Gerster, Davide Vodola, Pavel Hrmo, Thomas Monz, Philipp Schindler, Markus Müller

Published Mar 23, 2022• 9 citations
arXivHTMLPDF

Abstract

A major challenge for the realization of useful universal quantum computers is achieving high fidelity two-qubit entangling gate operations. However, calibration errors can affect the quantum gate operations and limit their fidelity. To reduce such errors it is desirable to have an analytical understanding and quantitative predictions of the effects that miscalibrations of gate parameters have on the gate performance. In this work, we study a systematic perturbative expansion in miscalibrated parameters of the M\o{}lmer-S\o{}rensen entangling gate, which is widely used in trapped-ion quantum processors. Our analytical treatment particularly focuses on systematic center-line detuning miscalibrations. Via a unitary Magnus expansion, we compute the gate evolution operator, which allows us to obtain relevant key properties such as relative phases, electronic populations, quantum state purity and fidelities. These quantities, subsequently, are used to assess the performance of the gate using the fidelity of entangled states as performance metric. We verify the predictions from our model by benchmarking them against measurements in a trapped-ion quantum processor. The method and the results presented here can help design and calibrate high-fidelity gate operations of large-scale quantum computers.

DOI
10.1103/physreva.105.032437

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A major challenge for the realization of useful universal quantum computers is achieving high fidelity two-qubit entangling gate operations. However, calibration errors can affect the quantum gate operations and limit their fidelity. To reduce such errors it is desirable to have an analytical understanding and quantitative predictions of the effects that miscalibrations of gate parameters have on the gate performance. In this work, we study a systematic perturbative expansion in miscalibrated parameters of the M\o{}lmer-S\o{}rensen entangling gate, which is widely used in trapped-ion quantum processors. Our analytical treatment particularly focuses on systematic center-line detuning miscalibrations. Via a unitary Magnus expansion, we compute the gate evolution operator, which allows us to obtain relevant key properties such as relative phases, electronic populations, quantum state purity and fidelities. These quantities, subsequently, are used to assess the performance of the gate using the fidelity of entangled states as performance metric. We verify the predictions from our model by benchmarking them against measurements in a trapped-ion quantum processor. The method and the results presented here can help design and calibrate high-fidelity gate operations of large-scale quantum computers.
The performance of quantum gate operations is experimentally determined by how correctly operational parameters can be determined and set, and how stable these parameters can be maintained.In addition, gates acting on different sets of qubits require unique sets of control parameters.Thus, an efficient multidimensional parameter estimation procedure is crucial to calibrate even medium-sized quantum processors.Here, we develop and characterize an efficient calibration protocol to automatically estimate and adjust experimental parameters of the widely used two-qubit Mølmer-Sørensen entangling gate operation in a trapped-ion quantum information processor.The protocol exploits Bayesian parameter estimation methods that include a stopping criterion based on a desired gate infidelity.We experimentally demonstrate a tune-up procedure that leads to a residual median gate infidelity due to miscalibration of 1.3(1) × 10 -3 , requiring 1200 ± 500 experimental cycles, while completing the entire gate calibration procedure in less than one minute, which provides a significant speedup over commonly used manual tune-up routines.This approach is applicable to other quantum information processor architectures with known or sufficiently characterized theoretical models.
High-fidelity two-qubit entangling gates are essential for the realization of useful quantum algorithms on quantum processors. The Mølmer-Sørensen (MS) gate has become a common choice for trapped-ion quantum computing due to its resilience to ion temperature and its demonstrated record fidelities. However, the spectral impurity of the driving laser field impacts gate performance, with phase noise influencing the qubit dynamics through mechanisms operating on different timescales. In this work, we present a comprehensive theoretical analysis of laser phase noise in the MS gate, identifying two spectral ranges which influence the gate fidelity the most: "fast" noise at frequencies near the motional mode spectrum, and "slow" noise at frequencies on the order of the inverse gate time. We derive the noise Hamiltonians for two common laser beam geometries and obtain analytical expressions for the average gate error in terms of the laser noise power spectral density and gate parameters. For slowly varying noise spectra, we provide simplified error estimates. In addition, we validate our findings against previously published numerical simulations.
Continuously parameterized two-qubit gates are a key feature of state-of-the-art trapped-ion quantum processors as they have favorable error scalings and show distinct improvements in circuit performance over more restricted maximally entangling gatesets. In this work, we provide a comprehensive and pedagogical discussion on how to practically implement these continuously parameterized Mølmer Sørensen gates on the Quantum Scientific Computing Open User Testbed (QSCOUT), a low-level trappedion processor. To generate the arbitrary entangling angles, θ, we simply scale the amplitude of light used to generate the entanglement. However, doing so requires careful consideration of amplifier saturation as well as the variable light shifts that result. As such, we describe a method to calibrate and cancel the dominant fourth-order effects, followed by a dynamic virtual phase advance during the gate to cancel any residual light shifts, and find a linear scaling between θ and the residual light shift. Once, we have considered and calibrated these effects, we demonstrate performance improvement with decreasing θ. Finally, we describe nuances of hardware control to transform the XX-type interaction of the arbitraryangle Mølmer-Sørensen gate into a phase-agnostic and crosstalk-mitigating ZZ interaction.
High-fidelity two-qubit entangling gates play an important role in many quantum information processing tasks and are a necessary building block for constructing a universal quantum computer. Such high-fidelity gates have been demonstrated on trapped-ion qubits; however, control errors and noise in gate parameters may still lead to reduced fidelity. Here we propose and demonstrate a general family of two-qubit entangling gates which are robust to different sources of noise and control errors. These gates generalize the renowned Mølmer-Sørensen gate by using multitone drives. We experimentally implemented several of the proposed gates on ^{88}Sr^{+} ions trapped in a linear Paul trap and verified their resilience.
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.
Abstract To achieve the entangling gate fidelity above the quantum error correction threshold, it is critical to suppress errors due to experimental imperfection. We consider the Mølmer-Sørensen gates in trapped-ion systems, and develop a general approach to suppress a family of noise sources that appeared as either symmetric or asymmetric errors. Using the time-average displacement minimization technique, both symmetric error and displacement-dependent part of the asymmetric errors are eliminated. Then, by analyzing the tangent space of displacement- independent errors, we obtain the analytic form of the generators of the correction operator to the remaining error terms. We then develop a compensation pulse to fully suppress the remaining displacement-independent errors. The effectiveness of our scheme is further verified by numerical analysis, through which we observe a significant reduction of entangling gate infidelity. Our findings enhance gate fidelity and robustness to noise for ion trap quantum computing.