Fast, robust, and high-fidelity quantum gates for trapped particle quantum computers

Abstract

The present disclosure relates to quantum computing related methods, systems, and devices for determining a phase modulation function (Φ(t)) for a laser pulse implementing an arbitrary quantum gate for a qubit implemented by a trapped particle, comprising obtaining a Rabi frequency (Ω), and a trap frequency (ω) for the laser pulse and the trapped particle, determining, based on the Rabi frequency (Ω) and the trap frequency (ω), the phase modulation function (Φ(t)) for the laser pulse such as to minimize an effect of a photon recoil of the trapped particle induced by said laser pulse on a gate fidelity of the quantum gate, as well as to a corresponding method for performing an arbitrary quantum gate as well as to a system for modulating laser pulses used to implement one or more quantum gates for a subset of qubits in a quantum register, and to a quantum computer.

Description

FAST, ROBUST, AND HIGH-FIDELITY QUANTUM GATES FOR TRAPPED PARTICLE QUANTUM COMPUTERSFIELD OF INVENTION

[0001] Aspects of the present disclosure relate to methods, systems, and apparatuses configured to suppress decoherence effects caused by photon recoil, imperfect vibrational state preparation and / or spatial inhomogeneities of qubit laser pulses etc.) in trapped particle (e.g., atoms, molecules, ions, etc.) quantum computing systems by - inter alia - temporally modulating the phase of a laser beam / laser pulses implementing arbitrary quantum gates. The present disclosure is particularly relevant for neutral atom-based optical qubits, which may suffer from recoil-induced decoherence, fundamentally limiting gate fidelity and / or fast operation of quantum gates. Aspects of the present disclosure allow to overcome such limitations at least partially, and to design fast, robust (e.g., against spatial inhomogeneities of gate lasers, or microwave radiation, etc.), and high-fidelity qubit gates that are, in some implementations, ~ to to 20 times faster than the state-of-the-art while maintaining or even increasing gate fidelity and / or gate robustness.INTRODUCTION

[0002] Quantum computers may outperform classical computing devices in specific applications such as data base search, finding the prime factors of an integer, quantum chemistry, etc. Quantum computers use physical qubits to store the basic unit of information and perform quantum gates on the qubits to process the stored information, e.g. according to processing instructions of a quantum computing algorithm. Running quantum algorithms typically requires performing single- and two- qubit gates, which are the basic computation operations acting on individual qubits and individual pairs or sets of qubits, respectively.

[0003] Neutral atom-based quantum computers, for example, typically trap neutral atoms in optical traps (e.g. in arrays of optical dipole traps or in optical lattices or similar traps), and two internal states, e.g., long-lived (with respect to the gate time) and / or external states of the atoms can be used to form the qubits of a quantum register.

[0004] In such systems, single-qubit gates can be implemented via, e.g., optical or microwave, transitions between two stable or metastable internal states of the neutral atoms, e.g. between two hyperfine states of the electronic ground state coupled by a microwave transition or between an electronic ground state coupled via laser radiation to an electronically excited, long-lived state of the atom (e.g., the well-known clock states of88Sr or similar atoms). Two- and multi-qubit gates, such as a controlled phase gate can be realized, e.g., using optical transitions to highly excited Rydberg states (for a comprehensive overview see M. Morgado and S. Whitlock, Quantum simulation and computing with Rydberg-interacting qubits; AVS Quantum Sci. 3, 023501 (2021)) that allow for qubit-qubit interaction and entanglement in the optical trap array.

[0005] Details on neutral atom quantum computers, quantum register formation and operation, as well as associated quantum gate protocols are described in applicants own PCT / EP2024 / 066941, titled HARDWARE-EFFICIENT NEUTRAL ATOM QUANTUM COMPUTING METHOD AND DEVICE, PCT / EP2023 / 074768, titled APPARATUS AND METHOD FOR TRAPPING AND MANIPULATING LARGE NUMBERS OF INDIVIDUAL NEUTRAL ATOMS, and PCT / EP2023 / 074768 titled APPARATUS AND METHOD FOR CONTINUOUS LOADING OF NEUTRAL ATOM REGISTERS, which, for conciseness, are incorporated herein in their entirety. The following prior art references further illustrate the technological background of the present disclosure:[1] K. Wintersperger, F. Dommert, T. Ehmer, A. Hoursanov, J. Klepsch, W. Mauerer, G. Reuber, T. Strohm, M. Yin, and S. Luber, “Neutral atom quantum computing hardware: Performance and end-user perspective,” EPJ Quantum Technology, vol. 10, pp. 1-26, Dec. 2023.[2] L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, “Quantum computing with neutral atoms,” Quantum, vol. 4, p. 327, Sept. 2020.[3] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, “Trapped-ion quantum computing: Progress and challenges,” Applied Physics Reviews, vol. 6, p. 021314, May 2019.[4] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys., vol. 75, pp. 281-324, Mar 2003.[5] A. M. Kaufman and K.-K. Ni, “Quantum science with optical tweezer arrays of ultracold atoms and molecules,” Nature Physics, vol. 17, no. 12, pp. 1324-1333, 2021.[6] J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Recoilless optical absorption and Doppler sidebands of a single trapped ion,” Phys. Rev. A, vol. 36, p. 428, 1987.[7] T. Ido and H. Katori, “Recoil-Free Spectroscopy of Neutral Sr Atoms in the Lamb- Dicke Regime,” Phys. Rev. Lett., vol. 91, p. 053001, 2003.[8] D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., “Logical quantum processor based on reconfigurable atom arrays,” Nature, vol. 626, no. 7997, pp. 58-65, 2024.[9] S. Pucher, V. KTusener, F. Spriestersbach, J. Geiger, A. Schindewolf, I. Bloch, and S. Blatt, “Fine-structure qubit encoded in metastable strontium trapped in an optical lattice,” Phys. Rev. Lett., vol. 132, p. 150605, Apr 2024.

[0010] I. S. Madjarov, A. Cooper, A. L. Shaw, J. P. Covey, V. Schkolnik, T. H. Yoon, J. R. Williams, and M. Endres, “An atomic-array optical clock with single-atom readout,” Phys. Rev. X, vol. 9, p. 041052, Dec 2019.

[0011] A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, P. Scholl, T. H. Yoon, J. Choi, and M. Endres, “Multi-ensemble metrology by programming local rotations with atom movements,” Nature Physics, pp. 1-7, 2024.

[0012] M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, J. Ye, and A. M. Kaufman, “Seconds-scale coherence on an optical clock transition in a tweezer array,” Science, vol. 366, p. 93, oct 2019.

[0013] S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li,A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuleti'c, and M. D. Lukin, “High-fidelity parallel entangling gates on a neutral-atom quantum computer,” Nature, vol. 622, pp. 268-272, Oct. 2023.

[0014] F. Robicheaux, T. M. Graham, and M. Saffman, “Photon-recoil and laser-focusing limits to Rydberg gate fidelity,” Physical Review A, vol. 103, p. 022424, Feb. 2021.

[0015] I. Lizuain, J. G. Muga, and J. Eschner, “Motional frequency shifts of trapped ions in the Lamb-Dicke regime,” Physical Review A, vol. 76, p. 033808, Sept. 2007.

[0016] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano,B. Jelenkovi'c, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature, vol. 422, pp. 412-415, Mar. 2003.

[0017] K. Molmer and A. Sorensen, “Multiparticle Entanglement of Hot Trapped Ions,” Physical Review Letters, vol. 82, pp. 1835-1838, Mar. 1999.

[0018] C. Monroe, “Primer on Molmer-Sorensen Gates in Trapped Ions,”

[0019] H. Azuma, “A simplified Molmer-Sorensen gate for the trapped ion quantum computer,” Physica Scripta, vol. 99, p. 045107, Mar. 2024.

[0020] T. Choi, S. Debnath, T. A. Manning, C. Figgatt, Z.- X. Gong, L.-M. Duan, and C. Monroe, “Optimal quantum control of multi-mode couplings between trapped ionqubits for scalable entanglement,” Physical Review Letters, vol. 112, p. 190502, May 2014.

[0021] S.-L. Zhu, C. Monroe, and L.-M. Duan, “Arbitrary-speed quantum gates within large ion crystals through minimum control of laser beams,” Europhysics Letters, vol. 73, p. 485, Jan. 2006.

[0022] P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan, L.-M. Duan, and C. Monroe, “Phase Control of Trapped Ion Quantum Gates,” Journal of Optics B: Quantum and Semiclassical Optics, vol. 7, pp. S371-S383, Oct. 2005. Comment: 21 pages 12 figures.

[0023] P. Scholl, A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, J. Choi, and M. Endres, “Erasure-cooling, control, and hyper-entanglement of motion in optical tweezers,” arXiv preprint arXiv:23ii.i558o, 2023.

[0024] J. W. Lis, A. Senoo, W. F. McGrew, F. Ronchen, A. Jenkins, and A. M. Kaufman, “Midcircuit Operations Using the omg Architecture in Neutral Atom Arrays,” Physical Review X, vol. 13, p. 041035, Nov. 2023.

[0025] H. K. Cummins, G. Llewellyn, and J. A. Jones, “Tackling systematic errors in quantum logic gates with composite rotations,” Physical Review A, vol. 67, p. 042308, Apr. 2003.

[0026] G. Dridi, M. Mejatty, S. J. Glaser, and D. Sugny, “Robust control of a not gate by composite pulses,” Physical Review A, vol. 101, p. 012321, Jan. 2020.

[0027] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herb ruggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms,” Journal of Magnetic Resonance, vol. 172, pp. 296-305, Feb. 2005.

[0028] T. E. Skinner and N. I. Gershenzon, “Optimal control design of pulse shapes as analytic functions,” Journal of Magnetic Resonance, vol. 204, pp. 248-255, June 2010.

[0006] In addition, EP 4115352 Al relates to quantum computing technology using a nuclear spin of atoms to implement qubits. US 11,293,851 B2 relates to a quantum register of trapped atoms and seems to disclose mitigating a photon-recoil during a detection of the atomic state. In addition, WO 2021 / 178038 Al relates to a qubit implementation using hyper-fine structure of atoms. Further, WO 2023 / 130114 Al relates to correcting errors in quantum computation with trapped atoms mentioning a recoil effect during atom state detection for a qubit implementation using nuclear spin of atoms. Finally, WO2O19 / O14589 Al also relates to qubit implementations using hyper-fine structure of atoms.SUMMARY

[0007] One of the main challenges in neutral atom-based quantum computing is the execution of qubit gate operations on individual atoms (or ions, molecules, etc.) with high fidelity in a fast and robust manner for quantum register comprising a large number of qubits (e.g., > 1000). Typical single and two-qubit gate schemes may require the use of laser light (or MW radiation) with at least some degree of local spatial control, as well as some relative phase control between the atomic quantum states. Thus, typically, similar control is needed for laser fields at the positions of the qubits of the quantum register, used for the respective quantum gate operations. For example, known schemes for quantum computing with neutral atoms may use at least two different laser systems, each equipped with individual-atom addressing units for operating local single and two-qubit gates.

[0008] As briefly mentioned above, trapped atoms are currently among the most promising platforms for quantum information processing (see refs. [1-3]). These particles are typically confined within electromagnetic or optical fields, and lasers, microwaves, or radiofrequency signals are typically used to manipulate their atomic internal states for coherent control of quantum operations. A Hilbert space of such a trapped particle is thus composed of two subspaces: an (internal) qubit subspace, which is used for quantum computation, and a motional state subspace, which describes the spatial vibrations (confined motion) of the particle in the trap. The qubit subspace and the motional subspace maybe coupled, such that a vibration of the particle can cause a loss of qubit gate fidelity, as well as a loss of coherence due to entanglement (see Fig. 1 for details).

[0009] The coupling typically depends on / can be expressed through a parameter r| referred to as the Lamb-Dicke parameter which corresponds to the ratio between the recoil frequency shift (the energy an atom would have after emitting a photon, if initially at rest, divided by the reduced Planck constant h) and the trap frequency c (note: for low enough temperatures of the trapped particles the optical trapping potentials exhibit quantum harmonic oscillator behavior characterized by co). If a frequency corresponding to the recoil is large compared to the trap frequency a>, the emission or absorption of a photon by a trapped atom (or ion, molecule, etc.) will substantially populate motional excited states of the trapped particles and thereby cause decoherence, negatively affecting quantum gate fidelity.

[0010] Ideally, r| should tend to o, in which case the qubits essentially behave like a spin-1 / 2 particle while the motional states evolve independently (i.e., no entanglement) and can be factored out. To make r| very small, in theory, it is possible to make the trap frequency co very high but doing so in practice is limited due to power limitations of the trapping lasers, in particular for large quantum registers. Unlike trapped-ion hardware where ions vibrate at frequencies up to the MHz range (see ref. [4]), neutral atoms, which carry no electric charge, can only be confined by their AC polarizability in optical lattices or optical tweezers created by lasers with resulting trap frequencies of about too kHz or less (see ref. [5]).

[0011] In ref. [6], the authors demonstrate spectroscopy of individual trapped ions on the so-called carrier transition, which is largely insensitive to recoil. Their demonstration extends the work of Robert Dicke on narrowing spectroscopic lines and the work of Rudolf MbBbauer on recoil-free emission and absorption of gamma rays. The requirement for such recoil-free spectroscopy is that the Rabi frequency fl is much smaller than the trap frequency co in order to resolve motional sidebands. This requirement would translate into very slow quantum gates, if such ideas were directly applied to quantum computing with neutral atoms. For neutral atom quantum computing, it is thus important to realize recoil-free gates that operate much faster and, ideally, suppress the effect of photon recoil as good as possible.

[0012] In this context, the authors of ref. [7] extend the previous ideas by from trapped ions to neutral atoms trapped in an optical trap. They show that by choosing a suitable wavelength for the trapping laser, it is possible to realize a magic trappingcondition, where both states involved in an optical transition experience the same trapping potential. This magic trapping condition mimics the conditions realized in ref. [6] with trapped ions, which, being ions and not neutral atoms, naturally experience the same trapping potential regardless of their internal state. This idea of magic trapping is key for the implementation of so-called MbBbauer spectroscopy of an optical transition for trapped neutral atoms. The Lamb-Dicke parameter depends mainly on how the qubit gate, made by operating on specific states of the atom, is chosen. They can be categorized into hyperfine qubits (ref. [8]), fine-structure qubits (ref. [9]), and optical qubits (ref.

[0010] ).

[0013] The energy level differences of hyperfine qubits go up to the GHz range, and the effects of vibration are practically rendered negligible by using, e.g., Raman lasers applying two co-propagating waves to realize Raman transitions in such a way that the Lamb-Dicke parameter can be made extremely small. However, the energy difference between the states of an optical qubit corresponds to the energy of visible spectrum photons, and when manipulated, the vibrational excitations can be significant. For this reason, while hyperfine qubits can achieve fidelity levels of 99.99% under MHz-level driving (ref. [8]) without the need to apply a strategy to suppress recoil, optical qubits, e.g., implemented by88Sr atoms or similar elements can, to date, only achieve -99% gate fidelity at a kHz-level driving (ref.

[0011] ).

[0014] Further, for optical qubits, such performances are conventionally obtained by applying laser pulses with low Rabi frequency to excite more selectively the qubit transition. However, this typically involves slow gates, e.g. a few hundreds of ps for a88Sr atom. The motional excitations are thus a strong limitation particular for optical qubits. This is unfortunate because the optical states are particularly suitable for storing quantum information due to their very long coherence times, on the order of several seconds (ref.

[0012] ). For such optical qubits, typically having a natural linewidth of less than 10Hz, or preferably of less that 1Hz, on the optical qubit transition, ref.

[0024] relates to a composite pulse sequences that allow to perform a certain shelving operation with less heating.

[0015] In view of the above, finding / designing a fast, robust and recoil-free quantum gate scheme that can be applied to arbitrary quantum gates and / or arbitrary initial qubit states, preferably across large quantum registers are thus of primaryimportance to fully exploit the potential of optical qubits for advanced quantum computers.

[0016] As discussed in detail below, aspects of the present disclosure allow to efficiently suppress the photon recoil, errors due to laser inhomogeneity and / or mixed vibrational states using a combination of essentially three central ideas. First, the photon recoil is suppressed even using a high Rabi frequency on the optical transition with a simple pulse design, allowing one to perform qubit gates at least 15 times faster than the current state of the art. Second, it is possible to design very fast gates that are robust against inhomogeneities using an innovative pulse sequence involving local addressing lasers. The technique can be combined with the recoil-free gate method to design fast, robust and recoil-free qubit gates. The resulting gates being recoil-free, the entanglement is naturally mitigated, but some entanglement may remain since photonrecoil is not the only source of entanglement. Aspects of the present disclosure thus allow to mitigate entanglement even further by increasing the pulse duration.

[0017] More specifically, in a first aspect, the present disclosure relates to a method for determining a phase modulation function 4>(t) for a laser pulse implementing an arbitrary quantum gate for a qubit implemented by a trapped particle, preferably an arbitrary single qubit gate, comprising: obtaining a Rabi frequency 2, and a trap frequency co for the laser pulse and the trapped particle, and determining, based on the Rabi frequency £2 and the trap frequency co, the phase modulation function 4>(t) for the laser pulse such as to minimize an effect of a photon recoil of the trapped particle induced by said laser pulse on a gate fidelity of the quantum gate.

[0018] In some implementations, determining the phase modulation function (p (t) for the laser pulse may comprise defining, using the Rabi frequency £2 and the trap frequency co, a quantum control problem based on a Hamiltonian (for examples of such a Hamiltonian see Enclosure A, equation (1) and / or Enclosure B, equations (8) and (14)) describing the dynamics of the trapped particle when subjected to the laser pulse, and determining the phase modulation function < >(t) for the laser pulse, based on solving the quantum control problem, such that a gate infidelity measure associated with the photon recoil is minimized below a predefined threshold. As explained in more detail in Enclosure B, the control problem may be solved using a gradient algorithm based on GRAPE (ref.

[0027] ). For instance, the phase modulation function < >(t) maybediscretized in very small timesteps which can take any value (unconstrained problem). In addition, longer and longer pulse durations T can be used until a numerical solution of the control problem is found with a precision of ~ io-10or similar. For further details see Enclosure B, section Time-optimal recoil-free pulses.

[0019] As discussed with reference to Fig. 3 below, in some implementations, the quantum control problem may comprise a gate control condition corresponding to performing the quantum gate, and one or more phase control conditions corresponding to an effect of the photon recoil on gate fidelity. For example, defining the quantum control problem may comprise approximating the Hamiltonian for the dynamics of the trapped particle when subjected to the laser pulse as a power series in the Lamb-Dicke parameter r| up to at least a first order in q, preferably up to a second order in q, wherein an nthphase control condition of the quantum control problem corresponds to an nthorder term of the power series in q.

[0020] In this manner, a first phase control condition may correspond to a first order effect of the photon recoil, a second phase control condition may correspond to a second order effect of the photon recoil, and / or a third phase control condition may correspond to a vibration mode entanglement effect of the trapped particle.

[0021] In other implementations, determining the phase modulation function(t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse may comprise determining, based on a Hamiltonian describing the dynamics of the trapped particle when subjected to the laser pulse (for an example see equation (1) in Enclosure A), a cost function J, that depends, via the Hamiltonian, on the phase modulation function (t), and that quantifies a gate infidelity corresponding to the photon recoil induced by said laser pulse, and determining the phase modulation function 4>(t) by minimizing the cost function J below a predefined threshold or until a predefined halting condition is met.

[0022] For example, as explained in more detail in “Numerical optimization for a recoil-free pulse” of Enclosure A , determining the phase modulation function 4>(t) by minimizing the cost function J may comprise: selecting a basis representation, especially a basis comprising Fourier components or piecewise-constant functions, representation of the phase modulation function (f>( ), and, optionally, a regulation mask for avoiding discontinuities of the phase modulation function < >(t), and iterativelyadjusting a set of basis coefficients that constitute the weights of each basis component in the basis representation of (t) such that the cost function J is minimized below the predefined threshold or until the predefined halting condition is met.

[0023] In addition, to better understand which effects have a significant contribution on gate fidelity, in some embodiments, the cost function J may comprise a first contribution Juniquantifying a deviation of the quantum gate from a desired ideal, unitary quantum gate, and on or more second contribution J ent? Jmot quantifying an effect of the photon recoil on gate infidelity. Further, the cost function J may comprise a weighted sum of the first and the one or more second contributions, wherein the respective weights are selected such that one or more of the second contributions dominate by at least a factor of 5, preferably by a factor of 10 over the first contribution.

[0024] In order to determine fast and recoil free laser pulses, in some implementations, determining the phase modulation function < >(t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse may comprise determining a pulse duration and / or a pulse area for the laser pulse to be modulated by the phase modulation function < >(t). In this manner, so called time optimal recoil free (TORF) gates can be implemented (for further details see Enclosure B).

[0025] To render such quantum gates as discussed above also more robust against spatial laser inhomogeneities (e.g., caused by spatial laser amplitude variations of a gaussian laser pulse across the quantum register, or by residual electromagnetic fields locally shifting qubit resonance frequencies in the quantum register, or by a probe shift effect caused by a qubit state transition laser etc.), in some implementations, defining the quantum control problem or the cost function J may comprise including, into the Hamiltonian, an effect of a Rabi frequency deviation 60, and / or an effect of a laser frequency detuning 5A, and wherein determining the phase modulation function < >(t) for the laser pulse comprises determining the phase modulation function < >(t) for the laser pulse such that the photon recoil is minimized below a fist predefined threshold, and such that the effect of the Rabi frequency deviation 60, and / or the effect of the laser frequency detuning 5A on a quantum gate fidelity is minimized below a second predefined threshold.

[0026] To also address negative effect of imperfect motional ground state preparation in the quantum register, in some implementations, determining the phase modulation function < >(t) for the laser pulse may comprise determining a pulse duration and / or a pulse area of the laser pulse such that a quantum gate infidelity measure for a decoherence caused by vibrational entanglement is minimized below a third predefined threshold (for details see Enclosure A and Enclosure B).

[0027] A further aspect of the present disclosure relates to a method for performing an arbitrary quantum gate for a qubit implemented by a trapped particle, comprising determining a phase modulation function < >(t) for a laser pulse implementing the quantum gate such that a photon recoil of the trapped particle induced by said laser pulse is minimized, and applying the laser pulse to the trapped particle based on the determined phase modulation function < >(t). In some implementations, the phase modulation function < >(t) maybe determined carrying out the steps of the method discussed above and below with reference to Fig. 3. In this manner, quantum gate fidelity can be improved while also making the gate operation faster and more robust against inhomogeneities typically present in large quantum registers as discussed above.

[0028] A further aspect of the present disclosure relates to a method for implementing quantum gates on a subset of qubits in a quantum register, the quantum register being implemented by an array of trapped particles. The method comprises illuminating, each qubit in the subset, with one or more respective local laser pulses configured to induce one or more local z-rotations (e.g., in a Hilbert space spanned by a qubit ground state ground state |0) and a qubit excited state 1 1)) for each illuminated qubit based on inducing local light shifts caused by the local laser pulses (for further details see applicants own PCT / EP2024 / 066941, incorporated herein in its entirety), and illuminating, each qubit in the subset, with one or more global spatially inhomogeneous laser pulses configured to induce one or more spatially dependent qubit state transformations (e.g. a transformation transforming the north pole of the Bloch sphere to its equator; for further details see section “Composite pulse protocol for universal one-qubit gates” of Enclosure A), wherein the one or more local laser pulses are configured to compensate for spatial variations, across the subset of qubits, of the spatially dependent qubit state transformations caused by the one or more global spatially inhomogeneous laser pulses. In some cases, the global spatiallyinhomogeneous laser pulses may be single or multi-photon pulses, e.g., to optically induce one or more spatially dependent qubit state transformations for nuclear or hyperfine state qubits, e.g., via a multi-photon Raman transition, or using a three- photon transition for an optical qubit.

[0029] For example, illuminating each qubit in the subset with the one or more respective local laser pulses may comprise, determining, based on the arbitrary quantum gates to be implemented, a respective effective pulse area for the one or more local laser pulses to compensate for the spatial variations, across the subset of qubits, of the spatially dependent qubit state transformations caused by the one or more global spatially inhomogeneous laser pulses, and illuminating, each qubit in the subset, with the one or more local laser pulses, each having the determined respective effective pulse area. In this manner multiple gates can be performed on multiple qubits in the subset with high fidelity and essentially simultaneously in a robust manner. For example, determining the respective effective pulse area may comprise, determining a respective illumination time for each of the one or more local laser pulses, and / or determining a respective laser frequency for each of the one or more local laser pulses, and / or determining a respective laser intensity for each of the one or more local laser pulses.

[0030] The inventors have found that in some implementations, robustness and gate fidelity can be further improved by illuminating each qubit in the subset with a respective first local laser pulse configured to induce a respective first local z-rotation for each qubit in the subset, illuminating each qubit in the subset with a first global spatially inhomogeneous laser pulse configured to induce a respective first spatially dependent qubit state transformation for each qubit in the subset, and illuminating each qubit with a respective second local laser pulse configured to induce a respective second local z-rotation for each qubit in the subset.

[0031] Moreover, in some implementations (for an example see Fig. 4 of Enclosure A) the method may further comprise, illuminating each qubit in the subset, with a second global spatially inhomogeneous laser pulse configured to induce a respective second spatially dependent qubit state transformation which essentially reverses the first spatially dependent qubit state transformation for qubits not illuminated by a local laser pulse (for details see equations (12) and (13) as well as “Thecomposite pulse scheme” of Enclosure A), and illuminating each qubit with a respective third local laser pulse configured to induce a respective third local z-rotation for each qubit in the subset. In this manner, it can be ensured that gate infidelity accumulated over more than 1000 subsequent quantum gates can be below 1% for an initial motional ground state population p0of 99% (see Fig. 5 of Enclosure A).

[0032] As explained in detail in Enclosure A, such a Mikado pulse sequence allows to perform the desired arbitrary quantum gates on the subset of qubits in a fast, high-fidelity, and robust manner, because local deviations in the effective Rabi frequency and / or local frequency shifts (e.g., probe shifts) can be compensated for by the local laser pulses. Further optional implementation details and embodiments are discussed below with reference to Fig. 5.

[0033] The present disclosure also relates to a system for modulating laser pulses used to implement one or more quantum gates for a subset of qubits in a quantum register, comprising a first laser source for coupling a qubit ground state |o) and a qubit excited state | 1) (e.g. to generate a superposition state as discussed above), a first set of optical components, coupled to the first laser source and configured to illuminate (e.g., globally as discussed above and illustrated in Fig. 2 below) the subset of qubits, and one or more optical modulators for modulating an output of the first laser source to generate one or more laser pulses to implement the one or more quantum gates for the subset of qubits, wherein at least one of the optical modulators is configured to apply a phase modulation function < >(t) for the one or more laser pulses such that a photon recoil induced by said laser pulses is minimized. In some implementations, the system may further comprise processing circuitry (such as a computer, FPGA, ASICs, a quantum computing control system, etc.), coupled to memory and configured to determine the phase modulation function 4>(t) for the one or more laser pulses by carrying out the steps of the methods discussed above and below with reference to Fig.3-

[0034] The system may also comprise a second laser source for inducing a light shift on at least one of the qubits in the subset, and second optics coupled to the second laser source and comprising an addressing unit configured to locally and selectively illuminate the subset of qubits with laser pulses inducing a qubit specific light shift. Thus, the system maybe configured to generate one or more local laser pulses and / orspatially inhomogeneous global laser pulses as discussed above and below with reference to Fig. 5. Further optional implementation details are discussed below with reference to Fig. 2 and Fig. 5.

[0035] The present disclosure also relates to a quantum computer (see Fig. 6 below), comprising a quantum register formed by an array of trapped particles, and a quantum gate laser system comprising a system as discussed above and below with reference to Fig. 2. Such a quantum computer may further comprise processing and control circuitry configured to receive, via a first network (e.g., the internet) from a user device, instructions for performing a quantum algorithm using a subset of qubits of the quantum register; and a qubit state readout system, wherein the processing and control circuitry may be further configured to control the quantum gate laser system and the qubit state readout system, such as to perform the quantum algorithm based at least in part on performing any of the methods discussed herein, to determine a result of the quantum algorithm by measuring a state of the subset of qubits (e.g., by state selective fluorescence imaging, e.g., as discussed in PCT / EP2024 / 066941, PCT / EP2023 / 074768, and PCT / EP2023 / 074768), and to send the result of the quantum algorithm, via the first network to the user device.

[0036] The present disclosure also relates to a method for quantum computing (as shown in Fig. 7), comprising receiving, via a first network from a user device, instructions for performing a quantum algorithm using a subset of qubits of a quantum register, performing the quantum algorithm based at least in part on any of the methods discussed herein, determining a result of the quantum algorithm by measuring a state of the subset of qubits, and sending the result of the quantum algorithm, via the first network to the user device.BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Further details of the apparatuses, systems, and methods described above are discussed in the following with reference to exemplary implementations illustrated by the drawings. The foregoing broadly outlines the features and technical advantages of examples in accordance with the present disclosure in order that the detailed description that follows may be better understood. Additional features and advantageswill be described hereinafter. The conception and specific examples disclosed may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present disclosure. Characteristics of the concepts disclosed herein, both their organization and method of operation, together with associated advantages will be better understood from the following description when considered in connection with the accompanying figures. Each of the figures is provided for the purposes of illustration and description, and not as a definition of the limits of the claims. The same reference numbers in different drawings may identify the same or similar elements.

[0038] Fig. 1 illustrates how an absorption of a photon by a trapped particle may induce a photon recoil, which may lead to entanglement between internal and motional degrees of freedom and vibrational heating, thereby reducing a gate fidelity for trapped particle qubits;

[0039] FIG. 2 illustrates a system for modulating laser pulses used to implement one or more quantum gates for a subset of qubits in a quantum register according to aspects of the present disclosure;

[0040] Fig. 3 illustrates a flow diagram of a method for determining a phase modulation function 4>(t) for a laser pulse implementing a quantum gate for a qubit implemented by a trapped particle according to aspects of the present disclosure;

[0041] Fig. 4 illustrates a flow diagram of a method for performing a quantum gate for a qubit implemented by a trapped particle according to aspects of the present disclosure;

[0042] Fig. 5 illustrates a flow diagram of a method for implementing quantum gates on a subset of qubits in a quantum register that are robust against spatial inhomogeneities of global gate laser pulses according to aspects of the present disclosure;

[0043] Fig. 6 illustrates a quantum computer according to aspects of the present disclosure;

[0044] Fig- 7 illustrates a method for quantum computing according to aspects of the present disclosure.DETAILED DESCRIPTION OF SOME EXEMPLARY EMBODIMENTS

[0045] Various aspects of the present disclosure are described in more detail hereinafter with reference to the accompanying drawings. The present disclosure may, however, be implemented in many different forms and should not be construed as limited to any specific structure or function presented herein. Rather, these aspects are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present disclosure to those skilled in the art. Based on the teachings herein one skilled in the art should appreciate that the scope of the present disclosure is intended to cover any aspect of the present disclosure disclosed herein, whether implemented independently of or combined with any other aspect of the present disclosure. For example, a device or system maybe implemented, or a method maybe practiced using any number of the aspects set forth herein. In addition, the scope of the present disclosure is intended to cover such a device, system or method which is practiced using other structure, functionality, or structure and functionality in addition to or other than the various aspects of the present disclosure set forth herein. Any aspect of the present disclosure disclosed herein may be implemented by one or more elements of a claim. While specific feature combinations are described in the following with respect to certain aspects of the present disclosure, it is to be understood that not all features of the discussed examples must be present for realizing the technical advantages of the devices, systems, methods and computer programs disclosed herein. Disclosed aspects may be modified by combining certain features of one aspect with one or more features of other aspects. A skilled person will understand that features, steps, components and / or functional elements of one aspect can be combined with compatible features, steps, components and / or functional elements of any other aspect of the present disclosure.

[0046] Several aspects of quantum computation e.g., with neutral atoms, ions, or molecules will now be presented with reference to various devices, systems and methods that are described in the following detailed description and illustrated in the accompanying drawings by various blocks, modules, components, circuits, steps,processes, algorithms, and / or the like (collectively referred to as “elements”). These elements may be implemented using hardware, software, or combinations thereof. Whether such elements are implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system.

[0047] While aspects of the present disclosure are presented in the following using optical qubits, e.g. implemented via trapped bosonic Strontium isotope88Sr atoms (as discussed, for example, in applicants own PCT / EP2024 / 066941, PCT / EP2023 / 074768, and PCT / EP2023 / 074768), it is to be understood, that any other species of neutral atom, ions and / or molecules with a suitable internal level structure may also be used as qubits in other implementations of the present disclosure.

[0048] FIG. 1 illustrates how irradiating a trapped particle qubit induces a photon recoil and how this induced photon recoil may lead to a reduced gate fidelity for trapped particle qubits. Panel a) of Fig. 1 illustrates differences between an ideal qubit 110 and a trapped qubit 120 such as an optical qubit. The ideal qubit 110 may be represented by an ideal two-dimensional Hilbert spacewhich is spanned by two orthogonal states |0)qand |l)q. The states |0)qand |l)qmay correspond to a ground state and an excited internal state of the particle. Generally, the Hilbert space 3~Cqof the ideal qubit 110 serves as a resource for performing quantum computation, e.g., for performing ideal gate operations acting on the Hilbert space 3~Cq. For example, performing gate operations may comprise to manipulate the ideal qubit state 110 by lasers and / or microwaves and / or radiofrequency signals such as to achieve a coherent control, e.g., by the laser pulse 140. For example, irradiating the ideal qubit with a laser pulse may change the internal state of the qubit. Particularly, when the state is initially 10)q, the laser pulse 140 maybe performed such as to implement a Hadamard gate H on the qubit, e.g., the laser pulse 140 may transform the qubit to the superposition state illustrated in panel b) of Fig. 1.

[0049] However, in realistic scenarios, the ideal qubit 110 is confined within electromagnetic and / or optical fields. For example, the ideal qubit maybe confined within a trap, e.g., an optical trap. Therefore, the state of the ideal qubit 110 within the Hilbert space 3~Cqdoes not constitute a proper description of the corresponding trappedparticle as also motional degrees of freedom (in addition to the internal degrees of freedom describe by the Hilbert spacehave to be considered. Thus, the ideal qubit no maybe turned to a trapped qubit 120. Consequently, the Hilbert space describing the trapped particle may comprise a tensor product space with a first factor corresponding to the internal degrees of freedom and a second factorcorresponding to motional degrees of freedom, i.e.,For example,may describe the spatial vibrations of the trapped particle in the trap. Typically, it is assumed that the motion of the trapped quantum particle is quantized, such that the motional Hilbert spacemay comprise a countable basis with basis elements |n) for N B n > 0.

[0050] For example, the trapped quantum particle 120 may be trapped in a trapping potential 130. In particular, the trapping potential 130 may comprise an essentially harmonic trapping potential. The motional state of the qubit may comprise a motional ground state |0)msuch that the state | V of the trapped particle qubit 120 is given by IV = |0)q ® |0 )m. For example, when the trapping potential 130 is essentially harmonic, the motional ground state may comprise a Gaussian shape 135, e.g., may correspond to a ground state of a harmonic oscillator.

[0051] Generally, the two subspaces Mqandmay be coupled, meaning that the vibrations of the trapped particle may lead to qubit-motion entanglement. Specifically, the coupling between the subspaces maybe quantified by a Lamb-Dicke parameter 17. For example, if 77 = 0 , the trapped qubit 120 may essentially behave like the ideal qubit 110, e.g., the motion of the trapped qubit 120 evolves independently and can be factored out. The Lamb-Dicke parameter 17 depends inversely on the trap frequency. Thus, for a given particle and qubit transition, 17 maybe diminished by increasing the trap frequency. For hardware based on trapped ions, the trap frequency can comprise frequencies in the MHz regime which allows for treating the trapped qubit 120 as an ideal qubit 110.

[0052] However, for hardware based on neutral atoms, the trap frequency typically comprises frequencies about 100 kHz due to power limitations of the trapping lasers, such that the Lamb-Dicke parameter 17 remains relatively high. Thus, for trapped particle qubits based on neutral atoms, the coupling of the internal degrees of freedomand the motional state cannot be neglected a priori, such as for trapped particle qubits based on ions.

[0053] The problem of the coupling betweenbecomes particularly relevant for trapped particle qubits based on neutral atoms when state transformations of the trapped qubit 120 should be performed. In particular, manipulating the trapped qubit 120 by lasers and / or microwaves and / or radiofrequency signals may cause the trapped qubit 120 to absorb a photon momentum when excited, wherein the magnitude of the absorbed photon momentum may depend on the energy difference between the internal qubit states |0)Qand |1)Q. Specifically, the square of the Lamb-Dicke parameter, ?j2, may represent a ration between the recoil energy, e.g., the energy the particle would have after emitting a photon if initially at rest, and the energy spacing between two motional states. Generally, atomic qubits, e.g., qubits based on alkaline metals or alkaline-earth metals, may be categorized according to the energy difference. For example, the atomic qubits may be categorized into nuclear spin qubits, fine- structure qubits, hyperfine-structure qubits and optical qubits. While hyperfine qubits operate in a microwave regime, e.g., energies in the GHz regime, optical qubits typically comprise energy differences in the regime of 10-100 THz, e.g., at much larger energy differences.

[0054] Among the atomic qubits, for nuclear spin qubits, the Lamb-Dicke parameter maybe essentially neglectable, meaning that the qubit can be driven with standard control schemes for two-level quantum systems without being affected by motion. For the fine-structure qubits, a Raman based scheme maybe used for their control. In particular, the Raman based scheme may comprise applying two copropagating waves to realize two-photon transitions, allowing to reach an effective Lamb-Dicke parameter 77 in the range of IO-2.

[0055] Generally, the coupling between the motion and the internal state becomes significant for optical qubits based on e.g., an optical transition betweenJS0and 3P0, for which 77 may be in the range of IO-1. Lamb-Dicke parameters of this magnitude may cause significant interactions between the qubit and the motion. In particular, the coupling may affect a gate fidelity, e.g., the fidelity of performedoperations, through photon recoil and implies undesired entanglement between the two subspaces 3~Cqand Km, resulting in a loss of coherence.

[0056] For example, with reference to panel b) of Fig. 1, for the case of the trapped qubit 120 with a large Lamb-Dicke parameter 77, irradiating the trapped particle qubit 120 with the laser pulse 140 may not result in the superposition state |x+) as in the ideal case, e.g., irradiating the ideal particle qubit 110 with a corresponding laser pulse. In particular, the absorption of a photon by the particle maybe such that the momentum of the absorbed photon causes a photon recoil, e.g., a change of the motional state of the particle from |0)mto \rec)m. For example, the motional state of the trapped qubit 120 after the photon recoil may comprise the wave function 138, e.g., an excited mode and / or a superposition of modes of the trapping potential 130. More precisely, the absorption of the photon may transform the internal qubit state from |0)Qto |1)Qand simultaneously the motional state of the particle from |0)mto |rec)m. Therefore, the laser pulse 140, intended to transform the initial state of the qubit from the ground state |0)Qto the superposition state |x+) = (| 0)q+ 11>Q), induces insteadthe transformation 9 However,this state may comprise an entangled state, e.g., the reduced density operator on the system J~Cqmay be mixed.

[0057] Fig. 2 illustrates a system 200 for modulating laser pulses used to implement one or more quantum gates for a subset of qubits 205 in a quantum register 210. As discussed in detail in applicants own PCT / EP2024 / 066941, PCT / EP2023 / 074768, and PCT / EP2023 / 074768, the quantum register 210 maybe implemented by an array of trapped neutral atoms placed inside a vacuum cell 215 used for isolating the quantum register 210 from background gas collisions and other sources of decoherence and / or particle loss. For example, the quantum register 210 may be initialized by laser cooling of a plurality of neutral atoms and subsequently trapping and further cooling them (e.g., via sideband-cooling or similar techniques known in the art) to essentially occupy the vibrational ground state of the respective trapping potential. The trapped particles 205 may also be rearranged to obtain a quantum register 210 of essentially unity filling wherein each trapped particle 205 ideally occupies the motional ground state of the respective trapping potential (see Fig. 1). In such a configuration, the motion of the trapped particles 205 can be described tohigh accuracy by a motion in a harmonic oscillator potential characterized by a trapping frequency a> as it is well known in the art.

[0058] The system 200 further includes a first laser source (not shown) generating a global laser beam 220, for coupling a qubit ground state | o) and a qubit excited state |1) of the qubits 205, e.g., to perform single qubit x-rotations via transferring population between the qubit ground state | o ) and a qubit excited state | 1). For example, illuminating a qubit 205 with a (near) resonant laser pulse 222 generated from the laser beam 220 of the first laser source having an effective pulse area that corresponds to a — Rabi pulse, transfers an illuminated qubit 205 initially in the ground state |o) to a superposition state | s) = 1 / V2 (| 0) + | 1)), as known in the art.

[0059] The system 200 further comprises a first set of optical components (e.g., mirrors, lenses, etc.), coupled to the first laser source and configured to illuminate a subset of qubits 205 with laser pulses 222 obtained from the first laser source, and one or more optical modulators 225 (such as an AOM, an EOM, a photonic integrated circuit, a combination thereof, etc.) for modulating the laser beam 220 of the first laser source to generate the one or more laser pulses 222 to implement the one or more quantum gates for the subset of qubits 205.

[0060] At least one of the optical modulators 225 is configured to apply a phase modulation function < >(t) for the one or more laser pulses 222 such that a photon recoil induced by said laser pulses 222 is minimized (for details see Fig. 3 and Fig. 4 below). The optical modulators 225 may be driven by an RF drive signal 230 that, in some implementations, may be generated by an arbitrary waveform generator 235 or a similar drive signal source. The drive signal source 235 maybe controlled by a computer, a dedicated quantum computing control system, or similar processing circuitry 240 (coupled to memory) and be configured to apply the drive signal 230 to the optical modulator 225 such as to apply the phase modulation function < >(t) (e.g., determined by the method discussed above and below with reference to Fig. 3) to minimize a photon recoil induced by said laser pulses 222 derived from the first laser source. For example, the processing circuitry 240 may receive a Rabi frequency fl and a trap frequency (e.g., obtained by a preceding calibration of the quantum register 210), determine the phase modulation function < >(t) as disclosed herein, and apply acorrespondingly modulated RF drive signal 230 to the optical modulator 225 phase modulate the laser pulse 222 implementing a quantum gate for the subset of qubits 205.

[0061] In some implementations, the processing circuitry / control system 240 may thus be configured to receive a Rabi frequency fl, and a trap frequency a) for the laser pulse 222 and the trapped particles 205, and to determine, based on the Rabi frequency fl and the trap frequency a>, the phase modulation function < >(t) for the laser pulse 222 such as to minimize a photon recoil of the trapped particles 205 induced by said laser pulse 222. Further, the computer, control system, or processing circuitry 240 may further be configured to apply a laser intensity modulation function A(t), such as a square pulse or a gaussian pulse shape, or similar pulse shapes via the drive signal source 235 and the optical modulator 225, or a separate intensity modulator (not shown) to the laser beam 220, e.g., to generate one or more global laser pulses 222 having a corresponding pulse area.

[0062] The inventors have found that in some systems, an optical phase modulation applied by typical optical modulators 225 such as AOMs, EOMs, etc. may deviate from a desired phase modulation function < >(t), e.g., determined as disclosed herein, to suppress the photon recoil of a laser pulse 222 implementing a desired arbitrary quantum gate. To address this problem, e.g., to reduce a discrepancy between the desired phase modulation function < >(t) and the actual optical phase modulation applied by the optical modulator 225, the system 200 may, in some implementations, include an optical phase detector system (that in the illustrated example may be implemented by a fast photodiode 242, and an RF phase discriminator 245), for detecting an optical phase modulation applied by the one or more optical modulators 225 to the one or more laser pulses 222, as well as a control feedback loop, e.g., comprising the quantum computing control system / processing circuitry 240 for adjusting the drive signal 230 of the one or more optical modulators 225 such that a difference between an output 247 of the optical phase detector (serving as an error signal for the feedback loop) and the desired phase modulation function 4>(t) is reduced.

[0063] For instance, as illustrated in the example of Fig. 2, a heterodyne phase detection scheme may be used. Generally, a homodyne scheme would also work, but the heterodyne scheme is preferable as detecting the phase of an oscillating signal is easier. For example, the optical modulator 225 maybe an AOM, which both modulates the phase and shifts the frequency of the laser pulse 222 by an amount corresponding to the carrier RF frequency of the drive signal 230. Before the optical modulator 225, a small part of the laser beam 220 can be collected by an imperfect mirror or glass plate 250 to let this part of the laser light 252 interfere with a part of the phase modulated and frequency shifted laser light 224 that has passed through the optical modulator 225. Doing so allows to detect an interference signal by the fast photodiode 242 having a bandwidth that should be larger than the RF modulation frequency applied to the optical modulator 235. The output signal 244 of the fast photodiode 242 thus oscillates at the difference frequency between the laser beam 220 and the frequency shifted output of the optical modulator 225. Since the output of the optical modulator 225 is phase modulated the output of the fast photodiode 240 is as well.

[0064] For example, a phase-frequency discriminator can be used as the RF phase detector 245, which takes a RF reference signal 249 as the local oscillator, e.g., provided by the drive signal source 235 as show in Fig. 2 and compares it to the interference signal 244 oscillating at the AOM shift frequency. The detected phase signal 247 can then processed by the computer / processing circuitry 420. After, in some instances, removing an arbitrary time shift related to the delay in the transmission of the signals, the difference to the desired phase modulation function 4>(t) can be determined and used to modify the output 230 of the drive signal source 235 to modify the operation of the optical modulator 225 to reduce the difference. To improve feedback loop convergence, in some implementations, the signal representing the correction can be deconvolved to correspond to the modulated phase profile by a linear response of regime of the optical modulator 235 (which could be precharacterized with conventional techniques such as measuring the step-response function, etc.).

[0065] In this way, the correction applied to the signal source 235 can be first software deconvolved and then later physically convolved with the linear response function of the optical modulator 235 itself. As these two operations tend to cancel eachother out, the programmed corrections may actually result in a corresponding correction in the optical signal 222.

[0066] To obtain the correction signal from the difference between the recorded phase profile and the desired one, in some examples, a PID-type control scheme can be employed. Here it can be sufficient, to use the proportional (P) and integral (I) parts of the PID algorithm. The integral part is important, as in any PID scheme, to remove any final offset from the desired signal. In some examples, a so called PI2D control scheme with a double integrator can be used to remove any offset in the linear gradient of the phase as known in the art of control system theoiy. A PID control scheme is a one-time operation to optimize the phase profile of the pulses. This operation does not need to be repeated often. Only each time the system has drifted away from the calibrated state (system misalignment, temperature drift, any other aging factor of the experimental apparatus).

[0067] To implement a method as discussed below with reference to Fig. 5 (e.g., a spatially robust Mikado pulse sequence) the system 200 may further comprise a second laser source (not shown) providing a second laser beam 260 for generating local laser pulses 262 inducing an individually controllable light shift for the qubits 205 in the quantum register 210, as well as a second set of optical components (e.g., comprising a 4F telescope 267 and a large NA objective 269) coupled to the second laser source as well as a qubit addressing unit 265 (e.g., a multi-channel 2D acoustooptic deflector, a multi-channel PIC-based addressing unit, etc.) configured to locally and selectively illuminate a subset of qubits 205 in the quantum register 210 with local laser pulses 262 inducing a qubit specific controllable light shift inducing controllable local z-rotations for the illuminated qubits 205 as discussed in more detail with reference to Fig. 4 in Enclosure A. In particular, the addressing unit 265 maybe configured to generate, based on corresponding drive signals 270, e.g., controlled by the computer / quantum computing control system 240 one or more local laser pulses 262 according to any of the methods described herein.

[0068] As a person skilled in the art understands, a system such as the system 200 of Fig. 2 might be part of a trapped particle quantum computer (see Fig. 6), comprising a quantum register 210 formed by an array of trapped particles 205, and aquantum gate laser system comprising a system for modulating laser pulses used to implement one or more quantum gates for a subset of qubits 205 in the quantum register 210.

[0069] Fig. 3 shows a flow diagram of a method 300 for determining a phase modulation function < >(t) for a laser pulse implementing an arbitrary quantum gate for a qubit implemented by a trapped particle, preferably an arbitrary single qubit gate, such as the qubits 205 shown in Fig. 2. The method 300 may, in some implementations, be executed by the computer / quantum control system / processing circuitry 240 of the system 200 of Fig. 2 or similar apparatus.

[0070] The method 300 comprises obtaining 310 a Rabi frequency fl, and a trap frequency a) for the laser pulse and the trapped particle, e.g., by performing a calibration of the quantum register 210 illuminated by the laser pulses 222 shown in Fig. 2. For example, by varying the pulse area and / or the detuning of the global laser pulses 222 and subsequently detecting the state of the qubits (e.g., via state-selective fluorescence imaging) a maximal Rabi frequency fl and, optionally, spatial variations of the Rabi frequency fl as well as a resonance frequency of the qubit transition (as well as, optionally, spatial variations thereof) can be determined. In step 320 the phase modulation function < >(t) for the laser pulse is determined, based on the Rabi frequency fl and the trap frequency a>, such as to minimize an effect of a photon recoil of the trapped particle induced by said laser pulse on a gate fidelity of the quantum gate.

[0071] For example, in some implementations, determining 320 the phase modulation function (t) for the laser pulse may comprise defining a quantum control problem based on a Hamiltonian describing the dynamics of the trapped particle when subjected to the laser pulse, and determining the phase modulation function < >(t) for the laser pulse, based on solving the quantum control problem, such that a gate infidelity measure associated with the photon recoil is minimized below a predefined threshold

[0072] Generally, as explained above, the Hilbert space of the trapped particle may be described by a tensor product between the internal qubit states 10)qand | l)qand the motional states |n)mfor n > 0, which are of infinite (but countable) number. A general state of such a system may be written as |tp) = 2n>oan® |n)m+ Pn |l)q ® |n)m, wherein |n)mdenotes the nlhmotional state. Without loss of generality, the motion of the trapped particle maybe restricted along one spatial dimension, e.g., the spatial dimension in the same direction as the driving laser pulse. In particular, the driving laser maybe configured to drive transitions between |0)Qand |1)Qat a Rabi frequency fl (t). In addition, the phase 0(t) of the driving laser maybe modulated over time such as to control the system. In a given rotating frame, the dynamics of the particle maybe described by the Hamiltonianwherein <J+= 1 l)(01, fl(t) is the Rabi frequency of the laser beam, is the trap frequency, e.g., a> = 2n x 100Hz, a and ad are the creation and annihilation operators,?7 = is the Lamb-Dicke parameter with m the mass of the particle and A(t) is thedetuning of the laser frequency from the qubit transition.

[0073] Specifically, if the interaction is between the trapped particle and a resonant laser, e.g., the detuning A(t) = 0, and the Rabi frequency is essentially constant fl(t) = fl (e.g., neglectable raising time), the Hamiltonian maybe simplified towherein the control parameters in the Hamiltonian are the positive Rabi frequency fl, the phase 0(t) of the laser field modulated in time t, and the trap frequency a>.

[0074] Generally, the above Hamiltonian may be expanded in terms of 17. In particular, the Hamiltonian may be expanded in a polynomial in 17. For example, when the Lamb-Dicke parameter is large, the Lamb Dicke-regime, e.g., an expansion of the Hamiltonian H up to the first order in 77, may not capture all the complexity of the original, full Hamiltonian. It may therefore be necessaiy to extend the expansion beyond the Lamb-Dicke regime by expanding the Hamiltonian up to the second order in 17. In fact, the second-order Hamiltonian maybe written aswherein the terms hq(t) and hp(t) maybe expressed in the Pauli basis. In particular, the operator hq(t) may describe a Hamiltonian of an ideal two-level quantum system and the operator hp(t) may comprise an operator orthogonal to hq(t), e.g., the terms hq(t) and hp(t) may be given by

[0075] As illustrated above, the Hamiltonian may be decomposed into three parts. For example, a first part may describe the dynamics in the qubit subspace only. In addition, a second part may describe the coupling between the two subspaces, e.g., the coupling between the internal subspace Mqand the motional subspace Km. In addition, a third part may act on the motional states only. The coupling part maybe subdivided into a recoil contribution, mixing the motional states through terms as a, a2, ad and (ad) and a pure entanglement part, acting on cd a. The factor (1 - ??2 / 2) on the qubit part may slow down the operation in the qubit subspace, regardless of the level of motional excitation. In other words, it may be a direct effect of the motion of the qubit dynamics.

[0076] Generally, the average Hamiltonian Theory (AVH)

[0020] may allow for approximating the dynamics of an evolution operator under a non-constant, e.g., timedependent, Hamiltonian. Given thatU = —IHU, wherein H may comprise a givenHermitian Hamiltonian, the AVH states that I / (t) maybe given bywherein H is the average and / or effective Hamiltonian. The average and / or effective Hamiltonian may comprise an infinite series whose first terms maybe given byIn some embodiments, the approximation requires that | \H(t) 11 • t « 1, e.g., with respect to the Frobenius norm 11 • 11.

[0077] For the Hamiltonian in the second order approximation in r / one may show, e.g., by using the AVH, that the evolution operator of the system under said Hamiltonian is given bywherein the terms HRec(t), HEnt(t) are defined byandParticularly, the termsandVRec(t)as weh as VEnt(t) are given by

[0078] In the scenario described above, the fidelity of the gate is not only affected by the photon recoil but also by aa^Vent, which may increase the level of entanglement. Surprisingly, the expression of VEntmaybe independent of the trap frequency. Unlike photon recoil, this additional entanglement may not be mitigated in the resolved sideband regime. This can be understood by assuming a constant pulse, in which case the operators Uqand hqmay commute. This, VEntmaybe proportional to the length flT of the pulse. Generally, to reach a given target qubit gate, flT « 0Tarmust hold, independently of whether the resolved sideband regime is used or not, so this entanglement term is the same.

[0079] Generally, the matrix Uqmay correspond to the evolution operator of an ideal spin— particle driven by the Hamiltonian hqand the termsand VR^ may comprise non-unitary 2 x 2 matrices generating transitions between the motional states through a and ad. The term exp(— ia)d*at) may act purely on the motional subspace, e.g., the term may not influence the qubit gate and maybe ignored. As the terms VR^Cand VR^Cdepend on the qubit dynamics, it is possible to control the qubit in away that may suppress photon recoil by shaping the phase (t) of the driving laser beam overtime. Based thereon, a quantum control problem maybe defined. In particular, a quantum control problem may be defined such as to implement a recoil- free condition. Generally, the particular quantum control problem may depend on the magnitude of 17.

[0080] For example, if 17 is in the Lamb-Dicke regime, the quantum control problem, e.g., the quantum control problem enforcing recoil-free pulses, may be formulated aswherein UTaris the target qubit gate, e.g., the qubit gate intended to be implemented. Specifically, the recoil-free condition V^c= 0 may add complexity to a qubit control mechanism compared to standard procedures used for ideal two-level systems. Importantly, when fl « a>, the term eia>tmay oscillate fast compared to the term UqhpUq. Therefore, the term V^cmay average out over several oscillations and may become quite small. This may mitigate at least a part of the effects of the photon recoil.

[0081] In some embodiments, where 17 is large, e.g., not within the Lamb-Dicke regime, including only the first order condition, e.g., V^c= 0, may not be sufficient.Therefore, a second-order quantum control problem, e.g., a second order recoil free quantum control problem, maybe defined as

[0082] Now, instead of dealing with an infinite-dimensional system that is in general complex to analyze and difficult to simulate, the above defined quantum control problems (only comprise 2 x 2 complex matrices. This reduces the problem to that of a standard spin ¥2 particle with additional constraints, making the problem more suitable for optimization. In particular, one may leverage the extensive research on optimal controlof two-level quantum systems, such as gradient algorithms, shooting algorithms, geometric curves and / or the maximum principle.

[0083] In particular, the above defined quantum control problems (Uq(?) == 0 or Uq(T) = UTar, V^c= 0,^2 = 0) may be solved using a gradient decent algorithm, e.g., a gradient decent algorithm based on GRAPE. For example, the phase modulation function < >(t) maybe discretized in small time steps, which may take any values (unconstrained problem). Generally, one may consider longer and longer pulse durations T during the optimization. For example, one may consider longer and longer pulse durations T until the quantum control problem is satisfied with a precision in the regime IO-10. Specifically, the numerically found solution of the quantum control problem may comprise the phase modulation function < >(t).

[0084] Thus, in some implementations, the quantum control problem may comprise a gate control condition (e.g., Uq(l") = UTar) corresponding to performing the quantum gate, and one or more phase control conditions (e.g., V^c(?) = 0, Vc(?) = 0, etc.) corresponding to at least a first order effect of the photon recoil induced by the laser pulse

[0085] Further, in some implementations, defining the quantum control problem may comprise approximating the Hamiltonian for the dynamics of the trapped particle when subjected to the laser pulse as a power series in the Lamb-Dicke parameter r| up to at least a first order in q, preferably up to a second order in q, wherein an nthphase control condition of the quantum control problem may correspond to an nthorder term of the power series in q (for details see Enclosure B). Such an approach, when setting up and (numerically) solving the quantum control problem facilitates a fast determination of the phase modulation function (t) up to a desired accuracy / gate fidelity in a computing resource efficient manner.

[0086] Further, as discussed for the example above, a first phase control condition may correspond to a first order effect of the photon recoil, a second phase control condition may correspond to a second order effect of the photon recoil, and / or a third phase control condition may correspond to a vibration mode entanglement effect of the trapped particle / qubit.

[0087] Further, in some implementations, determining the phase modulation function (t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse may comprise determining a pulse duration and / or a pulse shape for the laser pulse to be modulated by the phase modulation function 4>(t). In this manner, gate fidelity and fast gate operation can be optimized jointly to ensure that recoil free gates can be executed as fast as possible for a given desired gate fidelity.

[0088] In some implementations, determining the phase modulation function (t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse may comprise, determining, based on a Hamiltonian describing the dynamics of the trapped particle when subjected to the laser pulse, a cost function J, that depends, via the Hamiltonian, on the phase modulation function < >(t), and that quantifies a gate infidelity corresponding to the photon recoil induced by said laser pulse, and determining the phase modulation function < >(t) by minimizing the cost function J below a predefined threshold or until a predefined halting condition (e.g., a maximal number of numerical iteration steps, etc.) is met.

[0089] For instance, determining the phase modulation function < >(t) by minimizing the cost function J may comprise selecting a Fourier series representation of the phase modulation function < >(t), and, optionally, a regulation mask for avoiding discontinuities of the phase modulation function (p(t), and iteratively adjusting a set of Fourier coefficients of the Fourier series representation of < >(t) such that the cost function J is minimized below the predefined threshold or until the predefined halting condition is met. Further, the cost function J may comprise a first contribution quantifying a deviation of the quantum gate from a desired ideal quantum gate, and on or more second contribution quantifying an effect of the photon recoil on gate infidelity.

[0090] Further, the cost function J may comprise a weighted sum of the first and the one or more second contributions, wherein the respective weights are selected such that one or more of the second contributions dominate by at least a factor of 5, preferably by a factor of 10 over the first contribution.

[0091] Additionally, determining the phase modulation function < / >(t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse may comprise determining a pulse duration and / or a pulse shape for the laser pulse to be modulated by the phase modulation function < / >(t) corresponding to the quantum gate.

[0092] In some implementations, defining the quantum control problem may comprise including, into the (approximated) Hamiltonian for the trapped qubits, an effect of a Rabi frequency deviation 80,, and / or an effect of a laser frequency detuning 8A, wherein determining the phase modulation function 4>(t) for the laser pulse may comprise determining the phase modulation function (t) for the laser pulse such that the photon recoil is minimized below a fist predefined threshold, and such that the effect of the Rabi frequency deviation 80, and / or the effect of the laser frequency detuning 8A on a quantum gate fidelity is minimized below a second predefined threshold.

[0093] For example, a corresponding Hamiltonian including such effects (e.g., caused by spatial variations of the Rabi Frequency and / or the detuning across the quantum register) may be defined asFor such a Hamiltonian the following quantum control problem maybe defined:'UQ(T) = UrargetUllec = 0' iC = 0UDet = 0, I-1Amp=0 with:and numerically solved (e.g. using a gradient decent algorithm such as the one described in ref.

[0027] . For instance, the phase rp(t) can be discretized in very small timesteps which can take any value (unconstrained problem). Considering a longer and longer pulse duration T allows to find a numerical solution with a precision of ~ io-10. The result is assumed to be a time-optimal pulse (multiple initializations are test) to determine a phase modulation function 4>(t) that minimized a first and second order effect of the photon recoil as well as ensures robustness of the quantum gate for a large range of spatial variations of the Rabi frequency and / or the laser frequency (see Fig. 11 of Enclosure B).

[0094] In some implementations, determining the phase modulation function(t) for the laser pulse may comprise determining a pulse duration and / or a pulse area of the laser pulse such that a quantum gate infidelity measure for a decoherence caused by vibrational entanglement is minimized below a third predefined threshold. For example, as illustrated in Fig. 3 of Enclosure A, the pulse duration and / or a pulse area of the laser pulse may be determined such, that the effect of residual motional entanglement on gate infidelity can be reduced to below io-4for pulse durations larger than -15 ps. Further details are provided in Enclosure A.

[0095] Fig. 4 illustrates a method 400 for performing a quantum gate for a qubit implemented by a trapped particle, (e.g., performed by a system such as the system of Fig. 2) comprising determining 410 a phase modulation function 4>(t) for a laser pulse implementing the quantum gate such that a photon recoil of the trapped particle induced by said laser pulse is minimized, and applying 420 the laser pulse to the trapped particle based on the determined phase modulation function 4>(t). For example, in some implementations, determining 410 of the phase modulation function (t) may comprise determining the phase modulation function (t) by carrying out the steps of the method discussed above, e.g., with reference to Fig. 3. In this manner, as explained above, fast, high-fidelity, and robust quantum gates can be implemented for large arrays of trapped particle qubits.

[0096] Reducing the effects of spatially varying Rabi frequency and / or spatially varying laser frequency can be further improved by designing a qubit pulse sequence including global rotations and local, individually controllable z rotations (for details seeEnclosure A, section “Composite pulse protocol for universal one-qubit gates”). Based on these insights, Fig. 5 illustrates a flow diagram of a method 500 for implementing quantum gates on a subset of qubits in a quantum register that are robust against spatial inhomogeneities of global gate laser pulses according to aspects of the present disclosure. The method 500 includes illuminating 510, each qubit in the subset, with one or more respective local laser pulses configured to induce one or more local z- rotations for each illuminated qubit based on inducing local light shifts caused by the local laser pulses, and illuminating 520 each qubit in the subset, with one or more global spatially inhomogeneous laser pulses configured to induce one or more spatially dependent qubit state transformations, wherein the one or more local laser pulses are configured to compensate for spatial variations, across the subset of qubits, of the spatially dependent qubit state transformations caused by the one or more global spatially inhomogeneous laser pulses.

[0097] Generally, the idea behind the method 500 for implementing quantum gates comprises to map the spatial variations across the subset of qubits of one or more global spatially inhomogeneous laser pulses, e.g., site-dependent variations of the Rabi frequency and / or a detuning 8 onto additional parameters provided by the one or more local laser pulses. For example, the spatial deviations may be mapped onto parameters associated with a rotation angle of one or more local z-rotations induced by the one or more respective local laser pulses. Thereby, the method 500 for implementing quantum gates renders the one or more spatially dependent qubit state transformations induced by one or more global spatially inhomogeneous laser pulses robust against spatial inhomogeneities in the Rabi frequency fl and / or the detuning 6, which in realistic scenarios have to be taken into account. Further details, on how an arbitrary quantum gate can be decomposed into local z-rotations and global qubit state transformations are explained in section “Composite pulse protocol for universal one-qubit gates” of Enclosure A.

[0098] For example, illuminating each qubit in the subset with the one or more respective local laser pulses may comprise determining, based on the quantum gates to be implemented, a respective effective pulse area for the one or more local laser pulses to compensate for the spatial variations, across the subset of qubits, of the spatially dependent qubit state transformations caused by the one or more global spatiallyinhomogeneous laser pulses, and illuminating, each qubit in the subset, with the one or more local laser pulses, each having the determined respective effective pulse area.

[0099] In some implementations, determining the respective effective pulse area may comprise determining a respective illumination time for each of the one or more local laser pulses, and / or determining a respective laser frequency for each of the one or more local laser pulses, and / or determining a respective laser intensity for each of the one or more local laser pulses.

[0100] In some implementations, the method 500 may comprise illuminating each qubit in the subset with a respective first local laser pulse con figured to induce a respective first local z-rotation for each qubit in the subset, illuminating each qubit in the subset with a first global spatially inhomogeneous laser pulse configured to induce a respective first spatially dependent qubit state transformation for each qubit in the subset, and illuminating each qubit with a respective second local laser pulse configured to induce a respective second local z-rotation for each qubit in the subset. The method may further comprise illuminating each qubit in the subset, with a second global spatially inhomogeneous laser pulse configured to induce a respective second spatially dependent qubit state transformation which essentially reverses the first spatially dependent qubit state transformation for qubits not illuminated by a local laser pulse, and illuminating each qubit with a respective third local laser pulse configured to induce a respective third local z-rotation for each qubit in the subset. Further details are discussed in Enclosure A with reference to the composite gate pulse sequence shown in Fig. 4c of Enclosure A.

[0101] In some cases, the first of the global spatially inhomogeneous laser pulses may be configured to transform a qubit state | o) or 11) to a superposition state of | o) and 11), in particular a superposition state with equal modulus square of the amplitudes for |o) and|i), and the second global spatially inhomogeneous laser pulse maybe configured to transform the superposition state back to the state |o) or |i).

[0102] Further, illuminating the qubits in the subset with one or more global spatially inhomogeneous laser pulses may comprise applying a phase modulation function < >(t) to one or more global spatially inhomogeneous laser pulses, wherein thephase modulation function < >(t) is determined by carrying out the method as disclosed herein (e.g. Fig. 3).

[0103] In some cases, the method may further comprise determining the spatial inhomogeneity of the one or more global laser pulses across a portion of the quantum register comprising the subset of qubits, and determining the respective effective pulse area based on the estimated spatial inhomogeneity, e.g., by performing a calibration procedure for the quantum register to determine laser intensity variations, probe shifts etc.

[0104] As discussed in Enclosure A, the method of Fig. 5 allows to maintain the high fidelity of single gates over long sequences of gates, as shown by the randomized benchmarking in Fig. 5 of Enclosure A. In particular, the cost function element Jmot in Fig. 5 (a) shows an improvement by at least three orders of magnitude in heating suppression compared to MbBbauer pulses. Such a strong suppression is key to achieve excellent minimization of the cost function elements Jent and Junifor a scalable number of gates, as shown in Fig. 5(b) and Fig. 5(c) of Enclosure A. Remarkably, Jent exhibits a saturation effect to (i-p0) / 2, derived in Appendix L of Enclosure A. This asymptotic behavior underlines the importance of preparing sufficiently cold atoms to carry out high-fidelity gates on optical qubits. Remarkably, even in the conservative scenario of low quality motional ground-state preparation with a motional ground-state population p0= 90 %, the crossover point defined by Juni > Jem is reached for > 1000 gates showing that recoil-free gates as disclosed herein (cf. Fig. 3, e.g., enhanced by the Mikado pulse scheme of Fig. 5, and, for example, implemented via the system of Fig. 2) enable application of optical qubits for quantum computing and entanglement-enhanced quantum metrology with superior gate fidelities.

[0105] Fig. 6 illustrates a typical implementation of a quantum computer 600 comprising a quantum register 610, e.g. formed by an array of trapped particles, and a quantum gate laser system 620. In some implementations, the quantum computer can be controlled by a user device 650, possibly via a network 660. In some implementations, the quantum gate laser system 620 may be configured to manipulate and / or cause a controlled quantum state evolution of one or more atomic objects within the quantum register 610. For example, the quantum gate laser system 620 maycomprise one or more lasers, which provide one or more laser beams to atomic objects in the quantum register 610. In some implementations, the laser beams maybe continuous or pulsed laser beams.

[0106] In some implementations, the qubit state readout system may be configured to collect and / or detect photons generated by qubits (e.g., during reading procedures). The optics collection system may comprise one or more optical elements (e.g., lenses, mirrors, waveguides, fiber optics cables, and / or the like) and one or more photodetectors. In various embodiments, the photodetectors may be photodiodes, photomultipliers, charge-coupled device (CCD) sensors, complementary metal oxide semiconductor (CMOS) sensors, Micro-Electro-Mechanical Systems (MEMS) sensors, and / or other photodetectors that are sensitive to light at an expected fluorescence wavelength of the qubits of the quantum computer. In various embodiments, the detectors may be in electronic communication with the processing and control circuitry 640.

[0107] In some implementations, the user device 650 is configured to allow a user to provide input to the quantum computer 600 and receive, view, and / or the like output from the quantum computer 600. The user device maybe in communication with the processing and control circuitry 640 of the quantum computer 600 via one or more wired or wireless networks 660 and / or via direct wired and / or wireless communications. In an example embodiment, the user device 650 may translate, configure, format, and / or the like information / data, quantum computing algorithms and / or circuits, and / or the like into a computing language, executable instructions, command sets, and / or the like that the processing and control circuitry 640 can understand and / or implement.

[0108] In some implementations, the processing and control circuitry 640 may be configured to control, inter alia, the quantum gate laser system 620 and / or the qubit state readout system 630. For example, the processing and control circuitry 640 may be configured to cause a controlled evolution of quantum states of one or more atomic objects within the quantum register 610 to execute a quantum circuit and / or algorithm. For example, the processing and control circuitry 640 may cause a reading procedure comprising, possibly as part of executing a quantum circuit and / oralgorithm. In various embodiments, the atomic objects confined within the quantum register 610 are used as qubits of the quantum computer no.

[0109] Fig. 7 illustrates a method for quantum computing, comprising receiving, via a first network from a user device, instructions for performing a quantum algorithm using a subset of qubits of a quantum register, and performing the quantum algorithm based at least in part on performing the method of any of claims 1 to 14, and / or the method of any of claims 15 to 22. The method further comprises determining a result of the quantum algorithm at least in part based on measuring a state of the subset of qubits, and sending the result of the quantum algorithm, via the first network, e.g., via a web interface to the user device.

[0110] The scientific reports in Enclosure A and Enclosure B mentioned above form an integral part of the present disclosure.Recoil-free Quantum Gates with Optical QubitsZhao Zhang,1, 2’ * Leo Van Damme,3’2’ * Marco Rossignolo,4Lorenzo Festa,1’2Max Melchner,1, 5’2Robin Eberhard,1, 5’2Dimitrios Tsevas,1’5’2Kevin Mours,1’5’2Eran Reches,1’5’2Johannes Zeiher,1, 5’2Sebastian Blatt,1’5’2Immanuel Bloch,1’5’2Steffen J. Glaser,3’2and Andrea Alberti1, 5’2’ *1Max- Planck- Institut fur Quantenoptik, 85748 G arching, Germany2Munich Center for Quantum Science and Technology, 80799 Munchen, Germany3School of natural sciences, Technical University of Munich, 85747 Garching, Germany 4 Qruise GmbH, Saarbrucken, 66113, Germany5Fakultat fur Physik, Ludwig- Maximilians- Universitat Munchen, 80799 Munchen, Germany (Dated: August 9, 2024)We propose a scheme to perform optical pulses that suppress the effect of photon recoil by three orders of magnitude compared to ordinary pulses in the Lamb-Dicke regime. We derive analytical insight about the fundamental limits to the fidelity of optical qubits for trapped atoms and ions. This paves the way towards applications in quantum computing for realizing > 1000 of gates with an overall fidelity above 99 %.Introduction. — Ultranarrow optical transitions enable of the atom (MbBbauer effect). Both regimes are natutoday’s most accurate clocks due to their long coherence rally approached with increasing trap frequency [32, 33]. times and large energy level splitting compared to clocks However, there are limits to the maximum trapping frebased on atomic microwave transitions [1-11]. Can the quency for ions and especially for neutral atoms, where same ultranarrow optical transitions be used to define a for the latter the trapping frequency is limited by the qubit for quantum computing applications which is comlaser power and by photon scattering of the trapping light petitive in terms of speed and fidelity with other atomic

[0034] . For neutral-atom quantum computing, efforts have qubit implementations [12-15] in trapped ions [16-19] been made to understand the role of photon recoil

[0035] and neutral atoms [20-27]? The answer is not obvious and to mitigate its effects on optical qubits

[0028] . Howsince optical qubits and optical clocks work in entirely ever, this problem has not been studied systematically different parameter regimes. The most striking example nor have general solutions for arbitrary gates been found is the speed of operation. The optical qubit has to be maso far. Other solutions have been proposed to suppress nipulated on a very short time scale

[0028] , whereas optical photon recoil by driving a three-photon transition. This, clock transitions are naturally probed over a long period however, requires excellent phase coherence between the of time for a higher resolution [1]. This raises the chalthree laser beams and precise control of their beam dilenge of how to operate the optical qubit at high Rabi rection and polarization [36-38]. frequencies while controlling the light shift induced by In this paper, we derive a new understanding of the the driving laser (so-called probe shift [29, 30]) and supphysics underlying photon recoil, systematically quanpressing the effect of photon recoil in a regime where the tify its impact on the gate operations on optical qubits motional sidebands are not inhibited. The second funand, based on the new insights, design a recoil-free pulse damental difference in the requirements is the need for scheme for quantum gates. Leveraging the novel recoil- an optical qubit to operate a universal set of gates from free pulse, we develop a composite pulse protocol to drive arbitrary initial states

[0031] , as opposed to optical clocks arbitrary gates on the optical qubit. The new composfor which spectroscopy is performed from a well-defined ite pulse scheme provides a unifying solution to all three initial state [1]. In fact, the optical qubit requires underrelevant challenges: (1) we employ recoil-free pulses to standing and optimizing the quantum process of the gate, rather than only controlling a particular state evolution [ a c [it. One can work in the regime of low Rabi frequency (sideband resolved regime), and / or engineer deep traps FIG. 1. Ideal qubit vs. trapped optical qubit: (a) An ideal (Lamb-Dicke regime), for which the photon recoil is to a qubit is driven from |0) to a superposition state, whereas the great extent absorbed by the trapping potential instead optical qubit absorbs the photon momentum when excited, (b) Instead of realizing the intended superposition of |0) and |1) states, the photon momentum causes a recoil from the initial motional state |0)mto |rec)mfor the atom ending in* These authors contributed equally to this work. state 1) , thus producing detrimental entanglement between t andrea.alberti@mpq.mpg.de internal and external degrees of freedom.s s f a l d qu t carr es a momentum t at may not e neg g e compared to the width of the momentum distribution of t t t g t eresonant laser, not only suppresses the recoil for the particular state in = the figure, but (i) for all initial qubit states, (ii) for motional Fock states, and (iii) even for coherent states of any (1) amplitude a. This constitutes one of the main results ofthis paper. which acts on the product space Q ® A4 defined by the To explain the statements (i)-(iii), we derive equations qubit (Q) and motional states (A4) . In / / . we omit terms of motions for the x(f) and p(f) position and momendealing with inhomogeneities of the probe shift, which tum operators in the Heisenberg picture. These are the are discussed later, and only focus on the photon-recoil Newtonian equations of motion for a driven harmonic effect, which is truly fundamental. The control parameoscillator, ters in the Hamiltonian are the positive Rabi frequency fi, the phase < / ?(£) of the laser field modulated in time dtp(f) t, and the trap frequency L>J . The Hamiltonian also condfxff)tains the Lamb-Dicke parameter 77 = kx<z= ky / hf (2maf) with . / '() being the zero-point width of the atom in the dt<rz= cos ip(f) — axsin < / ?(!)] + Ofq) (4) trap and m its mass, the Pauli matrices a with stanwhere Eqs. (2) and (3) follow from an approximation dard index notation, and the annihilation operator a. of 7F[y(| in Eq. (1) to the first order in 77, and Eq. (4) In Eq. (1) , a series expansion of H is provided to the describes the qubit dynamics to zero order in 77 (Aporder of rf , where / IQ = <jxcos ep(t) + hysin ep(t) and pendix A). From these equations, an intuitive, semiclas- hi = aycos tpft) — axsin tpft) are operators on Q, whose sical picture emerges, where the trajectory in phase space relevance will become clear in the following. is determined by the interplay between the harmonic os¬We obtain a basic understanding of the effect of cillator force and the recoil force applied to the atom, photon-recoil by studying the motion of an atom in phase which is given by the excitation rate dt fxf) / 2. In the space. To present our results, we will consider, withfollowing, we sketch a proof of three statements (Apout loss of generality, the specific case of a TT / 2 crypendix C for details): (i) The temporal phase profile rotation Ry(7r / 2) , and first realize it with a constant pft) is designed to suppress recoil for the three initial phase tpft) = 0. We call this a MbBbauer pulse because states: |0), (|0) + | l)) / v / 2 and (|0) + i | 1)) / A / 2. Owing the transferred momentum in the Lamb-Dicke regime is to the linearity of the Newtonian equations above and suppressed to the order of hk (fi / w) / \ / 6 according to the the time-reversal symmetry of H = — T1 / / T. with T the MbBbauer effect (Appendix B) . This residual recoil effect time-reversal operator, if |0) evolves under the gate in a can be completely eliminated by introducing recoil-free recoil- free way, so does T |0) = |1) too. Combining these gates, which modulate the phase pit) to bring the moresults, we conclude that any qubit superposition state tional state back to the origin in phase space: a formal is also recoil free, (F(T)) = (p(T)) = 0. (ii) Because the definition of recoil-free pulses and how to compute them expectation value of x and p vanishes for any Fock state, will be provided later. In Fig. 2, we show a comparison of and the zeroth-order qubit dynamics is independent of the MbBbauer and recoil-free gates for a particular initial motion, the driving force is independent of the initial qubit state. To the first order of 77, the recoil-free pulse Fock state, (iii) Given the linear structure of Eqs. (2)a o t a s t w y t b a t oa function of pulse duration T, for UJ = 2TT X 100 kHz, p = 0.22, where Ei are the Kraus operators and Xi are the probabilPo = 0.95. (a) Juni, (b) Jmot , (c) Jent- (d) Jent as a function ities of the different quantum channels, where the domiof the ground state probability pa for T = 15 ps. nant channel with index 0 obeys 1— xo <S 1. In addition, pmdenotes the initial motional state, which we choose to be a thermal state with ground state probability p$. To explain the results above, we expand the unitary By applying the formalism in Ref.

[0039] , we compute the U acting on <2 ® Al to the second order in r / : U(T) = fidelity of the process 8 averaged over the initial qubit E / Q (T) [1 + pVi(T) + p2V2(T) + ©(T / 3)] , with the following state as (J7) = [1 + 2 xA.|Tr(UttarEA.) / 2|2] / 3. Impordefinitions: tantly we find that the infidelity J = 1 — (J7) can be b bJum (7) Here, Uq(t) is the unitary generated by theHamiltonian hg (t) , which yields the being interpreted as the infidelity by entanglement andzeroth-order evolution of the ideal qubit unby systematic deviations from t / tar- Note that to derive der a renormalized Rabi frequency. In addithe bound, we only consider EQ. While J captures the process infidelity, it does not account for motional heattion,= -if fg Uq^hQ^U^T^e^dr and ing, which, if not suppressed, leads to a lower pg (i.e.,higher temperature) for the subsequent gates and, thus, sible for the first- and second-order motional heating to a higher infidelity. Hence, we introduce an additional with an exchange of one and two motional quanta, cost function Jmotdefined as the change of (a* a) in abrespectively. They can be understood as the Fourier solute value, averaged over the four initial qubit states transform of h \ and h$ in the rotating frame, evaluated used in the process tomography (Appendix D). at frequency w and 2w. In contrast, the operator Vgnt'(t)To obtain the recoil-free pulses, we modulate p(t) while does not change the motional state and is responsible for minimizing the weighted sum of the cost terms , Jententanglement, which will be discussed later. To obtain and Jmotfor Utar = RX(K / 2). We present in Figs. 3(a- Jmot= 0 + O(?75) (i.e., suppress recoil to the order of / / 1j c) the three terms as a function of the pulse duration T requires the conditions Free (71) = V^ec (T) = 0. To fulfill for different values of Q. The first relevant observation these conditions, T must be larger than one trap period: is that the three cost terms drop for durations > 2TT / W, T > 2TT [UJ, since phase-modulated pulses with duration indicating that the quantum speed limit is determined by T can only control the spectrum of the operator at the trap frequency. Second, we find that Jmotis increasfrequencies larger than 2TT / T. Hence, this establishes ingly suppressed as we decrease the Rabi frequency, as is the quantum speed limit TQSL of recoil-free pulses. expected in the sideband resolved regime. However, the same behavior is not observed for the other cost functions When the recoil-free condition V^ec (T) = V^c (T) = Jent and -Am- which plateau once the recoil (i.e., Jmot) 0 is held, Jent is only determined by Knt W = is suppressed. § Jo(T)dT. Using process tomography, wee p t d t t p p f t f t o a f, p p , $ Uq, implying that l would be proportional to the area of the pulse, which needs to be greater than FIG. 4. (a) Laser configuration with global recoil-free pulses TT / 2 and thus and local crz-rotations by local light shifts, (b) Composcannot vanish. Importantly, this fact shows that even ite pulse scheme based on global Mikado pulses and sitewhen operating deep in the sideband resolved regime, dependent Rz(ff) rotations. Inset: phase modulation, with constant-phase pulses cannot reduce Jent below JgjJateau. duration 65 % longer than zeroth-order quantum speed limit The second important result we obtain from Eq. (11) is TQSL = TT / (2C). (C) Bloch vector trajectory of Mikado pulse the linear scaling with 1 — p0, provided a sufficiently small under intensity deviation 61 / I = {—0.025, 0, 0.025}. The temperature, as shown in Fig. 3(d). dashed lines represent the rotation axis of the effective ro¬Composite pulse protocol for universal one-qubit tation by the Mikado pulse for the three intensities, respecgates. — We show that recoil-free pulses enable fast, arbitively. (d) Three cost functions vs. intensity deviation for the trary quantum gates for quantum computing with optical Mikado pulse. In the figure, the Mikado pulse is optimized for88Sr atoms; see Appendix I. qubits. The purpose is to implement an arbitrary unitary gate Ug<E SU(2). Naively, this could be implemented with a Euler decomposition

[0040] , RK(03)Ri / (02)A2,(0i) , the popular children’s game: where each rotation is recoil free. This can be realized with parallel addressing of the atoms, relying on scalable -RMik [<F(f)] Rz(a)Rx^IT)Rz^(. (13) opto-electronic solutions [41-44], which is within reach, but not demonstrated with atoms yet. Instead, we proMikado pulses are generated like recoil-free pulses by pose an alternative scheme that avoids addressing the modulating 99(f) to minimize the three cost functions atoms sequentially and, thus, avoids long execution time Tent and Jmot, where J“]kis adapted from Junipulses because of the relatively small Rabi frequency, (Appendix II). Thereby, Mikado pulses inherit the same typical of ultranarrow optical transitions. The proposed recoil-free characteristic. However, they allow for two scheme only uses two recoil-free gates per circuit step, loose parameters, a and / 3, in their optimization, which based on a global recoil-free pulse, together with fast, loserves as additional degrees of freedom. Compared to a cal oy-rotations Rz(ff) implemented by light shift on the recoil- free fixed rotation J?2, (TT / 2) , Mikado pulses are roindividual atoms; see Fig. 4(a). Inspired by the Euler debust against spatial inhomogeneities of and detuning composition, the composite scheme consists in applying fih oy / 2, which in realistic scenarios need to be added to the following operations: H in Eq. (1). An example of inhomogeneous detuning is the probe shift from the intensity inhomogeneity of theUg=Rz(6»3) RMik [<F(f) + 7r]Rz(02)AMik [<F(f)]Az (6*i ) - global pulse. The idea behind Mikado pulses is to map(12) site-dependent deviations of and 6 in H onto the deviaas schematically illustrated in Fig. 4(b) . Here, J?Mik [<F(f)] tions 6 a and 6(3 of the two loose parameters. The robustis a class of pulses transforming the north pole of the ness of Mikado pulses is exemplified by the three different Bloch sphere to its equator, which we call Mikado after Bloch-sphere trajectories shown in Fig. 4(c), correspond-FIG. 5. Randomized benchmarking of a sequences of SU(2) gates drawn from the Haar measure, realized with the composite pulse of Fig. 4(b), for different ground state probabilities pg (see legend), whereas the gray curves refer to constant-phase MoBbauer pulses for pa = 99 %. The saturation value of Jent(see text) is shown in both (b) and (c) as a point of comparison (dotted lines). Representative l-tr error bars are shown. ing to three different laser intensities. Figure 4(d) shows Appendix L. This asymptotic behavior underlines the imin addition that the recoil- free property (i.e., vanishing portance of preparing sufficiently cold atoms to carry out Jent and Jmot) is preserved despite the parameter inhohigh-fidelity gates on optical qubits. Remarkably, even mogeneity. Hence, knowing the parameters for each site in the conservative scenario of ground-state population allows correcting the deviations a + 6a and (3 + 6(3 in the Po = 90 %, the crossover point defined by Juni> Jent is Mikado gates by suitable <rz-rotation angles. reached for a number of gates above 1000.Conclusions. — In this paper, we have shown that our recoil-free pulses, enhanced by the Mikado scheme, pro¬The composite pulse scheme allows implementing any vide an affirmative answer to the opening question: they gate Ugpreserving comparable values of Jent and Jmot enable applications of optical qubits for quantum comas for Mikado pulses [Fig. 4(d)], while suppressing Juniputing and entanglement-enhanced quantum metrology to values below 10~6owing to the local Rzrotations; with state-of-the-art fidelities. see Appendix K. Importantly, the high fidelity of single Acknowledgments. — We acknowledge support from gates carries over to long sequences, as shown by the ranQRUISE in simulating and optimizing the recoil-free domized benchmarking in Fig. 5. In particular, Jmotin pulses. We thank Rainer Blatt and Xie-Hang Yu for inFig. 5 (a) shows an improvement by at least three orsightful discussions, and Tobias Olsacher for careful readders of magnitude in heating suppression compared to ing of the manuscript. We acknowledge funding from MuMoBbauer pulses. Such a strong suppression is key to nich Quantum Valley project TAQC, the BMBF project achieve excellent Jent and Junj for a scalable number of MUNIQC- Atoms, and the Munich Center for Quantum gates, as shown in Fig. 5(b) and Fig. 5(c). Remarkably, Science and Technology. M.R. acknowledges funding Jent exhibits a saturation effect to (1— p0) / 2, derived in from PASQUANS2.1.[1] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. [5] M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87, J. Ye, and A. M. Kaufman, “Seconds-scale coherence on 637 (2015). an optical clock transition in a tweezer array,” Science[2] M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, 366, 93 (2019). “An optical lattice clock,” Nature 435, 321 (2005). [6] I. S. Madjarov, A. Cooper, A. L. Shaw, J. P. Covey,[3] B. Bloom, T. Nicholson, J. Williams, S. Campbell, V. Schkolnik, T. H. Yoon, J. R. Williams, and M. EnM. Bishof, X. Zhang, W. Zhang, S. Bromley, and J. Ye, dres, “An Atomic- Array Optical Clock with Single- Atom “An optical lattice clock with accuracy and stability at Readout,” Phys. Rev. X 9, 041052 (2019). the 10- 18 level,” Nature 506, 71 (2014). [7] S. M. Brewer, J. S. Chen, A. M. Hankin, E. R. Clements,[4] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schaffer, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Mi- Leibrandt, “2A1+ Quantum-Logic Clock with a Systemlani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, atic Uncertainty below 10~18,” Phys. Rev. Lett. 123, “Atomic clock performance enabling geodesy below the 033201 (2019). centimetre level,” Nature 564, 87 (2018). [8] A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, and A. M.Kaufman, “Half-minute-scale atomic coherence and high tion,” Chin. Phys. B 30, 020305 (2021). relative stability in a tweezer clock,” Nature 588, 408

[0025] D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, (2020). S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara,[9] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. H. Pichler, M. Greiner, V. Vuletic, and M. D. Lukin, Robinson, E. Oelker, A. Staron, and J. Ye, “Resolving “A quantum processor based on coherent transport of the gravitational redshift across a millimetre-scale atomic entangled atom arrays,” Nature 604, 451 (2022). sample,” Nature 602, 420 (2022).

[0026] T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phutti-

[0010] A. Aeppli, K. Kim, W. Warfield, M. S. Safronova, and tarn, K. Jooya, P. Eichler, X. Jiang, A. Marra, B. Grinke- J. Ye, “Clock with 8 x 10 Systematic Uncertainty,” meyer, M. Kwon, M. Ebert, J. Cherek, M. T. Licht- Phys. Rev. Lett. 133, 023401 (2024). man, M. Gillette, J. Gilbert, D. Bowman, T. Ballance,

[0011] R. Finkelstein, R. Bing-Shinn Tsai, X. Sun, P. Scholl, C. Campbell, E. D. Dahl, O. Crawford, N. S. Blunt,S. Direkci, T. Gefen, J. Choi, A. L. Shaw, and M. Endres, B. Rogers, T. Noel, and M. Saffman, “Multi-qubit en“Universal quantum operations and ancilla-based readtanglement and algorithms on a neutral-atom quantum out for tweezer clocks,” arXiv (2024), arXiv:2402.16220. computer,” Nature 604, 457 (2022).

[0012] T. P. Harty, D. T. C. Allcock, C. J. Ballance, L. Guidoni,

[0027] D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. A. Janacek, N. M. Linke, D. N. Stacey, and D. M. H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, Lucas, “High-Fidelity Preparation, Gates, Memory, and D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, Readout of a Trapped-Ion Quantum Bit,” Phys. Rev. X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. SemeghLett. 113, 220501 (2014). ini, M. J. Gullans, M. Greiner, V. Vuletic, and M. D.

[0013] R. Srinivas, S. C. Burd, H. M. Knaack, R. T. Sutherland, Lukin, “Logical quantum processor based on reconfigA. Kwiatkowski, S. Glancy, E. Knill, D. J. Wineland, urable atom arrays,” Nature 626, 58 (2024). D. Leibfried, A. C. Wilson, D. T. C. Allcock, and

[0028] J. W. Lis, A. Senoo, W. F. McGrew, F. Ronchen, D. H. Slichter, “High-fidelity laser-free universal control A. Jenkins, and A. M. Kaufman, “Midcircuit Operations of trapped ion qubits,” Nature 597, 209 (2021). Using the omg Architecture in Neutral Atom Arrays,”

[0014] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Phys. Rev. X 13, 041035 (2023). Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletic,

[0029] A. V. Taichenachev, V. I. Yudin, C. W. Oates, C. W. H. Pichler, and M. D. Lukin, “Parallel Implementation Hoyt, Z. W. Barber, and L. Hollberg, “Magnetic Field- of High-Fidelity Multiqubit Gates with Neutral Atoms,” Induced Spectroscopy of Forbidden Optical TransiPhys. Rev. Lett. 123, 170503 (2019). tions with Application to Lattice-Based Optical Atomic

[0015] S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, Clocks,” Phys. Rev. Lett. 96, 083001 (2006).T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T.

[0030] L. Hu, E. Wang, L. Salvi, J. N. Tinsley, G. M. Tino, and Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, N. Poli, “Sr atom interferometry with the optical clock V. Vuletic, and M. D. Lukin, “High-fidelity parallel entransition as a gravimeter and a gravity gradiometer,” tangling gates on a neutral-atom quantum computer,” Class. Quantum Grav. 37, 014001 (2020). Nature 622, 268 (2023).

[0031] M. A. Nielsen and I. L. Chuang, Quantum Computation

[0016] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, and Quantum Information (Cambridge University Press, “Quantum dynamics of single trapped ions,” Rev. Mod. 2010). Phys. 75, 281 (2003).

[0032] J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Re¬

[0017] H. Haffner, C. F. Roos, and R. Blatt, “Quantum comcoilless optical absorption and Doppler sidebands of a puting with trapped ions,” Phys. Rep. 469, 155 (2008). single trapped ion,” Phys. Rev. A 36, 428 (1987).

[0018] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M.

[0033] T. Ido and H. Katori, “Recoil-Free Spectroscopy of NeuSage, “Trapped-ion quantum computing: Progress and tral Sr Atoms in the Lamb-Dicke Regime,” Phys. Rev. challenges,” Appl. Phys. Rev. 6, 021314 (2019). Lett. 91, 053001 (2003).

[0019] C. Monroe, W. C. Campbell, L. M. Duan, Z. X. Gong,

[0034] M. Saffman and T. G. Walker, “Analysis of a quantum A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. logic device based on dipole-dipole interactions of optiLinke, G. Pagano, P. Richerme, C. Senko, and N. Y. cally trapped Rydberg atoms,” Phys. Rev. A 72, 022347 Yao, “Programmable quantum simulations of spin sys(2005). tems with trapped ions,” Rev. Mod. Phys. 93, 025001

[0035] F. Robicheaux, T. M. Graham, and M. Saffman, (2021). “Photon-recoil and laser-focusing limits to Rydberg gate

[0020] M. Saffman, T. G. Walker, and K. Mqlmer, “Quantum fidelity,” Phys. Rev. A 103, 022424 (2021). information with Rydberg atoms,” Rev. Mod. Phys. 82,

[0036] T. Hong, C. Cramer, W. Nagourney, and E. N. Fortson, 2313 (2010). “Optical Clocks Based on Ultranarrow Three-Photon

[0021] M. Saffman, “Quantum computing with atomic qubits Resonances in Alkaline Earth Atoms,” Phys. Rev. Lett. and Rydberg interactions: progress and challenges,” J. 94, 050801 (2005). Phys. B: At. Mol. Phys. 49, 202001 (2016).

[0037] I. I. Ryabtsev, I. I. Beterov, D. B. Tretyakov, V. M.

[0022] C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg Entin, and E. A. Yakshina, “Doppler- and recoil-free atom quantum technologies,” J. Phys. B: At. Mol. Phys. laser excitation of Rydberg states via three-photon tran53, 012002 (2020). sitions,” Phys. Rev. A 84, 053409 (2011).

[0023] A. M. Kaufman and K.-K. Ni, “Quantum science with

[0038] D. S. Barker, N. C. Pisenti, B. J. Reschovsky, and G. K. optical tweezer arrays of ultracold atoms and molecules,” Campbell, “Three-photon process for producing a deNat. Phys. 17, 1324 (2021). generate gas of metastable alkaline-earth-metal atoms,”

[0024] X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, Phys. Rev. A 93, 053417 (2016). M. K. Tey, and L. You, “A concise review of Rydberg

[0039] L. H. Pedersen, N. M. Mqller, and K. Mqlmer, “Fidelity atom based quantum computation and quantum simulaof quantum operations,” Phys. Lett. A 367, 47 (2007).

[0040] Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Singlequbit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562 (2016).

[0041] B. Zhang, P. Peng, A. Paul, and J. D. Thompson, “Scaled local gate controller for optically addressed qubits,” Optica 11, 227 (2024).

[0042] T. M. Graham, E. Oh, and M. Saffman, “Multiscale architecture for fast optical addressing and control of large- scale qubit arrays,” Appl. Opt. 62, 3242 (2023).

[0043] A. J. Menssen, A. Hermans, I. Christen, T. Propson, C. Li, A. J. Leenheer, M. Zimmermann, M. Dong, H. Larocque, H. Raniwala, G. Gilbert, M. Eichenfield, and D. R. Englund, “Scalable photonic integrated circuits for high-fidelity light control,” Optica 10, 1366 (2023).

[0044] I. Christen, M. Sutula, T. Propson, H. Sattari, G. Choong, C. Panuski, A. Melville, J. Mallek, S. Hamilton, P. B. Dixon, A. J. Menssen, D. Braje, A. H. Ghadimi, and D. Englund, “An integrated photonic engine for programmable atomic control,” arXiv (2022), arXiv:2208.06732.

[0045] M. H. Levitt, “Composite pulses,” Prog. Nucl. Magn. Re- son. Spectrosc. 18, 61 (1986).APPENDIX A: ANALYZING ATOM MOTION IN an approximation of the dynamics in the Lamb-Dicke PHASE SPACE regime to the first order of 77.We parametrize the Bloch vector of the initial qubitIn this section, we derive the Heisenberg equations of state with spherical coordinates {0, ( / >}: motion for a trapped atom subject to an optical drive. We show that these take the form of semi-classical equa(MO)) = cos 0, (Bl) tions of motion of a driven harmonic oscillator, providing (M°)) = sin 0 sin (f>, (B2) us with a tool to intuitively understand the recoil-free (MO)) = sin 0 cos (f>, (B3) dynamics of quantum gates. Moreover, this also offers a straightforward explanation for extending the recoil-free with the . / '-axis defining the zenith direction. For the properties of optimized pulses from Fock states to coherevaluation of the driving force in Eq. (A5), we compute ent states. (<Mf)) under the application of a MbBbauer gate =The Hamiltonian of Eq. (1) up to first order in 77 is: 0):((7z(t)) = sin d cos(<)> — fit). (B4)H / h = — [axcos ip(t) + aysin < / ?(£)] + MaU i thi i i E (A4) d iThe Heisenberg equations of motion for the operators azwhere x(t) is the expectation value (x) and / (t) = ( / ) is and a under this Hamiltonian are: the expectation value of the driving force.With the initial condition x(0) = 0 and i(0) = 0, the cos < / ?(t) — axsin < / ?(t)] + 0(77), solution of Newton’s equation (B5) is: f ) )motion for x and p:We specialize the solution to the case considered in the(A4) text of a TT / 2 pulse defined by T = TT / (2Q) . The motional dtx = p / m state at the end of the pulse is: o i tUsing these solutions of the motion equation, we com¬APPENDIX B: MOBBAUER PULSES pute the amplitude of the coherent state average over the four qubit states considered in the quantum processWe refer to pulses with constant phase ip as MbBbauer tomography: pulses for the reasons provided in the text. These pulses provide a useful reference to be compared with the recoil- MT))2(p(T))2(ree gate. In this section, we compute the phase space <i« B10) f cni2> = (2m0)2(2p0)2dynamics associated with the atomic motional state dur¬3 £2- 2£ sin (^) + 1 ing a MbBbauer pulse. The computation makes use of the (Bit) equations of motions derived in Appendix A, which give 16 M - I)2Neglecting in this expression the small oscillating term, ~ '’ UT = U. Based on this, if a pair of initial qubit states the order of magnitude of the transferred momentum can are time-reversal conjugated, r |0) = |1) , then the two be approximated as 2po (|o(T) |2) =which is states are also time-reversal conjugated after time evolugiven in the text. tion, fU |0)q= f / | l)q. Hence, the forces / o(t) and / i (t) associated with the two initial states |0) and |1) are directly related, = — / 0(t), because of the definitionAPPENDIX C: SYMMETRIES OF PHASE SPACE of the force in Eq.= —az. Since / o(t) DYNAMICS is recoil-free, / i (t) is as well.We use the semiclassical equations of motion in Eqs. (2) and (3) (see Appendices A and B for the derivaWe prove claim (ii): if the recoil- free condition is full- tion) to prove the claims (i), (ii) , and (iii) in the text, filled for the ground motional state |0) , it is also full- namely: the recoil-free pulse condition obtained for sefilled for arbitrary motional Fock states |n)m. This dilected qubit states and the Fock state |0) extends directly follows from the fact that all Fock states give rectly to all qubit states and motional Fock and coherent the same initial condition in the semi-classical picture: states. These claims are based on the same assumptions x(0) = (n| £ |n) = 0, i(0) = (n\ p \n) / m = 0. Thereused to derive the semiclassical equations of motion in fore, according to the semiclassical equation of motion, Eq. (B5), notably retaining in the equations of motions the trajectory in phase space goes back to the origin for only first-order terms in 77. We also use the same notaall initial Fock states because they share the same initial tion introduced in Appendix B to denote the expectation conditions. value of x and f with x(t) and f(t) .Firstly, we prove claim (i): if the recoil- free condition Finally, we prove claim (iii): if a pulse satisfies the is held for the initial qubit state |0)q, (|0)q+ | l)q) / \ / 2 recoil- free conditions for the ground motional state |0) , and (|0)q+ i |l)q) / \ / 2, then it is held for all initial qubit its effect on a coherent motional state |a) is equivastates. The proof relies on the linearity of the Newtonian lent to a free propagation for a pulse duration T. The equation Eq. (B5) and of the Schrodinger equation. In phase space trajectory for an atom prepared in |0)mrepfact, by the linearity of Eq. (B5) , if Xi(t) is the solution resents a particular recoil-free solution of the inhomogefor the force fi(t) , then x(t) = CiXi(t) is the solution neous Eq. (B5), satisfying xground(0) = a^round^) = 0, for the sum of the forces, / (t) = Cifi(t). Therefore, if iground(O) = iground(T) = 0. The phase space trajecunder the force fi(t), the system is recoil- free at the end tory for an atom prepared in |o)mis also determined of the pulse (i.e., x(T) = 0 and x(T) = 0) , then under an by Eq. (B5), with the same driving force f(t) but with arbitrary linear superposition of force the system is also different initial conditions xcoh(0) = 2xoRe(a), icoh(0) = recoil- free. We also notice that the force / (t) is solely 2polm(o) / m. The solution of the motion equations of the determined by the zero-order dynamics in Eq. (4) , reprecoherent state can be expressed as the sum of the particusenting the Schrodinger equation for the qubit. Thus, if lar solution xgroundand a solution Xfree(t) of the homogethe initial density matrix pi results in the force fi (t) , then neous equation defined by the non-driven (i.e., f(t) = 0) the superposition of density matrices ^2j Ci / ?j(t) yields harmonic oscillator: xCoh(f) = a:ground(t) +xfree(t). Thus, the sum of the forces, / (t) = Therefore, if pi at the end of the pulse, the motional state of the atom produces a force fi with recoil-free dynamics, then the is a:coh(T) = ajground(T) T Xfree(T) = Xfree(T), since superposition of density matrices also produces a force the ground state dynamics is assumed to be recoil free, with recoil- free dynamics, according to the argument proAground (T) = 0. This represents the same point in phase vided above. Based on this result, we conclude that if the space that is reached by the coherent state after evolving system is recoil-free for the complete basis set of qubit for a time T in the harmonic trap, free of any drive. density matrices, that is, for the four density matrices defined by the pure states |0)q, |l)q(|0)q+ | l)q) / > / 2 andThe arguments provided above for (ii) and (iii) prove (|0)q+ f | l)q) / v / 2, then it is recoil-free for all initial qubit that for a recoil-free pulse, the final motional state in states. phase space (i.e, the centroid of the motional state dis¬Moreover, we can relax the previous requirements by tribution) is not affected by the pulse. This leaves open imposing the recoil-free condition on only three of the the possibility that the motional state distribution is disfour qubit states defined above: in fact, if the pulse is torted by the pulse, while its centroid follows the prerecoil free for the initial state |0) , we can show that it diction by the semiclassical equation Eq. (B5) . However, is as well for the initial state |1) This follows from the terms that distort the distribution are of the order of the symmetry of the Hamiltonian, f1 / / f = — H, under i / 2and higher. These terms are true quantum mechanical the time-reversal operation r = iayK, where K is the effects, which are not captured by the trajectory in phase complex conjugation operator in the basis where azis space and relate to higher-order recoil effects and qubitdiagonal. In turn, the antisymmetry of H implies that motion entanglement, both discussed in Appendices E the unitary evolution operator U is invariant under T, and F.APPENDIX D: NUMERICAL OPTIMIZATION expansion provides the basis for the analysis provided FOR A RECOIL-FREE PULSE in Fig. 3. The expansion of the Hamiltonian up to the second order in 77 isWe present the method used for optimizing recoil-free pulses in Fig. 2 and Fig. 3. The cost function is defined H(t) = H0(t) + pH^t) + p2fh(t) + ©(T?3) , (El) as the contribution of three different terms: where i i s ncontribution in Eq. (DI) is Jmot, representing the change of aia in absolute value, averaged over the four initial and qubit states used in the process tomography:sin 99(f),(E5) hi (f) = ffy cos 99(f) — oq, sin 99(f) . (D2)k — 1 n— 0 To understand the effect of recoil, we need to calculate the series expansion of U generated by this Hamiltonian. where p1;, = (7nl defines the density matrix of the For simplicity, we firstly discuss the first-order perturbainitial state |7n) = |7fc)q® ln)minspace Q ® A4. tion of the Hamiltonian in Eq. (El), rewriting it in the Here, |7fc)qrepresents one of the four qubit states, |0), form H(t) = Ho(t)+pH'(t) with H'(t) = Hi(t)+pH2(t). |1), (|0) + |l)) / V2, (|0) T zIlO / v^, whereas 177)mdenotes The expansion of the unitary to the first order in 77 is: the 77-th Fock state. The coefficients pn= p0(l ~ Po)nrepresent the occupation probability of the 77thmotional Fock state under the Boltzmann distribution. We use the U(T) = r weights went= 100, wun; = 1 and wmot = 100 for Fig. 2,and Went = 100, wunj = 1 and wmot = 10 for Fig. 3. ed as the = Uo(T)[7 + 77U'(E)] + 0(h (E6)The phase-modulation pulse is construct2)- product of a Fourier series and a regularization mask: where we factorized the zeroth-order unitary evolution 99(f) where UQ (T) and its first-order correction, described by V'(T).NcFrom the Schrodinger equation,we u(t) = ancos (77%^) + bnsin (77%^) , (D3) get the propagation of each order of U(T). n—1 The zeroth order term Uo(t) satisfies: while Ncis the cut-off number of Fourier coefficients, W0(f) = Ho(t)Uo(f), (E7) which is chosen between 40 and 60. The regulation mask avoids abrupt discontinuities in p(t) and is defined by: whose solution is:All the Fourier coefficients an, bnare numerically opis the propagation of an ideal qubit under the pulse with timized to minimize the cost function in Eq. (1)1) and, dressed Rabi frequency = fi(l — p2 / 2)

[0016] . From the thus, to obtain the recoil-free pulse. expression of UQ(T), we recognize that at the zeroth order in 77, the qubit and motion are decoupled and bothAPPENDIX E: PERTURBATIVE EXPANSION of them propagate as if there were no coupling between OF UNITARY UNDER RECOIL them. For the convenience in the later computation of the higher-order dynamics, we introduce the operators:In this section, we give a general perturbation expansion up to the second order in 77 of the unitary U gener^o(7 = U^t)h0(t)Uq(t),(E10) ated by the Hamiltonian in Eq. (1). This perturbationwhich represent h0(t) and hi(t) in the interaction picture The zeroth order dynamic [Z0(t) is same as provided in defined by Uq(t). the previous section in Eq. (E8). The first-order termThe propagation of V'(t) is defined by: Vi (t) obeys: idtV\t) = U^t)H\t)U0(t). (Ell) By integrating Eq. (Ell) and using the definitions inEqs. (E3) and (E4), we obtain a solution in the form: with the solution being:V'(T) = [«tyac)(T) _ H.c.] U1(T) = «1U«(T) - H.c., (F3)(E12) which describe the first-order recoil effect appearing in Eq. (E12). As expected, both zeroth- and first-order where the operators Wec\ We?, and V^{T') are defined terms, UQ (T) and Vi(T), obtained here coincide with below. We note that both operators Wee and We? are those derived from the calculation in the previous secresponsible for changes of motional states, and we refer tion, which is rigorous to the first order in 77. to them as recoil terms, whereas the operator Wnt'(T) The second-order term V2(t) in the unitary follows the maintains the same motional state and is responsible for equation: pure entanglement of the qubit with the motion.The expression of the first order recoil operator is: idtV2(t) =Combining this expression with Eq. (F2) , we obtain: which can be understood as a Fourier transform of hi in a rotation frame at frequency w. The effect of the second- ^tU2(t) = U^t')H2(t')U0(t') order recoil term Wee7is described by the operator: + (t), U1(t)], (F5)(E14) with the solution being:w t T c^(^ = 1 [(E15) The first line in Eq. (F6) includes the second-order recoil term and the atom-motion entanglement term in which is the de component of / / g in the rotating frame. Eq. (E12). The second line in Eq. (F6) contains terms to These results justify the series expansion provided in the the second order in 77 that are not contained in the pretext. vious section. However, we show below that these terms do not change the physical picture presented in the text.The first term, W(T)2 / 2, simply vanishes when theAPPENDIX F: RIGOROUS TREATMENT OF recoil- free condition, Vi(T) = 0, is fulfilled to the first SECOND-ORDER TERMS order in 77. The second term in Eq. (F6) is:We note that the unitary computed in Eq. (E6) is strictly valid only to the first order in 77, whereas the ex[ dt [atVi(t), Vi(t)] =Jo pression of V'(t) introduces terms, V^(T) and V^(T), which are of second order in 77. In the previous section, - 7<nX2) - 7«We(n2t)Z+ («t2U?2 / (T) - H.c.), (F7) the justification for keeping these higher-order terms in where we recognize three contributions associated with the expression of U(T) is based on physical intuition. the following operators: 6V is a motion-independent cor- Below, we provide a rigorous derivation of U(T) to the ~ (2V rection to the unitary, Wnt' is a purely motion-qubit en- second order in 77.We extend the procedure used to derive Eq. (E6) to tanglement term, and Wee (T) is a recoil term. the second order in 77, and obtain the expansion: The motion-independent correction contribution is:The purely motion-qubit entanglement term is: Note that Vn= cos(n0 / 2)<to + i sin(n(? / 2)<tnincludes only two orthogonal terms: the qubit identity operator <to and <tn. Therefore, quantum process in Eq. (G4) can44 (T) = ]odt2y dtr be represented as cos(w(ti(F9)The recoil term is:02 fT r-t-244 (T) = — Jodt2y, at! where we recognize two Kraus operators EQ = <TQ and (FEQ = <tn, with y being a 2 x 2 matrix defined by:10)( t l ) tquantum speed limit TQSL = 2TT / W.The diagonalization of the matrix % gives the two eigenvaluesAPPENDIX G: THERMAL QUANTUMWhen the recoil- fVrec (T) = 0, the propagator U(T) to the order of p2with (x+ + X-) = 1 and (1 — y+) X" 1 for 6p 1. Following the definition of Jent in Eq. (6), we obtain: can be simplified as:« E / 0(T) Uent + O(4)-~ 1^.02+ (J^2^ (G9)The term Uent is a motion-dependent unitary rotation on 6 p0the qubit and generate entanglement between qubit and motion: ss | sin2f 6p + O (6p2') (GIO) o \ " / Gent = expP^alaVjn? (T)] (G2) ~ ^(sP)e2+ o (hp2) + o (6P2e3) . (Gil)To understand the entanglement generated by E4nt, without loss of generality, we consider a motionwhere the approximations in Eqs. (G9) and (GIO) find dependent unitary acting on the qubit of the following applications in later expressions. form: To obtain an explicit expression for Jent, we evalu¬ / n \ ate 0 = — 2772|B(I / ?t (E))|, where B defines the map¬Uth = exp Ura® 4)4|, (G3) ping from a Hermitian operator A to a vector: B(A) ={Tr(AAr), Tr(Ad^), Tr(Acrz)} / 2. Therefore, we have obwhere V is a unitary on the qubit and is parametrized tained the important result that the entanglement infi- as V = exp (z(?<tn / 2), with the Hermitian operator an= delity of a recoil- free pulse depends solely on p0and n-xO’x +nyay + nzazand |n| = n2+ n2+ n2= 1. The through the expression: operator V produces a rotation of the qubit on the Bloch s t s t P aFrom f / th , we can derive the quantum process by tracing where the second equation is obtained from a series exout the motional states: pansion neglecting terms of order of 6p2pe.To develop an intuition about Jentin Eq. (G12), we£th(Pq)=Trm([ / th Pq ® Pm 4h) consider the simple example of a constant-phase pulse= Y, PoSpnVnpqV^n.(G4)(MbBbauer pulse) for a total duration T. In this case, n— 0 the zeroth-order ideal qubit dynamics is described by arotation on the Bloch sphere, Uq(t) = Rx(£lt), and the The infidelity -71k,1l)kcan be further simplified if corresponding entanglement operator is: we parametrize the Mikado unitary as I?Mik = J?Z ( / 3')-RK(T’’ / 2 + 60'}R.z(a1'), which relies on Euler’s de¬KntM = = ax™ , (G13) composition. In this parametrization, the error in theMikado pulse arises from 80. Thus, R(a', TT / 2, (3') is the nearest unitary in the Mikado family to I?Mik, resulting with the vector length |B(Ve^(T))| = QT / 2. Therein: fore, the contribution to the entanglement infidelity fromJent = 774(fiT)2+ Jent.rec- (G14)6 p0In the optimization of Mikado pulses, we minimize the cost function JMlk, which is obtained by replacing in where Jent,rec stands for the contribution to Jentfrom Eq. (1)1) the contribution from Junj with its generalized Tree* and V^ec • In fact, it should be noted that a form in Eq. (112). MbBbauer pulse is in general not recoil free, meaning that Static deviations of the Hamiltonian parameters arise that both motional heating operators V^ec and V?ec are from spatial inhomogeneities of the laser intensity. These nonzero and, thus, contribute to Jent- For the 7r / 2-pulse inhomogeneities contribute to site-dependent deviations scenario considered in the text, we obtain: of both the Rabi frequency 12 and detuning 8 (so-called probe shift) . To make Mikado pulses robust against such_ 7T28p 4 laser intensity inhomogeneities, we minimize',lf2 l p0' / '(G15)1Importantly, this expression is an approximation of the (JMik) = y £ jMikm (H4) plateau value in Fig. 3(c). where . / xllk(b~ / ) represents the Mikado infidelity for aAPPENDIX H: OPTIMIZATION OF MIKADO given intensity deviation 81 from the nominal intensity. PULSES In the numerical optimizations presented in the text, we choose 7V=11 relative intensity deviations, uniformly dis¬We discuss the optimization of the Mikado pulses, tributed in [—0.025, 0.025]. which are robust against deviations of the Hamiltonian parameters. Such pulses have been studied for nuclear magnetic resonance (NMR) applications (composite APPENDIX I: STRONTIUM-88 SETUP pulses of type B3

[0045] ) to transform one particular initial state to a statistical mixture of Bloch vectors whose The examples provided in the text are based on88Sr azimuthal angle is not fixed, but rather depends on the optical qubits. In general, the results apply to any other local imperfection. Note that in contrast to NMR aptrapped atom or ion with an ultranarrow transition. The plications, controlling the azimuthal angle is crucial to example of88Sr is special because intensity deviations realize quantum gates. Such a control of the azimuthal induce a stronger probe shift compared to other atoms angle is possible with the sequence introduced in Eq. (12) . and, thus, testing the recoil-free Mikado gates with thisThe error in the Mikado pulse is defined by the distance atom highlights their relevance and impact. between the Mikado unitary which is affected by The atom is driven at a Rabi frequency = 2TT x deviations in the Rabi frequency and detuning, and a 20 kHz, assuming that a homogeneous magnetic field of family of unitaries: B = 350 G is applied to the atom to enable the transition

[0029] . The probe shift is 8 A = 11.712 (61 / I). We note thatR(o, % / 2, / 3) = Rz(p)Rx(Hl) even small relative intensity deviations in the range of fewpercent induce significant probe shifts on the scale of 12. which generalize the simple TT / 2 pulse by including arbiThe Mikado pulse has a duration of T = 0.825TT / Q, which trary rotations around z-axis. Each unitary in this famis 65% longer than the zeroth-order quantum speed limit ily transforms the north pole of the Bloch sphere to a 7QSLfor a TT / 2 pulse. The cost function used in the nupoint on the equatorial plane. We quantify the above- merical optimizations has weights went= 100, wmot= 10, mentioned error with the infidelity measured as: wllm= 1, which is designed to strongly penalize entanglement and motional heating to achieve the recoil free’o condition. The unitary error is suppressed by optimiza¬= min -(1 - |Tr(<ikR(o, ^ / 2, / 3)) / 2|2) , (H2) o tion of the az-rotation angles discussed in the text. Averaging over the optimization interval [—0.025, 0.025] of which is obtained by generalizing the definition of in relative intensity deviations, we obtain the average cost Eq. (7) to the Mikado family in Eq. (Hl). functions (j“(k) = 5.61 x 10~5, (Je“jk) = 5.02 x 10~5,= 2.47 x 107for the optimal Mikado pulse. It Hence, we obtain the composite pulse scheme Ugdeshould be noted that while both and (-Z,)77) are scribed in in the text: of the same order of magnitude, A7,77is nearly independent of the intensity deviation SI, whereas h7,77displays a strong variation, as shown in Fig. 4(c). To the purpose Ugof quantum computing, it is key that all qubits (i.e., all sites) perform with comparably low value of the infidelity.whereAPPENDIX J: THE COMPOSITE PULSE SCHEME e1= e91- a, e2= e?> - a - is, 03= 093- p. (J5)We discuss the composite pulse scheme, which uses site-dependent azrotations in combination with the As discussed in Appendix II, the actual Mikado pulse Mikado pulses to achieve consistently low unitary infiAMU displays deviations from the unitaries R(a, TT / 2, (3) delities Junifor all qubits. in the Mikado family because of deviations in the Hamil¬We consider an arbitrary target unitary Ugfor the tonian parameters; these deviations are responsible for qubit and parametrize it using Euler’s decomposition as the unitary infidelity Au)7- The static error in the actual follows: Mikado pulse can be accounted for by the parametrization:where {69, 09, 0)} are the gate angles in this parametriza¬-RwikMf)] = R(a + 6a, — + 60, (3 + 6(3) (J6) tion. Noting Ry(0) = RX(—TT / 2)RZ(0)RX(TT / 2), we rewrite the parametrized target unitary as: -RwikM^) +7r] = -R(« + 6a, — — — 60, (3 + 6(3). (J7)where we introduce the deviation angles 6a, 6(3 and 60.Importantly, the Rxrotations in this expression can be We note that 60 is the same error angle appearing in substituted by unitaries in the Mikado family: in Eq. (113). To minimize the unitary infidelity introduced by 6a 6(3 and 60 the cy-rotation angles : )which can be derived using standard trigonometry. Here, ite pulse gate completely vanishes. In other words, the s = ±1 can be freely chosen. error induced by actual Mikado pulse can be fully cor¬For sufficiently small unitary infidelities • / ]),lllj7of the rected with cy-rotations. On the other hand, for larger Mikado pulse, the condition in Eq. (J 8) applies. When infidelities J^17, Eq. (J 9) applies. In this case, the error this is the case, the unitary infidelity . / llmof the composis only partially corrected, and . / llmcan be significantlye e g y y o o - ) s n >, y e r s . s e y g t lg. y e target unitary, , also improves for the Mikado pulses,FIG. 6. Color map showing the cost functions contributions being on average lower by a significant factor 6. This (top), Jent (middle), Jun; (bottom) as a function of the improvement is attributed to the Mikado pulses, which rotation angle (z-axis) and relative intensity deviation (t / - axis) for Mikado (left) and MbBbauer (right) composite pulses. are computed using optimal control and optimized to be robust against deviations of the Hamiltonian parameters. reduced below however, it will not entirely vanish. APPENDIX L: SATURATION OFWe can draw important insight from the previous reENTANGLEMENT INFIDELITY IN sults. The condition for vanishing unitary gate fidelity RANDOMIZED BENCHMARKING Junican no longer be satisfied when cos2($2 / 2) goes to zero. This is the case for ()g= TT. Thus, we conWe derive the asymptotic limit of the entanglement clude that the unitary infidelity . / llmof TT pulses depends infidelity Jentin the randomized benchmarking scheme more prominently on One straightforward way to presented in the text. achieve vanishing values of the . / llmeven in the case of TT Each random circuit is defined by a sequence of random pulses consists in applying twice a composite pulse gate unitaries acting on Q, targeting a TT / 2 gate instead TT gate; the price to pay is a doubling of the execution time. UC(N) = U^U^N-1\..U^1\ (LI) where each Ug^ is sampled from SU(2) according toAPPENDIX K: MIKADO VS. MOBBAUER the Haar measure. In the simulations, the individual GATES FOR COMPOSITE PULSES gates Ug^ are implemented based on the composite pulse scheme discussed in the text and illustrated in Fig. 4(b).We analyze the infidelity of the composite pulse scheme Like in Appendix K, for the global gates appearing, for an arbitrary gate Ugas defined in Eq. (J 2). For the J?2, (TT / 2) and R2,(— TT / 2) , which occur in the composite global gates, J?2, (TT / 2) and Rx( TF / 2), which occur in the pulse scheme, we consider either an implementation with composite pulse scheme, we consider either an implemenMikado gates, 4?Mik[<F(f)] and 4?Mik[<F(f) + ?r], or with tation with Mikado gates, 4?Mik[<F(f)] and 4?Mik[<F(f) + TT] , MbBbauer gates, defined by a constant phase < / ?(£) = 0and p(t) = 7T. Note that the implementation of <jz- rot at ions, the final qubit state is expected to be fully and of its counterpart G^gfiare simulated in the larger random with respect to the state evolved under the ideal space Q ® M space, which includes the motional states, circuit UC(N). Numerical simulations corroborate this throughout the entire circuit. Correspondingly, we deassumption. note the circuits realized by these pulses as: Thus, the evolution of the entire quantum circuit can be represented as:U^N) = U^> <-1}. . . G«fi, (L3) where Uris a random unitary sampled from the Haar measure. By tracing out A4, the effect on Q of the unialso acting on Q ® M.. Note that the simulation of tary above is the channel £{pq) = Uc{N')£r{pq)Uc{N')\ the quantum circuit keeps track of the motional states where throughout the whole quantum circuit and is not reduced to Q in the intermediate steps. Hence, the quantum process tomography is carried out on the entire simulated quantum circuit, using the same procedure developed in To derive the entanglement infidelity, we follow the prothe text for the individual gates Gj^ik- This approach cedure developed to compute Jent in Eq. (GIO) from allows us to quantify the evolution of the three figures of Gent in Eq. (G2) . For this purpose, it is convenient to merit, Jent, JUni, and Jmot, as a function of the number parametrize the random unitary as Ur= exp(— z0<tn / 2) , of gates sequentially applied in the circuit. The results with the rotation axis n and rotation angle 6 being ranare presented in Fig. 5 in the text. domly chosen. With this parametrization, we thus findTo obtain a quantitative expression for the asymptotic that the entanglement infidelity in the asymptotic limit limit of Jentin the randomized benchmarking, we use the of a large number of gates be expressed as results derived in Appendix G. We make the assumption that we can focus on the first two motional states |0) 2 f 0\ and | 1) and neglect the higher ones. This assumption is Jentiff) = - (1 - Po) sin o2- . (L7) \ Z J justified by the fact that the initial motional ground state probability is high, 1 — po 1, and that the recoil- free This expression can be further simplified by considering Mikado pulses preserve the probability of occupying the the distribution of 0, which according to the Haar meamotional ground state. With this assumption, we can sure on SU(2) behaves as p(0) = sin2(0 / 2) / 7r. By aversimplify U(T) in Eq. (G 1) , which describes the evolution aging Jent with this probability distribution, we obtain of a recoil- free Mikado pulse in Q ® M , to the following the following expression, expression:where / ?\iik represents an ideal pulse (i.e., an ideal which is the saturation level of Jentshown in Fig. 5. Mikado pulse transforming the north pole to the equator of the Bloch sphere) acting on Q and Rent=is derived from Eq. (G2) after expressing the entangling unitary as Gent = (-Rent)0“ • The process described in Eq. (L4) can be interpreted as follows: The Mikado pulse is perfectly implemented when the atom is in the motional state |0)m, while it suffers from a small unitary error J?ent when the atom is in the state |l)m;The unitary in Eq. (L4) shows that there are no motion changing terms in the dynamics of recoil- free Mikado pulses. If we assume, as in the rest of this paper, that the oy-rotations are ideally implemented for all relevant motional states, the motion-preserving dynamics of GMik allows us to easily calculate how the atomic state evolves for an increasing number of gates N: When the atom is in |0) state, the target circuit UC(N) is perfectly implemented. On the other hand, when the atom is in |l)mstate, due to the accumulation after each circuit step of an error caused by Gent conjugated with randomDesign of fast and high-fidelity gates for optical qubitsLeo Van Damme,1’ * Zhao Zhang,2’ * Amit Devra,1Steffen J. Glaser,1and Andrea Alberti21School of natural sciences, Technical University of Munich, Lichtenbergstrasse f, D-857J7 Garching, Germany2Max-Planck-Institut fur Quantenoptik, 857J8 Garching, Germany (Dated: August 6, 2024)Neutral atom-based optical qubits are highly valuable due to their exceptionally long lifetimes. However, they face challenges coming from spatial oscillations of the atoms in the trap, leading to qubit-motion entanglement and reduced gate fidelity. We have identified two main sources of this entanglement: photon recoil, which can be analyzed within the Lamb-Dicke regime, and atoms not being fully in the motional ground state due to imperfect cooling, requiring analysis beyond the Lamb-Dicke regime for a complete understanding. To address these issues, we developed solutions that eliminate recoil and disentangle the qubit from the motion by shaping the phase of the driving laser beam over time. Using time-optimal control, we show that simple bang-bang pulses eliminate photon recoil, reducing the gate duration by approximately 20 times compared to standard pulses in the resolved sideband regime (which requires a low Rabi frequency), while maintaining the same fidelity. Furthermore, we show that even in recoil-free processes, the gate fidelity is limited by a temperature-dependent bound — given in equation (22) — if the laser beam is unidirectional. This limit applies to the current state-of-the-art techniques, including the resolved sideband regime. To surpass it, we designed phase-modulated pulses that solve the disentangling control problem, increasing the fidelity by roughly an additional order of magnitude. Additionally, our method can be adapted to enhance the gate performance in multiple aspects, as shown through the design of robust gates against laser inhomogeneities. These results are validated through simulations of one- qubit gates operating on the optical clock transition of88Sr atoms trapped in an optical tweezers array.I. INTRODUCTIONTrapped atoms are currently among the most promising platforms for quantum information processing [1-3], due to their long coherence times, flexible connectivity, and fast gate times. These particles are confined within electromagnetic or optical fields, and their internal atomic states are manipulated using lasers, microwaves, or radiofrequency signals to achieve coherent control of quantum operations. The Hilbert space of a trapped atom is composed of two subspaces: the qubit, which is used for quantum computation, and the motion, which describes the spatial vibration of the particle in the trap. The two subspaces are coupled, meaning that the vibration of the particle can affect the gate fidelity and the coherence through qubit- motion entanglement. The strength of this coupling is measured by a parameter called the Lamb-Dicke parameter, denoted by 77. The square of this parameter, rj2, represents the ratio between the recoil energy (the energy the atom would have after emitting a photon if initially at rest) and the energy spacing between two motional states. If the recoil energy is relatively high, the emission or absorption of a photon will excite the motional states, increasing the entanglement between the qubit and the motion by a significant amount. Ideally, if 77 is 0, the qubit behaves like an ideal spin- 1 / 2 particle while the motion evolves independently and can be factored out. For a given atom and qubit transition, 77 can be diminished by increasing the trap frequency. Unfortunately, for optical qubits, 77 remains relatively high due to the power limitations of the trapping lasers. Unlike a trapped-ion hardware where ions vibrate at frequencies up to the MHz range [4], neutral atoms can be confined only about 100 kHz [5]. In hardware based on alkaline-earth atoms, the qubits can be categorized into nuclear spin qubits [6], fine-structure qubits [7], and optical qubits [8]. For the first one, the Lamb-Dicke parameter is negligible, meaning that the qubit can be driven with standard control schemes for two-level quantum systems without being affected by the motion. The fine structure qubits are often controlled with Raman schemes, consisting of applying two copropagating waves to realize two-photon transitions on the3FQ and ’’ / L states, allowing to reach an effective Lamb-Dicke parameter of / / -- 102[9]. For such a small 77, we can assume that the system evolves in the Lamb- Dicke regime, which corresponds to a first-order expansion of the dynamics with respect to 77, as we will discuss later. Although the motion causes only minor gate errors even without a recoil-free strategy, these errors can become important when many gates are applied. The motion is crucial for optical qubits based on e.g. the optical transitionThese two authors contributed equally to this work.1S'o o3P0, for which 77 < 101. causing significant interactions between the qubit and the motion. It affects the gate fidelity through photon recoil and implies undesired entanglement between the two subspaces, resulting in a loss of coherence. The photon recoil can be mitigated by driving the optical states using a relatively low Rabi frequency (compared to the trap) to avoid the excitation of the motional sideband transitions. This framework, referred to as the Resolved Sideband Regime, involves relatively slow gates, typically lasting hundreds of microseconds. Additionally, photon recoil is not the only source of qubit-motion entanglement. If the atom’s cooling is not perfect, the motion is not completely in the ground state even before the pulse is applied. Operating in the resolved sideband regime maintains the same motional state distribution before and after the pulse but does not cool the system to the ground state. Consequently, some entanglement remains, even without photon recoil, because the temperature is not exactly zero. A naive illustration of these concepts is shown in Fig. 1.FIG. 1. Naive overview of the problem. First row: The system consists of a qubit coupled to a quantum harmonic oscillator, with a few low energy states populated. The degree of qubit-motion entanglement, which is not depicted in this figure, depends on the number of populated motional states. Second row: Effect of a non-recoil-free qubit operation on the motion. Initially, the atoms are approximately 95% in the motional ground state. However, the quantum operation, driven by the laser, increases the population on higher motional states through photon recoil. Third row: Effect of a recoil-free operation, such as one performed in the resolved sideband regime, which maintains the same motional state distribution before and after the gate. In general, photon recoil excites the motion and is a source of entanglement. However, even with recoil-free processes, some motional states remain populated due to the initial temperature, also contributing to entanglement. Photon recoil can be well understood in the Lamb-Dicke regime (see Sec. Ill), while the effect of the temperature requires a deeper analysis (see Sections IV and V).For all these reasons, while hyperfine qubits can achieve fidelity levels of 99.99% under MHz-level driving

[0010] without any strategy to suppress recoil, strontium optical qubits can only achieve 99% fidelity at kHz- level driving

[0011] . The motional excitation is thus a strong limitation for optical qubits. However, optical qubits are particularly valuable for storing information due to their very long coherence times, on the order of several seconds

[0012] . Therefore, an efficient control strategy is essential to fully exploit the potential of optical qubits by designing fast and high-fidelity qubit gates.In this context, efforts have been made to understand the role of photon recoil

[0013] and to mitigate its effects on optical qubits. For example, the authors of Ref.

[0014] used a motional- state preserving pulse, which enables recoil-free 7T rotations (or NOT gates) on trapped171Yb atoms for a specific ratio between the trap and the Rabi frequency. However, this feature has been observed empirically under these particular conditions and is effective only for TT- rotations.In a parallel work

[0015] , we applied an optimization method based on state tomography to mitigate photon recoil and entanglement in a scalable sequence involving local z-rotations performed by fast and precise local addressing lasers. In this current work, we focus on achieving global recoil-free qubit gates using only the driving laser. We adopt a more mathematical approach based on perturbation theory. This method provides clear and interesting control problems and offers much deeper insights into the underlying physical processes.The paper is designed as follows. In section II, we describe the Hamiltonian governing the dynamics of trapped atoms and we give tools to characterize the photon recoil and the gate performances. We analyze the photon recoil under standard control schemes and its impact on the gate fidelity.In section III, we consider a system evolving in the Lamb-Dicke regime. We identify a recoil-free condition and use it to design any arbitrary recoil-free qubit gate in minimum time.In section IV, we study the system beyond the Lamb-Dicke regime to explore deeper effects of motion in optical qubits. We identify additional recoil-free conditions and derive very accurate recoil-free qubit gates.In section V, we study the effect of the temperature on the system. We provide a fidelity limit, given by Eq. (22), that the gate cannot exceed if the pulse phase is not modulated over time, even in the resolved sideband regime. We address this limitation using phase-modulated pulses solution to a specific control problem.In section VI, we include laser inhomogeneities in our model and derive additional robustness conditions. We present a control problem that can be solved to derive robust, recoil-free, and disentangling pulses for any arbitrary qubit gate.In section VII, we simulate our findings for an array of88Sr optical qubits and compare them to other pulse schemes.Finally, we summarize the findings and outline possible extensions of the method.II. MODELDynamics of trapped atoms We consider neutral atoms trapped in an array of optical lattices. Denoting \g) and |e) the internal states of an atom, used as ground and excited states of the qubit. The Hilbert space of the system is the tensor product between the qubit and the motional states |0), |1), |2), . . . , which are infinite in number. A general state is given by:where m denotes the 777thmotional state. We can restrict the motion to be along one spatial dimension in the same direction as the driving laser pulse. This latter drives transitions between \g) and |e) at Rabi frequency f2(t), and its phase ip(t) can be modulated over time to control the system. In a given rotating frame, the dynamics of an atom in the array is described by the Hamiltonian [4, 16] (fi = 1):where:• \g) and |e) are the internal qubit states,• fi(t) is the Rabi Frequency of the laser beam,• ip(t) is the phase of the laser beam, used to control the system,• LU is the trap Frequency, fixed to LU = 2TT X (100 kHz) in this paper,• a and of are the creation and annihilation operators,• r) = y / hk2 / (2mcu) is the Lamb-Dicke parameter (m is the atom’s mass and k = 2TT / A) ,• A(t) is the detuning of the laser frequency from the qubit transition.As explained in the introduction, the qubit is coupled to the motion through the Lamb-Dicke parameter 77, and rj2corresponds to the ratio between the recoil energy K2k2 / 2fh and the energy difference fiw between the motional states. The recoil energy depends on the atom species and the qubit transition frequency. For the sake of clarity, we focus on the optical transition1S'o o3F0(corresponds to A = 698 nm) of a88Sr atom, in which 77 ~ 0.2156 assuming a trap frequency of 100 kHz. The method applies to any other qubit in general, where 77 < 101.Photon recoil and gate fidelity The level of motional excitation corresponds to the population on the motional states |TT7 > 0). It is measured by the phonon number defined as n = (I / J\ a / a \if). This can be seen by computing n for a general state |?)) given in Eq. (1), which gives:where Pmis the population on the motional state \m). Note that n = 0 if the atom is 100% in the motional ground state. If the system experiences photon recoil while applying a laser beam, n(t) globally increases over time. Avoiding photon recoil consists of maintaining the same phonon number before and after the operation, i.e. performing a qubit gate while insuring n(T) — n(0) = 0. Since this quantity depends on the state and thus on IV’(O)), one has to nullify n(T) — n(0) for all possible initial states to fully suppress photon recoil. It is thus more meaningful to characterize photon recoil through the following operator describing the difference of motional excitation:where U is the evolution operator solution of the Schrodinger equation U = —iHU.For a given motional state m, the fidelity of the operation is measured as follows. We first define a set of four initial states IV’fcm) such that:corresponding to four initial qubit states per motional state m. Given the evolution operator U(T), each one of these initial states leads to a final state of the form = U(T) \ipkm}- Since the goal is to realize a target gate brar <= SU(2) on the qubit, we define a set of target states as IV’fem )=(%Tar ® IjifxAf) IV’fcm)- They correspond to ideal operations in the qubit subspace only while staying on the same motional state. Thus, we can define the gate fidelity associated to the motional state m as:An ideal qubit gate should correspond to J7!"1) = 1 for all m, but this is extremely restrictive and unrealistic because of infinitely many motional states. An essential property of ultra-cold atoms is that before any gate operation, the atoms are cooled up to very low temperatures, of the order of 1 pk [17-19]. This temperature determines the motional distribution before applying the pulse. We assume that it follows a Boltzmann distribution, determined by the population pg on the motional ground state according to:where M is the number of motional states, which is infinite in theory but can be truncated in the simulations. The ground state probability pg is related to the temperature via pg = 1 — e;' '; 7where kg is the Boltzmann constant and T the temperature. For example, a temperature of 1 pk corresponds to pg ~ 0.99 using a 100 kHz trap frequency. To evaluate the performances of the process, it is more insightful to weight the contribution of each W"'-1by this Boltzmann distribution and to construct an overall gate fidelity according to the formula:M has to be large enough to ensure that the simulation correctly describes the actual dynamics of the system. In this work, we choose M = 20 for all the simulated results, which is largely enough for the cases that we will consider.In all the sections concerning the analysis of photon recoil, we consider that 100% of the atoms are initially in the ground state, i.e. that pg = 1, in which case J7= J-4% . This is not true in practice because the temperature is not exactly zero. In a realistic experiment, the atoms are around 90 to 99% in the ground state. The choice pg = 1 aims to isolate the fundamental properties at the heart of the problem. The effect of the temperature will be studied in detail in a dedicated part, and a realistic pg will be used for the final analysis of our findings.III. RECOIL-FREE GATES IN THE LAMB-DICKE REGIMEAs a first approach, we can simplify the system by assuming that it evolves in the Lamb-Dicke regime, taking place when <gC 1. We will see later that this level of approximation is not enough to describe accurately theoptical transition described in the model section because 77 is not that small. However, this regime embeds the most important properties, is less complex to analyze, provides a lot of insights and gives more intuition on this kind of system. The results can be applied to systems in which 77 < 102. such as fine-structure qubit or trapped ions. We focus exclusively on the effect of motion by assuming A = 0 (no detuning) and Q = cat (negligible rising time). The inhomogeneities of the system will be studied in a dedicated section. The Hamiltonian is derived by using an expansion of the Hamiltonian (2) up to the first order in 77, leading to:where the second line is its expression in the Pauli basis, hqis a Hamiltonian of an ideal two- level quantum system and hpan operator that is orthogonal to it: hq(t) = 77 ( cos[< / ?(f)]oT + sin [< / ?(!)] cq,) ,(9) hpff) = f ( cos [< / ?(!)] cq, - sin^t)]^) , where op are the Pauli matrices.Effects of constant pulses To give a better intuition of the problem, we simulate the evolution operator U(T) using the Hamiltonian (8) with ip = 0, corresponding to a single atom driven by a constant pulse along the x axis of the Bloch sphere in the Lamb-Dicke regime. We study two target gates; a 90°-rotation about the x axis of the Bloch sphere, corresponding to a % / NOT gate, and a 180°-rotation, i.e. a NOT gate. For an ideal qubit, these gates can be realized with pulses of duration 7r / (2fi) and TF / Q, respectively. The figure 2 shows the effect of these two simple pulses on the gate fidelity as a function of the ratio w / fi.Even if we are focusing only on the Lamb-Dicke regime, Figure 2 already shows a lot of features, giving an idea of the complexity of this system. First, for all panels, the photon recoil and the gate error decrease globally as w / fi increases. This occurs because if w fi, the system evolves in the resolved sideband regime, in which the motion is not affected by the pulse. Since w is limited in practice (~ 100 kHz), this regime requires a relatively low Rabifrequency (few kHz) involving slow pulses of hundreds of / rs. Second, we can see that constant pulses perform very well for the NOT gate when w / fi is odd. This behavior is specific to NOT gates but, unfortunately, doesn’t appear for any other qubit gate. Nevertheless, it is very surprising that for these specific ratios, the gate fidelity is not limited by photon recoil. It means that the atom can absorb a photon without experiencing a momentum kick even far away from the resolved sideband regime.In the following, we will see that this feature actually exists for any arbitrary qubit gate and for any ratio w / fi, but requires controls beyond the constant pulse scheme.Recoil-free condition Since we are assuming that 77 is small, the unitary operator U(t), solution of U = —iHU, can be approximated using the Average Hamiltonian Theory

[0020] . More details are given in appendix A. We obtain:where Uqand VR6Care 2 x 2 complex matrices such that:with Uq(0) = I and VR6C(0) = 0 by definition. For clarity, we omitted the tensor product symbols. For a more rigorous expression, any operator acting on the qubit, e.g Uq, should be written as Uq® IMXM and an operator in the motional subspace as e.g 12 x 2 ® o' ■ All qubit and motion operators commute with each other.The matrix Uqcorresponds to the evolution operator of an ideal spin- 1 / 2 particle driven by the Hamiltonian hqand lAec isanon- unitary 2 x 2 matrix generating transitions between the motional states through a and of . The operatormotional subspace; it does not influence the qubit gate and can be ignored. We can show that the recoil operator R defined in Eq. (3) becomes R(t)the pulse, the effects of photon recoil is eliminated if the condition VR6C(T) = 0 is satisfied since it leads to R(T) = 0. VRec (T) = 0 is thus a rigorous way to formalize the recoil- free condition in the Lamb-Dicke regime. Note that if this condition is satisfied, we obtain U(T) = Uq(T) = U-fn-- where [ / Tar is the target qubit gate. In other words, in the Lamb-Dicke regime, eliminating photon recoil allows to reach exactly the desired qubit gate.Since VR6Cdepends on the qubit dynamics, it is possible to control the qubit in a way that suppresses photon recoil by shaping the phase ip(t) of the driving laser beam over time. To design it, we formalize a recoil-free control problem in the Lamb-Dicke regime as:The recoil-free condition adds complexity to the qubit control mechanisms compared to standard procedures used for ideal two- level systems. Importantly, il' Q < i1. the term / '~foscillates very fast compared to U^hpUq. Thus, the term VR6Caverages out over several oscillations and becomes quite small, which mitigates the effects of photon recoil. This scenario corresponds to the resolved sideband regime mentioned earlier. This regime is convenient but involves very long pulses (typically a hundred of ps) because the trap frequency is limited in practice.From a control perspective, it is highly beneficial that the original model governed by Eq. (8) has been simplified to the model of Eq. (11). Instead of dealing with an infinite-dimensional system that is complex to analyze and heavy to simulate, we now work with two 2 x 2 matrices under the constraint VRec(T) = 0. This reduces the problem to that of a standard spin-1 / 2 particle with an additional constraint, making it much more suitable for optimization. We can leverage the extensive research on optimal control of two-level quantum systems, such as composite pulses, inverse pulse engineering, gradient algorithms, shooting algorithms, geometric curves, maximum principle and more [21-30].From now on, we limit the study to a target gate of the form:UTar=e-^e^' (13) that is a rotation of angle hriiralong the . / -axis of the Bloch sphere. The problem can be solved for any arbitrary gate. The goal is to solve the control problem (12) in minimum time by finding the optimal shape ip(t) of the laser’s phase. The solutions are referred to as TORF (Time-Optimal Recoil-Free) pulses in the rest of the paper.Time -optimal recoil-free pulses In a first step, the control problem is solved using a gradient algorithm based on GRAPE

[0022] . The phase < / ?(£) is discretized in very small timesteps which can take any value (unconstrained problem). We consider a longer and longer pulse duration T until we find a numerical solution with a precision of ~ 10l 0. The result is assumed to be the time-optimal pulse (multiple initializations are tested).Remarkably, we obtain that the TORF pulses have always the same structure. For a given 0Tarand ratio w / fi, we obtain symmetric bang bang pulses made of segments of phase ip = 0 and < / ? = %. This is illustrated in Fig. 3.The pulse is determined by the three parameters (R which depend on the target gate and the ratio w / fi. This result is empirical; we don’t have a mathematical proof that there is no shorter solution

[0037] .Using a TORF pulse significantly simplifies the control problem. Since the dynamics of the qubit is only made of rotations about the . / '-axis of the Bloch sphere, we can show using the symmetries that the problem consists in finding the parameters (R solution a nonlinear system of equations (See Appendix B). One of these equation is obvious: since the qubit must reach the target gate, we have 20i — 202 + 03 = 0Tar, which can be understood by looking at the second panel of Fig. 3. The two other equations are more complicated and we didn’t find a solution without the help of a numerical algorithm. For instance, a TT / 2-TORF pulse designed for w / fi = 5 is such that 0i = 0.0840TT, 02 = 0.0269TT and 03 = 0.3858TT, leading to a pulse duration T = 0.60777r / fi. The same computation is made for all ratio w / fi, in the case 0Tar= TT (NOT gate) and 0Tar= ?r / 2 (A / NOT gate). The figure 4 shows the optimal pulse length 127’ as a function of w / fi.We obtain that for a NOT gate, if w / fi is odd, 0i = 02 = 0 while 03 = TF, which is precisely a constant vr-pulse. This is also evident in Figure 4, where the pulse duration T reaches the quantum speed limit of -K / LI under these conditions. This result matches beautifully the empirical findings outlined in Section II. Indeed, the minima visible in Figure 2 appear precisely when UJ / LL is odd, because in these conditions a constant vr-pulse is a TORF pulse. If ijj / Ll is not odd, the phase flips are necessary and the pulse is a bit longer than TT / Q. This characteristic never occurs for TT / 2 pulses. While we do observe minima of the pulse duration at A = 5, A = 9, A = 13, and so forth, it remains always higher than the quantum speed limit. This means that phase flips are always necessary to achieve recoil- free% / NOT gates. The phase flips are imperative for any qubit target gate such that ^target / TITT. Globally, we observe that in all cases, T decreases toward the quantum speed limit when w / fi — > oo. This is expected since it corresponds to the resolved sideband regime, in which photon recoil is naturally suppressed and standard pulses can be applied.IV. RECOIL-FREE GATES BEYOND THE LAMB-DICKE REGIMESince the parameter 77 is not very small in our model, the Lamb-Dicke regime does not capture all the complexity of the original Hamiltonian (2). It is thus necessary to extend the method beyond the Lamb-Dicke regime by expanding the Hamiltonian up to the second order in 77. For the sake of clarity, many details about the derivation of the following results are reported in Appendix A 3. We present here only the insightful formulas. We can show that the second-order Hamiltonian can be written as:H(t) = hq(t) (1 — yr)Qubitwhere hqand hpare given in Eq. (9). The Hamiltonian can be decomposed into three parts: i) a part describing the dynamics in the qubit subspace only, ii) one describing the coupling between the two subspaces, and iii) a part acting in the motional state only. The coupling part can be subdivided into a recoil contribution, mixing the motional states through a, a2, and al , and a pure entanglement part, acting on a^a. Note the factor (1 — r)2 / 2) on the qubit part. This term slows down the operation in the qubit subspace, regardless of the level of motional excitation. In other words, it is a direct effect of the motion on the qubit dynamics. A general expression of this factor can be found in Ref. [4],Effects of constant pulses Let’s analyze the effect of constant pulses onsimulated using the Hamiltonian (14) for a target NOT gate. We consider a standard constant vr-pulse of duration T = TT / Q and a slightly longer pulse of duration T = TF / [(1 — 772)fi] to compensate for the factor (1 — r / 2 / 2) on the qubit dynamics. The Figure 5 shows J7!0) as a function of w / fi.FIG. 5. Gate error of a target NOT gate after applying a constant pulse of duration T = TT / 'Q (blue dashed line) and of duration T = TT / [D(1 — T?2 / 2)] (solid red line) as a function of w / fi. The peaks appear for some ratios cj / fi = (2k + 1)(1 — rf / 2) with k an integer. In all cases, the Lamb-Dicke parameter is given by 77 = 0.2156 and the system is simulated using the Hamiltonian (14).We can see that using a slightly longer duration is necessary to obtain a better performance, as it compensates the lack of speed due the factor (1 — r / 2 / 2) on the qubit part, even in the resolved sideband regime. Note also that for this special case of a NOT gate, we still have the peaks of very low error as in the Lamb-Dicke regime but shifted by a factor (1 — r / 2 / 2). Like in the Lamb-Dicke regime, this property does not appear for any other qubit gate.Second-order Time-optimal recoil-free pulses Applying the Average Hamiltonian Theory, we can show that the evolution operator of the system under the Hamiltonian (14) is given by (see App. A 3):In this scenario, the fidelity is not only affected by the photon recoil but also by o / ayEnt ;which increases the level of entanglement. The expression of pEnt is remarkable since it does not depend on the trap frequency. Unlike photon recoil, this additional entanglement is not mitigated in the resolved sideband regime. This can be understood by assuming a constant pulse, in which case Uqand hqcommute and thus l is proportional to the the length LIT of the pulse. To reach a given target qubit gate, we must have LIT — 0Tarwhether we use the resolved sideband regime or not, so this entanglement term is the same. As we will see in the next section, if the two recoil- free conditions are satisfied and under our assumption pg = 1 (absolute zero temperature), this additional entanglement doesn’t affect the gate fidelity. We can thus ignore it at the moment. A second-order recoil-free control problem is defined as:By solving the problem in minimum time using a gradient algorithm, we obtain the surprising result that the pulse remains bang-bang and symmetric but with additional segments, as depicted in Fig. 6. For this kind of pulse, theduration is such that 127’ = 2(0j + 0$ + 02+ #4) + 05- One has to be careful that the qubit operations are slower than for a standard spin- 1 / 2 by a factor of 1 — 7?2 / 2. As a consequence, the pulse area is such that:The right panel of the figure shows the performances of a % / NOT gate using a second-order TORF pulse designed for w / fi = 5 and using a constant pulse. The TORF pulse is such that 0k = {0.0589%, 0.0313%, 0.1015%, 0.0097%, 0.2729%}. We can see that while the qubit gate error is -- ! x 103for the constant pulse, it reaches ~ l x 10Gusing the TORF one. This kind of performance can also be reached with a constant pulse if we operate in the resolved sideband regime, but in this case one must use a ratio w / fi ~ 130. Let us convert this into proper units to show how advantageous our technique is. For a trap of w / (2%) = 100 kHz, a ratio of 130 corresponds to a Rabi frequency of 12 / (2%) = 770 Hz, and thus a pulse duration T = % / [2O(l — T / 2 / 2)] = 332 pts. In comparison, the TORF pulse is designed for 12 / (2%) = 20 kHzand is such that T = 0.67517r / fi = 16.89 ps. In other words, the TORF technique allows one to reach the same fidelity with a -- 20 times shorter pulse!Remarkably, we can show that for the NOT gate, constant pulses of duration T = TF / [Q(1 — T / 2 / 2)] are still solutions to the problem when — ?y2 / 2)] is odd. Unlike any other gate, it is thus possible to reach performances similar to the one depicted in figure 6 for a NOT gate using these specific constant pulses.V. DISENTANGLING GATESEffect of the temperature In the above sections, we considered that the atoms are initially 100% in the ground state by setting pg = 1 in the definitions (7). This assumption has been made to capture the main effects coming from the photon recoil but is not true in practice because the atoms are not perfectly cooled to the motional ground state. The probability pg is related to the temperature of the atoms through pg = 1 — e , where kg is the Boltzmann constant and T is the temperature. In this work, we always assume a 100 kHz trap frequency, so pg depends only on the temperature. The population on higher motional states follow the Boltzmann distribution given in Eq. (6) which can be rewritten as:Currently,88Sr atoms can be cooled up to p0~ 0.98

[0019] . Generally, a small p0leads to high qubit- motion entanglement, thus to a lower gate fidelity and in fine a faster loss of coherence. This can be easily understood because when PQ is small, many motional states are populated, which increases the probability of entangling the qubit with the motion. The level of entanglement could be studied using a process tomography

[0038] , as in our parallel paper

[0015] . In this document, we prefer to study the effect of p0on the overall qubit gate defined in Eq. (7), which is quite insightful and does not require knowing the details of quantum process tomography.In the previous sections, we studied recoil-free gates that allow for keeping the same motional distribution after the pulse by canceling the excitation caused by the laser beam. However, they do not correct the entanglement coming from the temperature. Thus, whether we use the resolved sideband regime or TORF pulses, the gate fidelity remains limited. The figure 7 shows the overall error 1 — T for a target NOT gate and with a constant pulse of duration T = TF / [Q(1 — r / 2 / 2)\ as a function of w / fi (remember that this pulse is recoil-free if this ratio is odd) , taking into account various p0. As we can see, when the recoil-free condition is fulfilled (w ~ (2k + l)fi, or w / fi oo) , the gateFIG. 7. Left panel: Gate error computed for a NOT gate using a pulse of duration T = TT / [S7(1 — rf / 2)] as a function of ai / Q., for various probability po of the motional ground state. The curves are obtained by simulating the Hamiltonian (14) for g = 0.2156. The dashed lines depict the limit 1 — Him given in Eq. (22) that is reached using a recoil-free pulse or the resolved sideband regime. Right panel: Limit 1 — Him computed for a target NOT gate as a function of the temperature. error is bounded depending on PQ . AS we have seen in the previous sections, when pg = 1, this limit is 0 because the entanglement is solely due to photon recoil. However, if pg < 1, we see that even if the recoil is suppressed, the gate error cannot be reduced below a certain threshold 1 — Him , which is due to entanglement caused by the initial motional distribution.This limit can be derived using the expression of the fidelity (7) and of the evolution operator (15). If the recoil- free condition is true, i.e.= 0, if Uq(T) = Urar, and ignoring r '-" '"7' which does not play a role, theevolution operator given in Eq. (15) becomes, at the final time T:U(T) = UTare-l^a] aVE^ + o(773). (19)Using the formulae-'l'n2aVEat(T)2the contribution J7!"1) to the gate fidelity defined in (7) becomes:Note that for the ground motional state, we obtain= 1, i.e. it is not affected by VEnt- This additional entanglement concerns only higher motional states, which are more or less populated depending on the temperature of the system.A recoil-free pulse, whether it is a TORF shown in Fig 6, a constant pulse in the resolved sideband regime, or a special constant TT— pulse applied for an odd ratio w / fi (which is a special TORF), involves only rotations about x in the qubit subspace. As a consequence, hqand Uqcommute, and thus l is proportional to the area of the pulse, which is always given by 0Tar / [l —z / 2 / 2] . More explicitly, we obtain:Substituting this expression in (20) (being careful that (fymloxlfym) = 1), we obtain an upper bound per motional state:Using the full expression .F|nil= ^PmJ7^ , we obtain:As shown in the appendix D, this can be expressed as a sum of two geometric series, providing a simplification of the expression, which can be further simplified assuming that 77 is small. In the end, we obtain the fundamental limit reached by recoil- free gates:Note that for p0= 1, we obtain= 1. This shows that when the temperature is zero, the qubit gate can be infinitely accurate using a recoil-free technique. For a given target gate and trap frequency, this limit is only hw determined by the temperature of the atom, as shown by substituting po = 1 — e . This limit is very accurate, as can be seen by the dotted lines on the first panel of Fig 7.We emphasize the importance of Eq. (22) . It is impossible to break down this limit if we apply a pulse only along one direction of the Bloch sphere (usually the x direction) because in this case, Uqand hqcommute in the expression of VEnt, leading inevitably to gates that are bounded by this fidelity. As a consequence, in the current state-of-the-art techniques, i.e. using the resolved sideband regime, or using an MPP pulse

[0014] , or even a TORF pulse, the gate fidelity will always be limited by J-]imif no disentangling technique is applied. In this case, the only way to realize a very high fidelity gate is to perform an extremely accurate cooling (or to apply other techniques to decrease 77) , which is beyond the scope of this work. As shown in figure 7, at 2 / zk, the gate error is already bounded by 10 which is high, especially for a theoretical limit.Disentangling pulses Improving the cooling is thus the best way to avoid entanglement as much as possible in optical qubits. Nevertheless, we give in this section a technique to break down this limit using optimal control, at the price slower pulses. As we have shown, the entanglement caused by the temperature is described by the operator given in Eq. (16), and the one caused by photon recoil is givenWhile the photon recoil can be eliminated with bang-bang pulses depicted in Fig. 6, the effect of the temperature requires a more complex phase modulation. A disentangling pulse has to satisfy the following control problem:where all these operators are given in Eq. (16). The time-optimal solutions to this control problem are given by smooth shaped pulse phases that are difficult to understand intuitively. Such pulses are referred to as TOD (Time Optimal Disentangling) pulses. The figure 8 shows TOD pulses realizing NOT and % / NOT gates as a function of time, designed for w / fi = 5 and w / fi = oo. In the latter case, we ignored the constraints and V^cin the control problem (23). The solution can be applied in the resolved sideband regime in which photon recoil is naturally mitigated.FIG. 8. Upper panel TOD pulses realizing a NOT gate. Lower panel: TOD pulses realizing a yNOT gate. The case uj / Ul — > oo is optimized by ignoring the recoil-free constraints in the problem (23). It can be used in the resolved sideband regime. In all cases, the Lamb-Dicke parameter is r] = 0.2156.Note that these pulses have a much higher duration than the TORF pulses. For example, for w / (2%) = 100 kHz, a recoil- free \ / .\’OT gate designed for 0 / (2%) = 20 kHz requires a TORF pulse of duration T = 16.89 / zs, while the disentangling gate requires T = 48.25 / zs, which is almost 3 times longer.The figure 9 shows the performances of the TOD and the second-order TORF pulse designed for w / O = 5 assuming various p0.FIG. 9. Performances of a \ / NOT gate using a constant % / 2-pulse (CP) of duration T = % / [2fl(l — p2 / 2)], a second order % / 2-TORF pulse designed for uj / Ul = 5, and the % / 2-TOD pulse depicted in Fig 8 (designed for the same ratio), for various ground state probabilities. The solid blue line shows the limit of recoil-free gates given in Eq. (22).We can see that when pg is close to 1, the advantage of the TOD pulse compared to the TORF one is negligible.It makes a lot of sense since in this case, the entanglement is almost entirely due to photon recoil, already eliminated by the TORF technique. When pg decreases, however, we can see a gain of ~ 1 order of magnitude when using the TOD pulse, allowing for a fidelity going beyond the limit of recoil free gates.VI. ROBUST RECOIL-FREE PULSESRobustness conditions In practice, the driving laser is not homogeneous across the array of atoms. It mainly involves detuning and Rabi frequency inhomogeneities, which have a significant impact on the gate fidelity. They can be compensated by shaping the pulse phase ip(t) in such a way that the fidelity remains high over a wide range of detuning or pulse amplitude deviations. Robust control applied to inhomogeneous ensembles of spin-1 / 2 particles has been thoroughly studied in the past decades using all kinds of pulse engineering methods [21-30]. All these methods do not apply directly here because we also have photon recoil, but they can be adapted by adding the constraints given in Eqs. (16). An interesting technique consists of designing composite pulses made of TORF or TOD pulses. The idea is to use some composite pulses from the literature (or to derive them), designed on ideal spin-1 / 2 particles, and to replace the segments causing recoil by TORF pulses, with the correct phases. The method is efficient and very handy, but composite pulses are inherently slow compared to other optimal control techniques; we prefer to highlight a method for designing smooth-shaped pulses based on time-optimal control.The detuning and amplitude deviations act on the qubit subspace and can be taken into account in the model by redefining the qubit Hamiltonian hqgiven in Eq. (9) as:To design robust pulses, we choose here a technique based on the Average Hamiltonian Theory

[0020] , which allows us to perform gates that are insensitive to one (or more) order of the detuning <5A and amplitude deviations 4Q. We can show that the qubit operator subject to inhomogeneities is given by:where Uqis given in Eq. (16), while Voet anddescribe the perturbations caused by the inhomogeneities. They are given by:Remarkably, the expression of lAmp is exactly the same as VEnt- It implies that a disentangling pulse, designed for solving VEnt (I1) = 0, is also naturally robust against pulse amplitude inhomogeneities. We can interpret it the other way around: a pulse that is robust against pulse amplitude inhomogeneities is naturally disentangling. In total, a robust TORF pulse can be designed by shaping the phase ip(t) of the driving laser in order to solve the control problem:where are given in Eq. (16). We focus again on the case Urar =(rotatjon of $Tarabout x) without loss of generality. In this context, Uq(T) = UraTensures that the desired qubit gate is realized, Vftec(T) = 0 and V^2(T) = 0 ensures that the pulse is recoil free, Vbet = 0 ensures that the pulse is robust against detuning, and VAmp(T) = VEnt (I1) = 0 ensures that the system is disentangled and robust against pulse amplitude inhomogeneities. If we ignore the recoil-free constraints, the problem is already well-known. The expression of Vbet and Vkmp can be found, for example, in Ref.

[0029] , and several strategies have been made to suppress them. In our framework, this problem corresponds to the scenario where w / fi — > oo, since in this case the recoil operators are naturally suppressed and can be ignored. A pulse optimized by ignoring the recoil-free constraints can thus be applied in the resolved sideband regime with good accuracy. This kind of pulse will be referred to as a robust RS (Resolved Sideband) pulse.Robust time-optimal recoil-free pulses The problem (27) is solved in minimum time following the algorithm briefly described at the beginning of Sec. III. Fig. 10 shows robust TT and TT / 2 TORF pulses for a ratio w / fi = 5 and in the resolved sideband regime.FIG. 10. Upper panel: Phase of robust TT-TORF pulses solution to the problem (27) as a function of time (blue lines) and of robust RS pulses (dotted red lines). Lower panel: Phase of robust TT / 2-TORF and RS pulses. In all cases, the Lamb-Dicke parameter is r] = 0.2156.Note that whether we target a TT or a TT / 2 rotation, the pulses have a very similar length 127'. Additionally, we can see that even though these pulses are time-optimal solutions to the problem (27), their duration is relatively long for a given Rabi frequency. This is the price to pay in order to make the pulse robust against detuning and pulse amplitude inhomogeneities simultaneously. However, we see that the length of a robust TORF pulse is marginally higher than a robust RS pulse, and their shapes are very similar. For example, we can see that for w / fi = 5, the robust TT / 2-TORF pulse has a duration T = 3.70TT / Q versus 3.69TT / Q for the robust RS pulse. The shape is almost the same up to minor additional oscillations, which allows the suppression of the two additional recoil-free constraints. It clearly shows that it is very easy to eliminate photon recoil, i.e. this effect isn’t a problem to mitigate compared to laser inhomogeneities.VII. SIMULATION ON AN ARRAY OF TRAPPED STRONTIUM ATOMSThis section is aimed at evaluating the performances of our techniques using the full Hamiltonian given in Eq. (2) for the optical transition of88Sr atoms discussed in the model section. The detuning and the Rabi frequency deviations are assumed to be constant. The laser being inhomogeneous across the array of atoms, each of them experiences a certain detuning and Rabi frequency. We assume that the trapping lasers are homogeneous, but we could generalize our work and derive additional trap robust conditions. The dynamics of an atom in the array is described by the Hamiltonian:H(M, hfi,where w / 2?r = 100 kHz and 77 = 0.2156.The goal is to evaluate the gate fidelity of various pulses as a function of the detuning and the Rabi frequency deviations. Ideally, the fidelity should be 1 for all pairs (<5A, <5f2) but this is unrealistic. A robust pulse is such that ^(hAj hfi) ~ 1 for a relatively wide range of inhomogeneities. It can thus be evaluated by plotting F in the plane (<5A, <5f2). F is calculated using a ground state probability pg = 0.95. Figure 11 shows the error of a % / NOT gate using various pulses.First of all, we can see that the constant pulse (column a) shows a relatively high gate error (~ 103) even in the center of the plane, where the system is homogeneous. This comes from the qubit-motion entanglement, which is notsuppressed using such a pulse. Additionally, we see that the area where it works relatively well (< IO3) is small because this pulse is not robust against detuning and Rabi frequency deviations. In panel b), we apply an RS pulse, but not in the resolved sideband regime. Compared to the constant pulse, we can see that it is much more robust, but the fidelity is still limited by entanglement. This is what happens if we optimize a robust pulse while neglecting the motion in the model. When applied truly in the resolved sideband regime (panel c), the error is much lower since, in this case, photon recoil is naturally eliminated, and that entanglement coming from the temperature is also mitigated because VXmp = ^Ent is nullified. However, this pulse is excessively long (T = 920.5 / rs). In contrast, the robust TORF pulse (panel d) shows the same fidelity, using a duration that is 10 times shorter! Note that pulses b) and d) are very similar in shape and duration. It is quite remarkable that tiny differences in the pulse shape play a crucial role in suppressing the photon recoil.In the particular case of vr-gates and for a ratio of w / fi = 5 (fl = 20 kHz), the control problem (27) can be solved using composite pulses made of constant vr-subpulses with different phases. As we have shown e.g. in Fig 5, the photon recoil is suppressed in this case. To compensate for the factor 1 — rj2 / 2 on the qubit subspace, we can use a slightly higher Rabi frequency fl = fl / [l — rj2 / 2], i.e. f 1 / (2TT) = 20.48 kHz for each subpulse. Thus, we can use any composite vr-pulse from the literature that improves the robustness against detuning and Rabi frequency deviations. One of the pulses of Ref.

[0039] is precisely designed to satisfy the robustness conditions Vbetand (Eq. (16)). It is given in their table 1 (row 5(a)). It is made of 5 vr-subpulses of phase <pk = {240°, 210°, 300°, 210°, 240°}. This is the minimum number of segments necessary to satisfy the robustness condition. In other words, we need a duration T = 57r / fl = 125 / rs to solve the control problem using a composite pulse, against T = 3.77r / fl = 92.5 / rs in our case (Fig. 10).We observe that the composite performs quite well. The gate error is below 104over a wide range of <5 A / fl, but not so much along the vertical axis. Note that the area where the fidelity is high is slightly above <5fl / fl = 0 despite the correction that we made on the Rabi frequency. In contrast, the fidelity of the robust TORF pulse is well balanced and reaches an error of 103in the center of the plane, while the pulse is shorter. It shows that robust TORF pulses, even if they are more complex to optimize, perform better than other intuitive approaches.VIII. DISCUSSIONWe identified two main contributions to the lack of gate fidelity caused by the atom’s motion in the trap: the photon recoil and the imperfect cooling. These are particularly limiting for optical qubits in which the coupling between the qubit and the motion is relatively high. The photon recoil can be well understood in the Lamb-Dicke regime, even though this latter is not sufficient to capture all the effects of the motion, such as the fact that it slows down the qubit operation. After identifying recoil-free conditions, we have shown that photon recoil can be eliminatedFIG. 12. From top to bottom Pulse phase as a function of time and error for a target NOT gate using a composite pulse of Ref.

[0039] with an adapted amplitude Q / (2TT) = 20.48 kHz (left panel), and using the robust TORF pulse depicted in Fig. 10 (fi / (27r) = 20.48 kHz). using simple bang bang pulses that are very short, i.e. 20 times faster than a standard constant pulse applied in the resolved sideband regime for the same fidelity, and that in some cases even constant vr-pulses are naturally recoil-free (when w / fi is odd). Once the photon recoil is suppressed, the gate fidelity remains limited due to qubit- motion entanglement caused by the temperature of the atoms not being exactly zero after the cooling. This effect is more profound and requires a second-order expansion of the Hamiltonian in 77 to be characterized. This entanglement cannot be suppressed, whether by TORF pulses or in the resolved sideband regime, as long as the pulse is applied along a single direction of the Bloch sphere. We derived a disentangling control problem that we solved, and we showed that this additional entanglement can be suppressed at the price of slower pulses and more complex phase shapes.Later on, we included other sources of error coming from the laser inhomogeneities. We derived pulses that mitigate both types of entanglement and are robust against detuning and Rabi frequency deviations simultaneously and in a minimum time. They show excellent performances in the simulations compared to standard pulse schemes, although the inhomogeneities are costly to compensate in terms of pulse duration.We can expand our method in several directions. First, we can include several other types of inhomogeneities. For example, we can derive an additional condition to enhance the robustness against inhomogeneities of the trap frequency (see appendix C). It becomes particularly meaningful when the pulse duration becomes too long, as it may render the system recoil-free within a narrow range around the desired trap frequency. Subsequently, the method easily accommodates additional experimental effects, such as the Stark shift and the rising time of the Rabi frequency. Second, the control problem can be solved using several types of pulse schemes based on different techniques such as, among others, composite pulses, inverse engineering, Pontryagin Maximum Principle, or geometric curves. Third, the duration of robust pulses can be considerably reduced using local addressing laser beams performing rotations around the z-axis of the Bloch sphere, together with global Mikado pulses applied to all the atoms in the array

[0015] . Finally, the method might be generalizable for two-qubit gates, in which the series expansion can be made to isolate the two-qubit subspace and consider the sources of error as perturbations.Appendix A: Average Hamiltonian theory in the toggling frame1. BasicsThe average Hamiltonian theory

[0020] allows one to approximate the dynamics of an evolution operator under a non-constant Hamiltonian. Given U = —iHU where H is a given hermitian Hamiltonian, the AVH states that U(t) is given by:where H is the average (or effective) Hamiltonian given by an infinite serie whose two first terms are given by:The approximation requires |Lf(t)|t <?' 1 where | • | denotes the Frobenius norm. When the Hamiltonian is the sum of two part such that:H(t) = Ho(t) + Hi(t), (Al) it is common to apply the AVH in the Toggling Frame, which represents a frame associated to an operator UQ solution of UQ = —IHUQ . TO express the system in this new frame, we define an interaction operator Uj = UQ U where U(t) is the full solution such that U = —iHU. Expressing the derivative of UQ leads to:where Hf(t) is the interaction Hamiltonian. Ui can thus be approximated using the AVH. In this paper, we always use the first order of the AVH, leading to:In total, the evolution operator of the system is thus U(t) = Uo(t)Ui(t) with tfo(t) = — i Holt) Unit). Using the AVH in the toggling frame is more relevant since it is valid when | / / 1 1 is small, which is true in our cases.2. Application to an atom in the Lamb-Dicke regimeThe Hamiltonian expressed in the Lamb-Dicke regime — given by Eq. (8) — can be express as the sum H = UQ + 11 \ with:Lfo(t) = hq(t) + aoa^a, Hi(t) = r / hp(t)(a^ + a).HQ is the sum of two terms acting in independent subspaces. We can show that the operator UQ solution of UQ = —iHoUto is the product of two operators:where M — > oo is the number of motional states, and with Uq= —ihqUq. Using the commutation relationand the property (e-l“tat“)1 (al 4- a)e~l“tat“ =we can show that the interaction Hamiltonian is given by:Since hpand Uqcommute with a and aJ , the interaction operator U / defined by (A2) becomes:with Vj-tec = JQT(t')hp(tl)Uq(tl)eia’tdt' . Thus, using U = UQUI and UQCJOUQ = I, we obtain Eq. (11).The operator R given in Eq. (3) becomes:3. Application beyond the Lamb-Dicke regimeThe Hamiltonian (14) can be decomposed into H = HQ + Hi with:Applying the AVH in the toggling frame, we obtain:with HR Since HRec(f) and HEnt (f) are small, we haveAppendix B: Simplification of the TORE control problemUsing a piecewise constant pulse, the operator VR6Cbecomes:In this expression, the operator Uq(t) is given by:Uq(t) = Pn{t - tn)Pn^i(tn- tn-i) ■ ■ ■ F0(ti - t0)= Pn(t ~ tn)Uqn l. where Pk(r) = is the propagator associated to the fc-th segment of the pulse. Since the TORE pulse is only about x, we have hqk= (— l)fcf2cra, / 2, and thus Uqis made of rotations about x only. Additionnaly, we have hPkso we obtain the relations P^hPn= hPnPnand hPnUqn l= U%n lhPrl, leading to:Appendix C: Derivation of the robustness conditionsAppendix D: Limit of recoil-free gatesAppendix E: Miscellaneous[1] K. Wintersperger, F. Dommert, T. Ehmer, A. Hoursanov, J. Klepsch, W. Mauerer, G. Reuber, T. Strohm, M. Yin, and S. Luber, EPJ Quantum Technology 10, 1 (2023).[2] L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, Quantum 4, 327 (2020).[3] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Applied Physics Reviews 6, 021314 (2019).[4] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).[5] A. M. Kaufman and K.-K. Ni, Nature Physics 17, 1324 (2021).[6] K. Barnes, P. Battaglino, B. J. Bloom, K. Cassella, R. Coxe, N. Crisosto, J. P. King, S. S. Kondov, K. Kotru, S. C. Larsen, J. Lauigan, B. J. Lester, M. McDonald, E. Megidish, S. Narayanaswami, C. Nishiguchi, R. Notermans, L. S. Peng, A. Ryou, T.-Y. Wu, and M. Yarwood, Nature Communications 13, 2779 (2022).[7] A. Jenkins, J. W. Lis, A. Senoo, W. F. McGrew, and A. M. Kaufman, Phys. Rev. X 12, 021027 (2022).[8] I. S. Madjarov, A. Cooper, A. L. Shaw, J. P. Covey, V. Schkolnik, T. H. Yoon, J. R. Williams, and M. Endres, Phys. Rev. X 9, 041052 (2019).[9] G. Unnikrishnan, P. Ilzhofer, A. Scholz, C. Holzl, A. Gotzelmann, R. K. Gupta, J. Zhao, J. Krauter, S. Weber, N. Makki, H. P. Buehler, T. Pfau, and F. Meinert, Phys. Rev. Lett. 132, 150606 (2024).

[0010] D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., Nature 626, 58 (2024).

[0011] A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, P. Scholl, T. H. Yoon, J. Choi, and M. Endres, Nature Physics , 1 (2024).

[0012] M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, J. Ye, and A. M. Kaufman, Science 366, 93 (2019).

[0013] F. Robicheaux, T. M. Graham, and M. Saffman, Physical Review A 103, 022424 (2021).

[0014] J. W. Lis, A. Senoo, W. F. McGrew, F. Ronchen, A. Jenkins, and A. M. Kaufman, Physical Review X 13, 041035 (2023).

[0015] Z. Zhang, V. D. Leo, M. Rossignolo, L. Festa, M. Melchner, R. Eberhard, D. Tsevas, K. Mours, E. Reches, J. Zeiher,S. Blatt, I. Bloch, S. J. Glaser, and A. Alberti, in preparation (2024).

[0016] I. Lizuain, J. G. Muga, and J. Eschner, Physical Review A 76, 033808 (2007).

[0017] A. Heinz, Ultracold strontium in state-dependent optical lattices (2020).

[0018] T. Legero, J. S. R. V. Winfred, F. Riehle, and U. Sterr, in 2007 IEEE International Frequency Control Symposium Joint with the 21st European Frequency and Time Forum (2007) pp. 119-122.

[0019] R. Tao, M. Ammenwerth, F. Gyger, I. Bloch, and J. Zeiher, Phys. Rev. Lett. 133, 013401 (2024).

[0020] A. Brinkmann, Concepts in Magnetic Resonance Part A 45A, e21414 (2016).

[0021] G. Dridi, M. Mejatty, S. J. Glaser, and D. Sugny, Physical Review A 101, 012321 (2020).

[0022] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen, and S. J. Glaser, Journal of Magnetic Resonance 172, 296 (2005).

[0023] T. E. Skinner and N. I. Gershenzon, Journal of Magnetic Resonance 204, 248 (2010).

[0024] E. Dionis and D. Sugny, Physical Review A 107, 032613 (2023).

[0025] G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, Physical Review Letters 113, 043001 (2014).

[0026] L. Van-Damme, D. Schraft, G. T. Genov, D. Sugny, T. Halfmann, and S. Guerin, Physical Review A 96, 022309 (2017).

[0027] L. Van Damme, Q. Ansel, S. J. Glaser, and D. Sugny, Physical Review A 95, 063403 (2017).

[0028] H.-N. Wu, C. Zhang, J. Song, Y. Xia, and Z.-C. Shi, Physical Review A 107, 023103 (2023).

[0029] E. Barnes, F. A. Calderon- Vargas, W. Dong, B. Li, J. Zeng, and F. Zhuang, Quantum Science and Technology 7, 023001 (2022).

[0030] H. T. Nelson, E. Piliouras, K. Connelly, and E. Barnes, Physical Review A 108, 012407 (2023).

[0031] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, and D. J. Wineland, Nature 422, 412 (2003).

[0032] K. Mplmer and A. Sprensen, Physical Review Letters 82, 1835 (1999).

[0033] H. Azuma, Physica Scripta 99, 045107 (2024).

[0034] T. Choi, S. Debnath, T. A. Manning, C. Figgatt, Z.-X. Gong, L.-M. Duan, and C. Monroe, Physical Review Letters 112, 190502 (2014).

[0035] S.-L. Zhu, C. Monroe, and L.-M. Duan, Europhysics Letters 73, 485 (2006).

[0036] P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan, L.-M. Duan, and C. Monroe, Journal of Optics B: Quantum and Semiclassical Optics 7, S371 (2005).

[0037] The solution becomes singular when su / Q — > 0. This latter case is, however, not relevant because the length Q / ' of the solution explodes and our method is not valid since it is based on a perturbation theory, which is inaccurate when Q / ' is too big. Furthermore, when LJ / Q < 1, we find solutions that are very similar to the bang bang pulse of Fig. 3, but a bit smooth. Nevertheless, in this case, the smooth version is less than 0.1% shorter than the bang bang one, which is negligiblefrom a practical point of view. The singular and smooth solutions are not discussed since the first ones are irrelevant and the second are insignificantly shorter.

[0038] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (2010).

[0039] J. A. Jones, Physical Review A 87, 052317 (2013).

Claims

CLAIMS1. A method for determining a phase modulation function <p (t) for a laser pulse implementing an arbitrary quantum gate for a qubit implemented by a trapped particle, comprising: obtaining a Rabi frequency (2, and a trap frequency a) for the laser pulse and the trapped particle; determining, based on the Rabi frequency fl and the trap frequency a>, the phase modulation function < >(t) for the laser pulse such as to minimize an effect of a photon recoil of the trapped particle induced by said laser pulse on a gate fidelity of the quantum gate.

2. Method of claim 1, wherein determining the phase modulation function < >(t) for the laser pulse comprises: defining a quantum control problem based on a Hamiltonian describing the dynamics of the trapped particle when subjected to the laser pulse; and determining the phase modulation function < >(t) for the laser pulse, based on solving the quantum control problem, such that a gate infidelity measure associated with the photon recoil is minimized below a predefined threshold.

3. Method of claim 2, wherein the quantum control problem comprises a gate control condition corresponding to performing the quantum gate, and one or more phase control conditions corresponding to an effect of the photon recoil on gate fidelity.

4. Method of any of claims 2 to 3, wherein defining the quantum control problem comprises: approximating the Hamiltonian for the dynamics of the trapped particle when subjected to the laser pulse as a power series in the Lamb-Dicke parameter r| up to at least a first order in q, preferably up to a second order in q, wherein an nthphase control condition of the quantum control problem corresponds to an nthorder term of the power series in q.

5. Method of any of claims 2 to 4, wherein a first phase control condition of the quantum control problem corresponds to a first order effect of the photon recoil, a second phase control condition corresponds to second order effect of the photon recoil, and / or a third phase control condition corresponds to a vibration mode entanglement effect of the trapped particle.

6. Method of any of claims 1 to 5, wherein determining the phase modulation function < >(t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse comprises: determining, based on a Hamiltonian describing the dynamics of the trapped particle when subjected to the laser pulse, a cost function J, that depends, via the Hamiltonian, on the phase modulation function (p(t), and that quantifies a gate infidelity corresponding to the photon recoil induced by said laser pulse; and determining the phase modulation function < >(t) by minimizing the cost function J below a predefined threshold or until a predefined halting condition is met.

7. Method of claim 6, wherein determining the phase modulation function < >(t) by minimizing the cost function J comprises: selecting a basis representation, especially a basis comprising Fourier components or piecewise-constant functions, representation of the phase modulation function (p(t), and, optionally, a regulation mask for avoiding discontinuities of the phase modulation functioniteratively adjusting a set of basis coefficients that constitute the weights of each basis component in the basis representation of (t) such that the cost function J is minimized below the predefined threshold or until the predefined halting condition is met.

8. Method of any of claims 6 or 7, wherein the cost function J comprises a first contribution quantifying a deviation of the quantum gate from a desired ideal quantum gate, and on or more second contribution quantifying an effect of the photon recoil on gate infidelity.

9. Method of claim 8, wherein the cost function J comprises a weighted sum of the first and the one or more second contributions, wherein the respective weights are selected such that one or more of the second contributions dominate by at least a factor of 5, preferably by a factor of io over the first contribution.

10. Method of any of the claims 1 to 9, wherein determining the phase modulation function < / >(t) for the laser pulse such as to minimize the photon recoil of the trapped particle induced by said laser pulse comprises: determining a pulse duration and / or a pulse shape for the laser pulse to be modulated by the phase modulation function < / >(t) corresponding to the quantum gate.

11. Method of any of the preceding claims 2 to 10, wherein defining the quantum control problem or determining the cost function J comprises: including, into the Hamiltonian, an effect of a spatial Rabi frequency deviation 5 / 2, and / or an effect of a spatial laser frequency detuning 5d; and wherein determining the phase modulation function < / >(t) for the laser pulse comprises: determining the phase modulation function < / >(t) for the laser pulse such that the photon recoil is minimized below a fist predefined threshold, and such that the effect of the spatial Rabi frequency deviation 5 / 2 , and / or the effect of the spatial laser frequency detuning 5d on a quantum gate fidelity is minimized below a second predefined threshold.

12. Method of any of the preceding claims 1 to 11, wherein determining the phase modulation function < / >(t) for the laser pulse comprises: determining a pulse duration and / or a pulse shape of the laser pulse such that a effect of qubit-motion entanglement on gate fidelity is minimized below a third predefined threshold.

13. Method for performing an arbitrary quantum gate for a qubit in a quantum register formed by an array of trapped particles, the method comprising: determining a phase modulation function < / >(t) for a laser pulse implementing the quantum gate such that a photon recoil of the qubit induced by said laser pulse is minimized; and applying the laser pulse to the qubit based on the determined phase modulation function < / >(t).14- Method of claim 13, wherein determining the phase modulation function < >(t) comprises: determining the phase modulation function < >(t) by carrying out the steps of the method of any of claims 1 to 12.

15. Method for implementing arbitrary quantum gates on a subset of qubits in a quantum register, the quantum register being implemented by an array of trapped particles, the method comprising: illuminating, each qubit in the subset, with one or more respective local laser pulses configured to induce one or more local z-rotations for each illuminated qubit based on inducing local light shifts caused by the local laser pulses; illuminating each qubit in the subset, with one or more global spatially inhomogeneous laser pulses configured to induce one or more spatially dependent qubit state transformations; wherein the one or more local laser pulses are configured to compensate for spatial variations, across the subset of qubits, of the spatially dependent qubit state transformations caused by the one or more global spatially inhomogeneous laser pulses.

16. Method for implementing quantum gates according to claim 15, wherein illuminating each qubit in the subset with the one or more respective local laser pulses comprises: determining, based on the quantum gates to be implemented, a respective effective pulse area for the one or more local laser pulses to compensate for the spatial variations, across the subset of qubits, of the spatially dependent qubit state transformations caused by the one or more global spatially inhomogeneous laser pulses; and illuminating, each qubit in the subset, with the one or more local laser pulses, each having the determined respective effective pulse area.

17. Method for implementing quantum gates according to claim 16, wherein determining the respective effective pulse area comprises: determining a respective illumination time for each of the one or more local laser pulses; and / ordetermining a respective laser frequency for each of the one or more local laser pulses ; and / or determining a respective laser intensity for each of the one or more local laser pulses.

18. Method for implementing quantum gates according to one of the claims 15 to 17, comprising: illuminating each qubit in the subset with a respective first local laser pulse con figured to induce a respective first local z-rotation for each qubit in the subset; illuminating each qubit in the subset with a first global spatially inhomogeneous laser pulse configured to induce a respective first spatially dependent qubit state transformation for each qubit in the subset; and illuminating each qubit with a respective second local laser pulse configured to induce a respective second local z-rotation for each qubit in the subset.

19. Method of claim 18, further comprising: illuminating each qubit in the subset, with a second global spatially inhomogeneous laser pulse configured to induce a respective second spatially dependent qubit state transformation which essentially reverses the first spatially dependent qubit state transformation for qubits not illuminated by a local laser pulse; and illuminating each qubit with a respective third local laser pulse configured to induce a respective third local z-rotation for each qubit in the subset.

20. Method for implementing quantum gates according to claim 19, wherein at least a first of the global spatially inhomogeneous laser pulses is configured to transform a qubit state |o) or |1) to a superposition state of |o) and 11), in particular a superposition state with equal modulus square of the amplitudes for |o) and|i), and wherein the second global spatially inhomogeneous laser pulse is configured to transform the superposition state back to the state | o) or 11) .

21. Method for implementing quantum gates according to one of the claims 15 to 20, wherein illuminating the qubits in the subset with one or more global spatially inhomogeneous laser pulses comprises:applying a phase modulation function < >(t) to one or more global spatially inhomogeneous laser pulses, wherein the phase modulation function < >(t) is determined by carrying out the method of any of claims 1 to 12.

22. Method for implementing quantum gates according to one of the claims 15 to 21, further comprising: determining the spatial inhomogeneity of the one or more global laser pulses across a portion of the quantum register comprising the subset of qubits; and determining the respective effective pulse area based on the estimated spatial inhomogeneity.

23. System for modulating laser pulses used to implement one or more quantum gates for a subset of qubits in a quantum register, comprising: a first laser source for coupling a qubit ground state | o) and a qubit excited state 11>; a first set of optical components, coupled to the first laser source and configured to illuminate the subset of qubits; and one or more optical modulators for modulating an output of the first laser source to generate one or more laser pulses to implement the one or more quantum gates for the subset of qubits; wherein at least one of the optical modulators is configured to apply a phase modulation function < >(t) for the one or more laser pulses such that a photon recoil induced by said laser pulses is minimized.

24. System of claim 23, wherein the first set of optical components is configured to globally illuminate the subset of qubits.

25. System of any of claims 23 or 24, wherein at least one of the optical modulators is configured to apply a laser intensity modulation function modulating a corresponding Rabi frequency of the one or more laser pulses.

26. System of any of claims 23 to 25, further comprising: processing circuitiy, coupled to memory and configured to determine the phasemodulation function < >(t) for the one or more laser pulses by carrying out the steps of the method of any of claims 1 to 14.

27. System of any of claims 23 to 26, further comprising: an optical phase detector for detecting a phase modulation applied by the one or more optical modulators to the one or more laser pulses; and a control system for adjusting a drive signal of the one or more optical modulators such that a difference between an output of the optical phase detector and the phase modulation function < >(t) is minimized.

28. System of any of claims 23 to 27, further comprising: a second laser source for inducing a light shift on at least one of the qubits in the subset; a second set of optical components coupled to the second laser source and comprising an addressing unit configured to locally and selectively illuminate the subset of qubits with laser pulses inducing a qubit specific light shift.

29. System of claim 28, being configured to generate one or more local laser pulses according to the method of any of claims 15 to 22.

30. A quantum computer, comprising a quantum register formed by an array of trapped particles; and a quantum gate laser system comprising the system of any of claims 23 to 29.

31. Quantum computer of claim 30, further comprising: processing and control circuitiy configured to receive, via a first network from a user device, instructions for performing a quantum algorithm using a subset of qubits of the quantum register; and a qubit state readout system; wherein the processing and control circuitry is further configured to control the quantum gate laser system and the qubit state readout system, such as to perform the quantum algorithm based at least in part on performing the method of any of claims 1 to 14, and / or the method of any of claims 15 to 22;determine a result of the quantum algorithm based at least in part on measuring a state of the subset of qubits; and send the result of the quantum algorithm, via the first network to the user device.

32. Method for quantum computing, comprising: receiving, via a first network from a user device, instructions for performing a quantum algorithm using a subset of qubits of a quantum register; performing the quantum algorithm based at least in part on performing the method of any of claims 1 to 14, and / or the method of any of claims 15 to 22; determining a result of the quantum algorithm at least in part based on measuring a state of the subset of qubits; and sending the result of the quantum algorithm, via the first network to the user device.

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