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PHYSICAL REVIEW A 105, 022437 (2022)One- and two-qubit gate inf i delities due to motional errors in trapped ions and electronsR. Tyler Sutherland , 1,* Qian Yu , 2,3 Kristin M. Beck , 4 and Hartmut Häffner 2,3,51 Department of Electrical and Computer Engineering, Department of Physics and Astronomy, University of Texasat San Antonio, San Antonio, Texas 78249, USA2 Physics Department, University of California, Berkeley, California 94720, USA3 Challenge Institute for Quantum Computation, University of California, Berkeley, California 94720, USA4 Lawrence Livermore National Laboratory, 7000 East Avenue Livermore, California 94550, USA5 Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA(Received 2 November 2021; accepted 14 February 2022; published 23 February 2022)In this work, we derive analytic formulas that determine the effect of error mechanisms on one- and two-qubitgates in trapped ions and electrons. First, we analyze and derive expressions for the effect of driving f i eldinhomogeneities on one-qubit gate f i delities. Second, we derive expressions for two-qubit gate errors, includingstatic motional frequency shifts, trap anharmonicities, f i eld inhomogeneities, heating, and motional dephasing.We show that, for small errors, each of our expressions for inf i delity converges to its respective numericalsimulation; this shows that our formulas are suff i cient for determining error budgets for high-f i delity gates,obviating numerical simulations in future projects. All of the derivations are general to any internal qubit state,and any mixed state of the ion crystal’s motion that is diagonal in the Fock state basis. Our treatment of staticmotional frequency shifts, trap anharmonicities, heating, and motional dephasing apply to both laser-based andlaser-free gates, while our treatment of f i eld inhomogeneities applies to laser-free systems.DOI: 10.1103/PhysRevA.105.022437I. INTRODUCTIONThe highest-f i delity quantum computing gates are, atpresent, performed with trapped ions [1–3]. This, in com-bination with long coherence times, inherent uniformity,and all-to-all connectivity, is why trapped ions are one ofthe most promising quantum computing platforms to date[4–10]. The most common method for performing high-f i delity gates is to couple the internal states of the ions usinglasers. While laser-based gates have many advantages, strongspin-motion coupling, for example, they suffer from photonscattering and phase noise. Furthermore, the lasers necessaryfor high-f i delity gates are expensive and diff i cult to calibrate.Laser-free gates, however, offer a promising alternative to thisparadigm, where laser f i elds are replaced with microwavesthat directly couple internal states of the ions [1,10–17]. First,the use of microwave f i elds eliminates photon scattering. Sec-ond, the phase and amplitude of microwave f i elds are easierto control, thereby reducing decoherence due to noisy drivingf i elds, which is often a limiting factor in laser-based gates.However, the relatively slow gate times of microwave gates(compared with laser-based gates) exaggerate the effects ofmotional decoherence. Trapped electrons, albeit signif i cantlyless explored than ions, are another promising qubit plat-form and will likely have gate operations similar to thosein laser-free trapped ion setups [18–20]. Due to their lightmass, we expect trapped electrons will operate on much fastertimescales relative to laser-free trapped ion experiments. Un-fortunately, because sideband cooling is not possible, trapped* robert.sutherland@utsa.eduelectrons will have to operate at much higher temperaturesthan trapped ions, and, therefore, are likely to be sensitive tomotional decoherence as well.High-f i delity gates are critical for fault-tolerant quantumcomputation, which requires inf i delities ranging from 10 −2to 10 −4 [21], making it important to quantify sources of in-f i delity. While many error sources are general to trapped ion(and would be trapped electron) experiments, their effect ongate f i delity is typically calculated numerically; this leads toduplicate computational effort between research groups. Fur-thermore, it is diff i cult to determine how gate f i delities scalewith various experimental parameters (such as temperature)when working only with numerical simulations. In this work,we aim to ameliorate these issues by deriving analytic formu-las for likely sources of motional decoherence in trapped ionsand electrons. The formulas we derive make no assumptionsabout the initial qubit state of the ion and assume only thatthe motion is in an incoherent mixed state, diagonal in theFock state basis. We also assume that the qubit frequency isvery different from the motional frequency of all relevant mo-tional modes. We then compare every formula to its respectivenumerical simulation, showing that the two calculations con-verge in the high-f i delity limit for each source of inf i delity. Inshort, this work aims to expedite the formulation of error bud-gets in future experiments, providing analytic formulas wherenumerical simulations were needed. Moreover, the derivationsprovide insight into each error mechanism and show how theyscale with relevant experimental parameters. In this work, wefocus on static motional frequency shifts, heating, trap anhar-monicities, and motional dephasing, which are major sourcesofinf i delityinalltrappediontwo-qubitgates.Wealsoexploretheeffectsoff i eldinhomogeneitiesonone-andtwo-qubitgatef i delities, which are specif i c to laser-free systems.2469-9926/2022/105(2)/022437(13) 022437-1 ©2022 American Physical Society