Fast and high-fidelity quantum gates using dipole-dipole interactions

Abstract

The present disclosure relates to an apparatus for performing a quantum gate on a pair of atomic particles in a quantum register, comprising: a laser source, a microwave source, a set of optical elements configured to direct laser radiation generated by the laser source onto the pair of atomic particles, a microwave antenna, coupled to the microwave source, and configured to illuminate the pair of atomic particles with microwave radiation generated by the microwave source, wherein the microwave source is configured to control a duration, an intensity, a detuning and / or a phase of the microwave radiation, a laser modulator configured to control a duration, an intensity, a detuning and / or a phase of the laser radiation directed to the pair of atomic particles, and a control unit configured to control the laser modulator and the microwave source to illuminate the pair of atomic particles with modulated laser radiation and modulated microwave radiation to perform the quantum gate on the pair of atomic particles.

Description

FAST AND HIGH-FIDELITY QUANTUM GATES USING DIPOLE-DIPOLE INTERACTIONSFIELD OF INVENTION[oooi] The present disclosure relates to methods and devices for performing fast, high-fidelity, and robust quantum gates on atomic particles, such as neutral atoms, using radiation induced dipole-dipole interactions between pairs of atomic particles.INTRODUCTION

[0002] The ability to perform high-fidelity quantum gates in a fast and robust manner is a key requirement for building useful quantum computing devices. As known in the art, the computational advantage provided by quantum computing devices as compared to classical computers may be limited by the fidelity, speed and / or robustness of individual quantum gates. A sequence of an arbitrary number of such gates may implement a quantum algorithm. Such quantum gates generally act on a plurality of qubits of a quantum register. Such quantum registers may be realized, for instance, by trapping, inside a vacuum chamber, neutral atoms (such as rubidium, cesium, strontium or ytterbium atoms, etc.) or other types of atomic particles (molecules, ions, etc.) in arrays of optical tweezer traps or optical lattices or combinations thereof. Typically, realizing quantum gates for a group of two or more qubits requires interactions between qubits. For example, such interactions may be engineered using Rydberg states of neutral atoms (see e.g.: L. Henrit et al: Quantum computing with neutral atoms, Quantum 4, 327 (2020)).

[0003] Neutral atom quantum registers provide long coherence times, scalability and reconfigurable geometries for realizing arbitrary interaction connectivity between the qubits of the quantum register. For example, two-qubit quantum gates involving Rydberg states and van der Waals interactions have been experimentally realized using rubidium atoms (see S. J. Evered et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, pp. 268-272), achieving up to 99.5% gate fidelity. There remains a perpetual need for improving fidelity, speed and / or therobustness of two-qubit quantum gates for atomic particles. Any reference herein to “conventional”, “known”, “standard” or similar qualifications, as well as any citation of scientific literature, patent documents or Supplement A, is made solely for the purpose of providing background information. Unless explicitly stated, such references shall not be construed as an admission that the material forms part of the common general knowledge of the skilled person or that it was publicly available on the priority date of the present application.

[0004] For consistency, the symbols listed below are used uniformly throughout this description, the drawings and the claims:| o>, |i) - computational qubit states of a neutral atom.| n>, | r2> - first and second Rydberg states, respectively.J - resonance dipole-dipole (flip-flop) interaction strength; J > o unless stated otherwise.Vki- van der Waals interaction strength for the pair state | n< n); negative values denote attractive interactions.Ω0(t) - time-dependent laser Rabi frequency (“pulse amplitude”).n0,max - peak value of £lo (t) within a pulse; used as a normalisation constant for ranges given herein.nmw(t) - time-dependent microwave Rabi frequency.cpmw(t) - instantaneous phase of the microwave field; its derivative Amw(t)=dcpmw / dt acts as an effective detuning.Ao - laser detuning from the bare atomic 11>— > | ri> transition. Unless explicitly stated otherwise, Ao is held “essentially constant”, meaning it varies by not more than ±5 % of its absolute value during a gate pulse.“Essentially square” pulse - a pulse envelope whose instantaneous amplitude stays within ±5 % of its nominal plateau value over > 90 % of the programmed duration; ramp-on and ramp-off edges may be smoothed due to modulator bandwidth.SUMMARY

[0005] Realizing two-qubit gates for a pair of atomic particles typically includes coupling internal states of the atomic particles to a Rydberg level via electromagnetic radiation (such as laser radiation) such that the pair of atomic particles can interactwith each other through van der Waals forces. The length scale or effective range of this interaction is typically on the same order as the physical distance between the pair of atomic particles in a typical quantum register e.g., formed by trapping the atomic particles in an optical lattice or an optical tweezer trap array. Parameters of such electromagnetic radiation (e.g., duration, amplitude, frequency, phase, etc.), may be modulated in time in such a way that the desired two-qubit gate operation is realized. The modulation required to achieve the desired two-qubit gate is not unique, providing a degree of freedom that can be exploited to minimize execution time of the quantum gate and / or to maximize fidelity and / or the robustness of the quantum gate against, for instance, experimental imperfections, such as fluctuations of the intensity of the electromagnetic radiation. Fast execution times can be particularly relevant as the finite lifetime of the Rydberg state involved in the scheme above is considered a major source of decoherence that may lead to undesirable errors in the quantum computation.

[0006] The fidelity of quantum gates may be limited, inter alia, by the finite lifetime of the Rydberg states involved in mediating the van der Waals interactions between the atomic particles. Thus, reducing the execution time of a quantum gate can benefit its fidelity. Minimizing the quantum gate execution time may be facilitated by a large interacting strength. Further, the fidelity of the quantum gate may also depend on how robust execution of the quantum gate is to small variations in gate execution parameters, including but not limited to the amplitude and the phase of the electromagnetic radiation as well as the distance between the two atomic particles. Such variations may be caused by the unstable environmentally conditions, such that mitigating and / or compensating them during gate execution may be preferred over determining and eliminating their sources. Thus, reducing the sensitivity of the quantum gate fidelity to the gate execution parameters may benefit the experimental fidelity of the quantum gate. In exemplary implementations described in detail below and in Supplement A, specific Rydberg states are utilized, with | n> = |nP3 / 2, mJ = 3 / 2) and |r2) = |nSl / 2, mJ = 1 / 2) for both rubidium and cesium atoms, where n is the principal quantum number typically ranging from 40 to 50.

[0007] To improve speed, fidelity and / or robustness of multi qubit gates the present disclosure provides, inter alia, a method for performing a quantum gate on a pair of atomic particles according to claim 1, an apparatus for performing a quantum gate on a pair of atomic particles in a quantum register according to claim 14, a relatedmethod for quantum computing according to claim 16, and a neutral atom quantum computer according to claim 19. The corresponding dependent claims relate to further aspects of exemplary and / or advantageous implementations.

[0008] In some implementations, the atomic particles may be neutral atoms, such as cesium, rubidium, strontium or ytterbium atoms. The neutral atoms maybe part of a plurality of neutral atoms trapped in a plurality of optical tweezer traps or in an optical lattice or similar particle trap thereby forming a quantum register. The quantum register may be part of a quantum computing apparatus configured to execute quantum computing algorithms by performing a sequence of quantum gate operations on selected subgroups of atomic particles in the quantum register.

[0009] In some implementations, each of the pair of atomic particles may comprise two internal states that may serve as qubit states |0) and |1) and a pair of Rydberg states Ir and \r2) (see Fig. 1). An exemplary Hamiltonian H(t) describing such a two-qubit system in presence of the first and the second electromagnetic radiation may read as follows:where A and B in the subscript of a state |... ) or (... | indicate that the state belongs to particle A and B. Further, ft0(t) and 4>0(t) denote the time-dependent amplitude (e.g., expressed as a Rabi-frequency) and the time-dependent phase of the laser radiation, respectively, which couples the qubit state |1) to the Rydberg state Ir. Similarly, and ^mwC denote the amplitude and phase of the microwave radiation, respectively, which couples the two Rydberg states Ir and |r2) of each atomic particle. J denotes the strength of the dipole-dipole interaction, also known as dipolar exchange interaction, and V denotes the strength of the van der Waals interactions. As describedherein the induced dipole-dipole interaction between the atomic particles can be used for realizing improved two-qubit gates.[ooio] In some configurations, non-negligible van der Waals interactions may also be present and may have to be taken into account when designing pulse modulation protocols for two-qubit quantum gates with high fidelity, speed and robustness that are enabled by the induced dipole-dipole interaction. Further details are discussed in Supplement A and G. Giudici et al. Fast entangling gates for Rydberg atoms via resonant dipole-dipole interaction (submitted on 7 Nov 2024 and published on 11 Nov 2024 as arXiv: 2411.05073 [quant-ph]) hereby incorporated by reference in its entirety for all technical details, control algorithms and numerical parameters disclosed therein, to the extent that such matter is not inconsistent with the explicit disclosure of the present description. Throughout this disclosure, the (time-dependent) amplitude of the electromagnetic radiation is expressed in terms of Rabi frequency, e.g., denoted as £lo (t) for laser radiation and Qmw(t) for microwave radiation. As known in the art, the Rabi frequency is directly proportional to the electric field amplitude of the electromagnetic radiation and to the dipole moment of the corresponding atomic transition. For simplicity and consistency, we use l0and Qmw to denote both the Rabi frequencies and the normalized amplitudes of the respective electromagnetic radiation fields, with the understanding that proper normalization factors are implicitly included and clear from the context. This convention is maintained throughout all mathematical expressions, figures, and experimental protocols described herein.

[0011] In some implementations, a desired two-qubit gate to be realized based on induced dipole-dipole interactions may be a controlled-Z (CZ) entangling gate parametrized by an angle 9. In the two-qubit basis { 100), 101), 110), 111) } such a gate operation can be defined as:CZ(&) = diag(l, e10, e10, — el20)Such a CZ entangling gate may be realized using a sequence of pulses of electromagnetic radiation directed at the pair of atomic particles, comprising, for example, both laser and microwave radiation. In some implementations, radiation pulses delivering the first and the second electromagnetic radiation to the pair of atomic particles may be applied at least in part simultaneously. An exemplary protocol may involve two pulses of electromagnetic radiation, one comprising laser radiationcoupling the first qubit state 11) to a first Rydberg state liq), the other comprising microwave radiation coupling the two Rydberg states 1^) and |r2)- Both pulses may have the same duration (the gate execution time) and overlap completely in time. In other implementations the pulses may have different durations and / or may overlap partially in time. Further details can be found in Supplement A below. For example, some dipole-dipole interaction protocols described herein provide significant advantages over conventional van der Waals protocols, including up to 20% reduction in gate execution time and improved robustness against Rydberg decay.

[0012] Such a protocol may realize a CZ gate up to a single-qubit phase shift. This may be seen from, without loss of generality, from the Hamiltonian in Eq. (1) in a limit of strong dipolar interactions (e. g.,] » fl0, | Ao|, flmw, | Amw|), strong microwave detuning (e.g., Amw» fl0, Ao, flmw) and without considering, for ease of analysis, van der Waals interactions (F = 0). The Hamiltonian of Eq. 1 may then by simplified to:where W(t) = — qt is an effective interaction strength of a van der Waals type4 fl (tand A(t) = A0(t) —mwis the detuning of an effective electromagnetic radiation(combining the two pulses described above).Both, W(t) and A(t) may depend on the amplitude flmwand the detuning Amwof the microwave radiation. Eq. 2 can describe a system comprising two qubits interacting with each other through van der Waals forces and can be used to engineer a two-qubit CZ entangling gate, as described in, for example, H. Levine et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, PRL 123, 170503 ff.

[0013] Generally, realizing a two-qubit quantum gate with optimal fidelity may be achieved by determining an optimal pulse duration of each pulse and / or an optimal modulation of pulse parameters by numerical optimization procedures. This allows to implement fast, robust and high-fidelity gates also in less ideal situations, e.g., inpresence of non-negligible van der Waals interactions between the pair of atomic particles. Pulse parameters to be modulated may include radiation intensity ro amplitu de, phase and a detuning of the electromagnetic radiation (with respect to the relevant atomic transition frequencies) delivered by the radiation pulses.

[0014] In such a scenario, achieving the highest possible quantum gate fidelity may require optimizing the pulse duration and modulating several pulse parameters. For example, the gate fidelity maybe defined as a Bell state fidelity:2FBeii = |<^|texp(-^J0TH(t)dt) |+ +)|, where(3)- e2lG111)) is a Bell state that can be obtained when applying the CZ quantum gate to a product state |+ +) as known in the art (see for example M. A. Nielsen and I. L. Chuang: Quantum Computing and Quantum Information). Eq.3 allows to define a cost function that may be used for numerical pulse parameter optimization as discussed in more detail below and in Supplement A. For example, the following cost function C maybe defined:C — 1 — FBeu,(4) This cost function or a similar one may be minimized by determining an optimal modulation of the pulse parameters, such as n0(t), 4>0(t), flmw(t) andfor a set of pulse durations. Depending on the application scenario and the required gate performance, the Hamiltonian H(t) used in Eq.3 maybe the Hamiltonian H(t) of Eq. (1) or a suitable and simpler approximation that may be used to speed-up convergence of the numerical optimization procedure.

[0015] In some implementations, determining such time-dependent pulse parameters may comprise utilizing optimal control computational methods, such as Gradient Ascent Pulse Engineering, GRAPE, as detailed in, for example, B. Riaz, Optimal control methods for quantum gate preparation: a comparative study, Quantum Inf. Process. 18, 100, (Fig.2 for details). After determining the duration of each pulse as well as its modulation parameters, the electromagnetic radiation may bemodulated and shaped based on the determined modulation parameters and duration, such that the two-qubit gate is executed with optimized fidelity, speed and / or robustness. In this way, the negative impact of van der Waals interactions on the fidelity of the two-qubit quantum gate may be mitigated by finding optimal timedependent parameters for the pulses delivering the laser and the microwave radiation to the two-qubit system. Determining such time-dependent pulse parameters may comprise utilizing optimal control computational methods, such as Gradient Ascent Pulse Engineering (GRAPE), as detailed in Supplement A, which employs cost functions that account for both quantum gate fidelity and robustness against experimental imperfections such as interatomic distance fluctuations.

[0016] In the context of this disclosure, amplitude modulation of electromagnetic radiation refers to any time-dependent variation of the radiation intensity or amplitude, including but not limited to gradually varying the amplitude over time and switching the radiation completely on and off to create square pulse shapes. Square pulse modulation, where the amplitude is rapidly switched between zero and a constant value, represents a common form of amplitude modulation particularly suited for quantum gate operations due to its experimental simplicity and effectiveness.Throughout the description, references to amplitude modulation should be understood to encompass such square pulse implementations unless otherwise specified. Gaussian pulses or similar are also possible of course.

[0017] In some implementations, determining the optimal modulation of the pulse parameters may also comprise maximizing the robustness of the quantum gate against fluctuations, such as inter-particle distance fluctuations of the two atomic particles. For example, in a quantum register comprising a plurality of neutral atoms trapped in optical tweezer traps, the position of the tweezer traps, and, therefore, the distances in between the atoms, might fluctuate e.g., due to mechanical vibrations or trap laser instabilities. In addition, each atomic particle may not occupy the motional ground state of its trap. As a consequence, the position of the atomic particle in the trap is subject to thermal fluctuations. Such fluctuations in the distance between the two atomic particles may lead to fluctuations of the dipole-dipole interaction strength J and the van der Waals interaction strength V, since both interaction strengths depend on the distance between the two atomic particles. As a consequence, the fidelity of thequantum gate may be diminished as the interaction strength may not retain a constant value throughout the gate execution time.

[0018] In order to take such fluctuations into account while determining the optimal time-dependent parameters of each radiation pulse, the cost function in Eq. (4) may be replaced byWhere xMdesignates the maximum fluctuation of the distance between the two particles and FBeu (x) is the Bell fidelity obtained for the interaction strengths J and V governing the dynamics for the distance x. The second term in Eq. (5) may express a fidelity averaged over the inter-atomic distance within a typical range of distance fluctuations. Next, the optimal duration and modulation of the pulses maybe determined, based on the cost function C of Eq. 5 as described above and in more detail in of Supplement A. For neutral-atom tweezer arrays operated at room temperature we typically set the maximum modeled distance fluctuation to 8Rmax«3 % of the nominal trap separation, i.e. X_M = 0.03 in Eq. (5). This value corresponds to the root-mean-square spread of atomic positions achieved after Raman sideband cooling to 2 uK in traps of co__trap / 2n®ioo kHz.

[0019] In some implementations, determining the time-dependent pulse parameters comprises optimizing the fidelity of the quantum gate, the robustness of the quantum gate against inter-particle distance fluctuations, or both simultaneously. When optimizing robustness against inter-particle distance fluctuations, the optimization procedure explicitly accounts for how such fluctuations induce variations in both the dipolar interaction strength J and the van der Waals interaction strength Vij. Since both interaction strengths depend on the distance between the two atomic particles as J(R) ~ C3 / R3and Vy (R) ~ C6 / R6respectively, fluctuations in the interatomic distance R lead to fluctuations 8J / J = -38R / R and SVy / Vy = -68R / R. The optimization procedure can be configured to find pulse parameters that maintain high gate fidelity across a specified range of these interaction strength fluctuations, thereby making the quantum gateIOoperation robust against experimental imperfections in the positioning and trapping of the atomic particles.

[0020] In some preferred implementations, modulating the first and the second electromagnetic radiation comprises modulating the pulse amplitude £l0(t) of the first electromagnetic radiation at essentially constant detuning Ao, while modulating both the pulse amplitude ilmw(t) and the pulse phase cpmw(t) of the second electromagnetic radiation. For optimal gate performance, the pulse amplitude £20(t) of the first electromagnetic radiation may be modulated as an essentially square-pulse function, providing constant laser intensity during the gate operation. Similarly, the pulse amplitude Qmw(t) of the second el ectromagnetic radiati on may also be modulated as an essentially square-pulse (e.g., a square-pulse with smooth edges), which simplifies the experimental implementation. The essentially constant detuning Aoof the first electromagnetic radiation maybe selected to lie in the interval [-0.25 Qomax, o], where flomax is the maximal value of the pulse amplitude £0(t), as this range has been found to provide optimal balance between gate speed and fidelity. This configuration simplifies the experimental control requirements by eliminating the need for complex laser phase modulation while maintaining high gate performance through precisely controlled microwave modulation. A notable technical advantage of the disclosed quantum gate implementation is that it requires only microwave phase and amplitude modulation without optical phase modulation, significantly simplifying hardware requirements and reducing sensitivity to laser phase noise caused by the optical phase modulation.

[0021] Further details of the aspects described generally 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 advantages will 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 drawings. Each of the drawings is provided for the purposes of illustration and description, and not as a definition of the limits of the claims.BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Fig. 1 illustrates an exemplary system comprising two atomic particles for performing a quantum gate utilizing dipole-dipole interactions according to an exemplary implementation of the present disclosure.

[0023] Fig. 2 shows an exemplary result of determining the optimal modulation of the pulses delivering a first and a second electromagnetic radiation to the two atomic particles for realizing a quantum gate according to an exemplary implementation of the present disclosure.

[0024] Fig. 3 illustrates a method for performing a quantum gate on a pair of atomic particles according to aspects of the present disclosure.

[0025] Fig. 4 illustrates a method for quantum computing according to aspects of the present disclosure, e.g. by using an apparatus as described herein.

[0026] Fig. 5 illustrates a block diagram of an exemplary quantum computing device according to a possible implementation of the present disclosure.

[0027] Fig. 6 illustrates several components of a neutral atom quantum computing device according to a possible implementation of the present disclosure.

[0028] Fig. 7 illustrates a block diagram of an exemplary apparatus for performing a quantum gate on a pair of atomic particles according to aspects of the present disclosure.

[0029] Fig. 8 to Fig. 18 and Fig.21 refer to the research article of Supplement A.

[0065] FIG. 19 shows various pulse sequence implementations for the quantum gate protocols disclosed herein using dipole-dipole interactions.FIG. 20 illustrates a comprehensive conceptual overview of the dipole-dipole interaction quantum gate implementation as disclosed herein.DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0030] 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, an apparatus, a device or a system maybe implemented, or a method may be 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, apparatus, 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, apparatuses, 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.

[0031] Several aspects of trapping and manipulating (e.g., imaging, gate operations, spectroscopy, etc.) atomic particles will now be presented with reference to various devices, apparatuses, 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 maybe implemented using hardware, software, or combinations thereof. Whether such elements are implemented as hardware and / or software depends upon the particular application and design constraints imposed on the overall system. Further, the apparatuses and methods disclosed herein can be part of complex quantum technology systems such as neutral atom quantum computers. The skilled person will appreciate that in the following several components of such systems, such as certain laser sources, specifics of experiment control and timing units, optical setups, etc. may not explicitly be described.

[0032] FIG. 1 illustrates an exemplary system comprising a pair of atomic particles 100 that may be used as qubits for performing a quantum entangling gate utilizing dipole-dipole interactions. The two qubits, labelled A and B, are located at a distance x away from each other. Each atomic particle qubit has an internal level structure comprising four energy levels: the qubit states, designated by |0) and 11>, and two Rydberg states, designated as |r1) and |r2) used for inducing the dipole-dipole interaction. The system of two qubits may be illuminated by laser radiation and microwave radiation. The laser radiation has an amplitude of Ω0and a phase of φ0and may couple one of the qubit states, possibly 11), to one of the Rydberg states, possibly Ir. The microwave radiation has an amplitude of Ωmwand a phase of φmwand may couple the two Rydberg states 1^) and |r2). As a consequence of the illumination, the two qubits may interact with each other through dipole-dipole interactions of strength J(x) and V(x) and van der Waal interactions of strength V(x), both of which may depend on the physical distance x between the two qubits. The two-qubit system depicted in Fig. 1 may be described by the Hamiltonian in Eq. 1 or similar that may be used for numerically optimizing pulse parameters of the laser and microwave radiation used for performing the two-qubit gate of the pair of atomic particles.

[0033] FIG. 2 illustrates exemplary results of determining the optimal (timedependent) parameters of the pulses delivering laser and / or microwave radiation to the two atomic particles for realizing a quantum gate. The determination of optimal parameters comprises determining the duration of each pulse T as well as its timedependent parameters, possibly including amplitude (n0(t) and / or flmw(t)), phase(4>0(t) and / or c|)mw(t) ) and detuning (4>0(t) and / or c|)mw(t) ). The result of this determination is then used for modulating the pulses based on the determined timedependent parameters, such that the interactions between the two atomic particles realize a desired quantum gate operation. In each of the panels (a) - (d) the top part shows the result obtained for neutral cesium atoms and the bottom part for neutral rubidium atoms.

[0034] Panel (a) shows an infidelity (defined as 1 - FBell) as a function of the relative fluctuation 6R / R in the interatomic distance R, illustrating how the quantum gate fidelity may vary with the interatomic distance since as described above the interatomic interactions (dipole-dipole and / or van der Walls) depends on the interatomic distance. The presence of atomic distance fluctuations may be taken into account by a suitable pulse parameter optimization procedure based on, for example, the cost function in Eq.5.

[0035] Panel (b) shows the optimized phase of the laser radiation φ0as a function of time t, normalized with respect to the optimal gate duration T = T*.

[0036] Panel (c) shows the optimized phase of the microwave radiation (pmwas a function of time t, normalized with respect to the optimal gate duration T = T*.

[0037] Panel (d) shows the optimized ratio Ωmw / Ω0between the amplitude of the microwave radiation and the amplitude of the laser radiation as a function of time t, normalized with respect to the optimal gate duration T = T*.

[0038] The optimal pulse parameters shown in panels (a)-(d) may then be used to realize a quantum gate with the highest possible fidelity, robustness, and speed.

[0039] FIG. 3 illustrates a method for performing a quantum gate on a pair of qubits (for example, neutral atoms including cesium, rubidium, strontium and rubidium atoms), comprising: illuminating the qubits with laser radiation, such that in each qubit a first qubit state 11) is coupled to a first Rydberg state |r1). The method further comprises illuminating the qubits with microwave radiation, such that in each qubit the first Rydberg state |r1) is coupled to a second Rydberg state |r2), such that adipole-dipole interaction between the two qubits is induced. The method of Fig. 3 may also comprise one or more of the steps disclosed by the methods specified in appended claims 2 to 13.

[0040] The method is generally executed inside a vacuum chamber that houses a neutral-atom quantum register formed by an array of optical traps (e.g. optical tweezers or an optical lattice). Unless stated otherwise, reference is made to the illustrative level scheme of Fig. 1 in which the computational basis states are |o) and 11), the first Rydberg state is | n> and the second Rydberg state is | m).

[0041] In step 310 the pair of atomic particles is illuminated with a first electromagnetic radiation that resonantly or near-resonantly couples the qubit state 11) to the Rydberg state | n> of each atom. In preferred implementations the first electromagnetic radiation is laser radiation having an optical frequency in the range 320-800 THz, i.e. in the visible or near-infra-red. The laser radiation is delivered in the form of at least two radiation pulses, each pulse having a controlled pulse duration Tp. The two (or more) laser pulses may overlap in time such that they are applied at least partially simultaneously with the microwave pulses described below. Prior to emission, one or more time-dependent pulse parameters are determined, including at least the pulse amplitude Ω0(t) and, in some variants, a fixed detuning Ao and / or a time-dependent laser phase cpo(t). The parameter set may additionally comprise controlled ramp-on and ramp-off edges to account for finite modulator bandwidth. The determination of these parameters can be carried out by an iterative numerical optimization routine— preferably gradient-ascent pulse engineering (GRAPE)— that minimizes a cost function related to gate infidelity and / or maximizes robustness against experimental imperfections. In a particularly convenient embodiment, the laser detuning Ao is held essentially constant during the entire gate operation and is selected to lie in the interval -0.25 Qo,max< Ao < o, where 2o,maxdenotes the peak Rabi frequency of the laser pulse. The laser amplitude profile flo (t) is advantageously chosen as an essentially square pulse (i.e. flo (t) ® 2o,maxfor o < t < Tpapart from smooth leading and trailing edges). As used herein, “essentially square” means that the amplitude stays within ±5 % of 2o,maxover at least 90 % of the programmed pulse length. Concurrently with, or temporally interleaved with, step 310, step 320 applies a second electromagnetic radiation that couples the two Rydberg states | n> and | ofeach atom thereby inducing a resonant dipole-dipole interaction J between the pair. The second electromagnetic radiation is microwave radiation in the frequency range 5-70 GHz, generated for example by a direct digital synthesis (DDS) source followed by amplification and emission via a waveguide or chip-scale antenna or similar. In an analogous variant, the microwave amplitude Ωmw(t) is likewise modulated as an essentially square envelope (i.e. Ωmw(t)≈Ωmw,maxover >90 % of the pulse length) while the instantaneous microwave phase cpmw(t) is modulated to generate the required timedependent detuning Amw(t)=dcpmw / dt.

[0042] Like the laser field, the microwave field may be applied in the form of at least two radiation pulses that may overlap entirely or partially with the laser pulses. One or more time-dependent pulse parameters may be determined, most notably the microwave amplitude Ωmw(t) and the microwave phase φmw(t); optionally a microwave detuning Amw(t) can also be optimized. The optimization may pursue a target functional that (i) maximizes gate fidelity, (ii) maximizes robustness with respect to inter-particle distance fluctuations that would otherwise modulate J and the residual van-der-Waals terms Vy, or (iii) achieves a weighted compromise between (i) and (ii). The optimization procedure can again rely on GRAPE or on an equivalent gradient-based optimal-control solver, sampling the system Hamiltonian for a predefined distribution of distance variations (e.g. ±3 % around the nominal trap separation) to render the gate intrinsically robust. The method may also involve a computer-implemented control routine. As mentioned above, the routine may employ GRAPE or another iterative numerical method that successively refines the pulse parameters until a target gateerror threshold (e.g. 10-3) is met. The same routine can include a Monte-Carlo sampling layer that evaluates gate fidelity at multiple discrete inter-atomic distances Rk = Ro (1 + 8k) with 8k G [- 8\i;ix, +8Max] and feeds the averaged fidelity back to the cost function, thereby embedding robustness criteria directly into the optimization loop.

[0043] A representative optimisation uses N = 200-600 equidistant time steps, a second-order update rule (L-BFGS-B) and a stopping criterion of ε ≤ 10-6on the incremental reduction of the cost function. Control-field updates are constrained such that |ΔΩ| / Ω ≤ 2 % between successive iterations, ensuring smooth convergence and compatibility with analogue waveform generators of 250 MS s-1or higher.

[0044] When the modulated laser and microwave pulses terminate, the j oint state of qubits A and B may has undergone an entangling controlled-Z (CZ) or CZ-0 operation (0 ® n in the simplest case). This gate can be part of a universal gate set when complemented with arbitrary single-qubit rotations and therefore enables construction of larger quantum algorithms. In typical hardware the pair of atoms remains part of a larger register containing 102–105traps, so that the method of Fig. 3 can be executed in parallel on several disjoint atom pairs or sequentially on selected pairs according to the connectivity graph required by the algorithm. The foregoing method may be executed on the neutral-atom quantum computer illustrated in Figs. 5-7, where a laser source 705, laser modulator 725, microwave source 710 and microwave antenna 720 are under control of a central FPGA or controller 730. The trap array may typically be held at a base pressure below 1 × 10-10mbar, and site-resolved imaging allows verification of successful gate operation via repeated Bell-state analysis or quantum process tomography or similar. The control unit 730 may, by way of non-limiting example, comprise: a field-programmable gate array (FPGA) running at 250 MHz that synthesizes the digital control words for all analogue outputs, two 14-bit arbitrary waveform generators (AWGs) clocked at > 1 GS s-1— one driving the acousto-optic modulator in the laser path, the other driving a direct-digital-synthesis (DDS) microwave chain; a microwave up-conversion stage including an I / Q mixer and a power amplifier (> +30 dBm) feeding a horn antenna or on-chip coplanar waveguide situated inside the vacuum envelope; a synchronized photon-counting module or high performance CCD camera for qubit read-out.

[0045] FIG. 4 illustrates a method for quantum computing, comprising, optionally, obtaining 410 a set of instructions for performing the set of quantum gate operations on the selected subset of the trapped particle qubits of the quantum register, and trapping 420 a plurality of particles in an array of optical traps forming a quantum register of trapped particle qubits, and performing 430 a set of quantum gate operations of a quantum computing algorithm on a selected subset of trapped particle qubits by manipulating an internal and / or a motional state of the selected subset of trapped particle qubits. The method further comprises determining 440 a result of the quantum computing algorithm by measuring a state of the selected subset of trapped particle qubits, and, optionally, outputting 450 data corresponding to the result of thequantum computing algorithm. The method of Fig. 4 may also comprise one or more of the steps disclosed by the methods specified in appended claims 17 or 18.

[0046] FIG. 5 shows a typical implementation of a quantum computer 500 comprising a quantum register 510, e.g. formed by a plurality of trapped particles, and a quantum gate laser system 520. In some implementations, the quantum computer can be controlled by a (remote) user device 560, possibly via a network 550. In some implementations, the quantum gate laser system 520 may be configured to create plurality of trapped particles and / or manipulate and / or cause a controlled quantum state evolution of one or more atomic objects within the quantum register 510. For example, the quantum gate laser system 520 may comprise one or more lasers, which provide one or more laser beams to atomic objects (such as neutral atoms, molecules or ions) in the quantum register 510.

[0047] In some implementations, the quantum gate laser system 520 may comprise a laser source 520a, a laser modulator 520b, a microwave source 520c and a microwave antenna 520d. The quantum gate laser system may further comprise a set of optical elements configured to direct laser radiation generated by the laser source onto the the pair of atomic particles in the quantum register 510. The microwave source 520c may be configured such that a duration, an intensity or amplitude, a detuning and a phase of the microwave radiation is controllable. The laser source 520a may be configured such that a duration, an intensity or amplitude, a detuning and a phase of the laser radiation is controllable.

[0048] 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, thedetectors may be in electronic communication with the processing and control circuitry 640.

[0049] In some implementations, the user device 560 may be configured to allow a user to provide input to the quantum computer 500 and receive, view, and / or the like output from the quantum computer 500. The user device maybe in communication with the processing and control circuitry 540 of the quantum computer 500 via one or more wired or wireless networks 560 and / or via direct wired and / or wireless communications. In an example embodiment, the user device 550 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 540 can understand and / or implement.

[0050] In some implementations, the processing and control circuitry 540 may be configured to control, inter alia, the quantum gate laser system 520 and / or the qubit state readout system 530. For example, the processing and control circuitry 540 may be configured to cause a controlled evolution of quantum states of one or more atomic objects within the quantum register 510 to execute a quantum circuit and / or algorithm. For example, the processing and control circuitry 540 may cause a reading procedure comprising, possibly as part of executing a quantum circuit and / or algorithm. In various embodiments, the atomic objects confined within the quantum register 510 are used as qubits of the quantum computer 500.

[0051] FIG. 6 illustrates several components of a neutral atom quantum computing device according to a possible implementation of the present disclosure. The quantum computing device comprises a vacuum chamber 610 configured to house and isolate the quantum register from environmental disturbances. Within the vacuum chamber, a quantum register 620 is formed by an array of optical traps, such as optical tweezer traps or optical lattices, configured to trap and hold a plurality of atomic particles at well-defined positions. The device further includes a trap laser system 630 for generating optical traps to confine the atomic particles, and a quantum gate implementation system 640 comprising both laser and microwave sources for performing the quantum gate operations. The quantum gate implementation system640 is specifically configured to generate the first electromagnetic radiation (typically laser radiation) for coupling the first qubit state to the first Rydberg state, and the second electromagnetic radiation (typically microwave radiation) for coupling the first Rydberg state to the second Rydberg state in each atomic particle. An imaging system 650 is provided for detecting and measuring the states of the qubits, typically through fluorescence detection techniques. A comprehensive control system 660 coordinates the operation of all components and implements quantum computing algorithms through sequences of quantum gate operations based on the dipole-dipole interaction methods described herein. This configuration enables the execution of quantum algorithms using the fast, high-fidelity quantum gates that leverage the dipole-dipole interactions between Rydberg states of the atomic particles as detailed in the method claims.

[0052] FIG. 7 illustrates a block diagram of an exemplary apparatus for performing a quantum gate on a pair of atomic particles in a quantum register according to aspects of the present disclosure. The apparatus comprises a laser source 705 configured to generate the first electromagnetic radiation for coupling the first qubit state |o) to the first Rydberg state |ri) in each atomic particle. A microwave source 710 is configured to generate the second electromagnetic radiation for coupling the first Rydberg state | n> to the second Rydberg state | r2> in each atomic particle, thereby inducing the dipole-dipole interaction between the pair of atomic particles. A set of optical elements 715 is configured to direct the laser radiation generated by the laser source onto the pair of atomic particles in the quantum register. These optical elements may include mirrors, lenses, beam splitters, and spatial light modulators arranged to precisely focus the laser radiation on the target atomic particles. A microwave antenna 720, coupled to the microwave source, is configured to illuminate the pair of atomic particles with microwave radiation generated by the microwave source. The microwave source is configured such that a duration, an intensity or amplitude, a detuning and a phase of the microwave radiation are precisely controllable. A laser modulator 725 is configured to control a duration, an intensity or amplitude, a detuning and a phase of the laser radiation directed to the pair of atomic particles. The apparatus further comprises a control unit 730 configured to control the laser modulator and the microwave source to illuminate the pair of atomic particles with modulated laser radiation and modulated microwave radiation to perform thequantum gate. The control unit implements the optimization methods described herein to determine optimal pulse parameters, including the time-dependent modulation of pulse amplitude, phase, and detuning, to realize quantum gates with high fidelity, speed, and robustness against experimental imperfections such as inter-particle distance fluctuations. The components of this apparatus work together to implement the methods for performing quantum gates utilizing dipole-dipole interactions as detailed in the method claims.

[0053] Fig. 8 shows a schematic representation of an exemplary atomic level scheme utilized in this disclosure for realizing a CZ gate up to a local phase. A laser field with amplitude Ωoand phase cpo couples the qubit state |i) to the Rydberg state | n). A microwave field with amplitude Ωmwand phase cpmw couples the two Rydberg states | rd and |r2) enabling a flip-flop interaction J(|r₁r2⟩⟨r2r₁| + h.c.) between the two atoms (cf. Eq. (1) in Supplement A). Field amplitudes and phases may serve as timedependent control functions.

[0054] Fig. 9 shows a schematic representation of two relevant blocks of the Hamiltonian of Eq. (1) that encode the dynamics of the state 101) (panel (a)) and 111) (panel (b)), after the unitary transformation U = UA ® UB, with UA = UB = diag(1, eiφo, ei(φo+φmw)) mapping laser and microwave phases into detunings Δo= dφo / dt , Δmw= dφmw / dt. Because of the symmetry A ↔ B of the protocol, all antisymmetric states are not relevant for the time evolution of 111). For J ≫ Ωo, |Δo|, Ωmw, |Δmw| the states in the shaded dashed box in panel (b) are decoupled from the dynamics of 111).

[0055] Fig. 10 shows in panel (a) a Bell state infidelity as a function of the dimensionless gate time ΩoT obtained for the Hamiltonian of Eq. (10) of Supplement A (solid lines) and for the Hamiltonian of Eq. (1) with J / Ωo= ∞ for different values of Ωmw / Ωo(dashed lines). In the latter case, the optimization is performed for two different time step sizes dt = T / N, i.e. N = 300 (dashed lines) and N = 600 (solid lines), while in the former case N = 100. Panel (b) shows an optimal microwave phase obtained at finite Ωmwfor different values of and for floT = 5.9. Panels (c) to (d) of Fig. 10 show a comparison between the pulse shapes obtained for the Hamiltonian of Eq. (10) (solid lines) and for the Hamiltonian Eq. (1) with J / Ωo= ∞ (dashed lines). Therelations between the parameters of the two models are dφo / dt = Δ − V / 2, dφmw / dt = −Ω2mw / (2V), and imply that φmwdiverges when V vanishes (cf. panel (b)).

[0056] Fig. 11 shows in panel (a) a Bell state infidelity as a function of the dimensionless gate time ΩoT for several values of J / Ωoranging between 10 and 50 (cf. bar on the right of panel (d)). The number of time steps is set to N = 200 and the regularizing parameter ε = 10-3(cf. Eq. (13) and the text below). Panels (b) to (d) show optimal laser phase cpo, microwave phase cpmw and microwave amplitude Ωmwat the time T* for which the time-optimal exact gate is found by the GRAPE optimization as described herein.

[0057] Fig. 12 shows in panel (a) an optimal CZ gate execution time for the pulses shown in Fig. 11 as a function of J / Ωo(markers). The inset shows the relative speed-up w.r.t. the time-optimal van der Waals gate for which ΩoT* ≃ 7.61 (line). Panel (b) is the same as panel (a) with the time spent in the Rydberg subspace TR(see Eq. (14)) in place of the gate execution time T*.

[0058] Fig. 13 shows in panel (a) a Bell state infidelity as a function of the relative fluctuation in interatomic distance 8R / R. The solid line corresponds to the time-optimal exact gate obtained with the procedure outlined in Sec. Ill B of Supplement A. The dashed lines correspond to the robust pulses obtained from the cost function of Eq. (16) with ΩoδT* = 0, 0.1, 0.2, where 8T* is a slight increase of the time-optimal gate time T*. Here J / Ωo= 10 and the interaction parameters are in the first row of Table I for rubidium (top) and cesium (bottom) Rydberg states. The horizontal dashed line is FBell= 0.999. The control functions are discretized on a time grid of N = 200 points, while the integral in Eq. (13) is discretized on K = 11 points. The time-optimal pulses for this set of parameters have an execution time T* ~ 6.30 / Ωo. The pulse shapes for laser phase cpo, microwave phase cpmw and amplitude Ωmware plotted in panel (b), (c) and (d), respectively.

[0059] Fig. 14 shows in panels (a) to (b) a Bell state infidelity due to atomic motion and Rydberg decay as a function of the trap frequency ωtrapfor the first row of the interaction parameters in Table I for rubidium (a) and cesium (b). The optical Rabi frequency is Ωo / 2π = 5 MHz. The solid line and the dashed lines correspond to theexact time-optimal protocol and the robust protocols with a time increase δT*, respectively. Panels (c) to (d) show a Bell state infidelity as a function of the optical Rabi frequency Ωoobtained from the robust protocols with ΩoδT* = 0.1 for all the interaction parameters listed in Table I and a trap frequency ωtrap / 2π = 100 kHz. The horizontal dashed lines are the infidelities due to Rydberg decay only.

[0060] Fig. 15 shows in panel (a) optimal pulse times for the different branches of solutions for the intermediate pulse in the exact piecewise protocol (see inset). The horizontal lines correspond to the asymptotic values √2π and √3π. Panel (b) shows pulse shape for the different branches of solutions at J / Ωmw= 8 (star). Panels (c) to (d) shows a Bell state fidelity for the approximate piecewise protocol without turning off the laser (see inset). The different branches from panels (a) to (b) are shown as a function of the duration of the laser pulse area floT, for panel (c) fixed J / Ωmwand for panel (d) fixed Ωmw / Ωo. As both parameters are increased, the optimal time T* approaches 2π / Ωo(vertical line).

[0061] Fig. 16 shows in panel (a) optimal times for different sets of solutions at finite blockade strength V. The shortest pulse for each branch (highlighted with a star) is shown in panel (b). The dashed horizontal line corresponds to the values in the limit V / Ωo→ ∞.

[0062] FIG. 17 illustrates comprehensive results of GRAPE optimization for determining optimal pulse parameters for the dipole-dipole interaction quantum gate operations disclosed herein. Panel (a) shows the Bell state infidelity as a function of dimensionless gate time ΩoT for several values of J / Ωoranging between 10 and 70, with the colorbar on the right indicating the corresponding J / Govalues. The sharp drop in infidelity to values below 10-10indicates the minimum time T* at which the exact gate can be realized with exceptional precision. Panels (b)-(c) show the corresponding optimal microwave phase φmw, normalized microwave amplitude Ωmw / Ωo, and laser detuning Δo / Ωoas functions of normalized time t / T at the optimal time T* for different values of the dipole-dipole interaction strength J / Ωo. Panel (d) shows the optimal fixed laser detuning Δo / Ωoas a function of the dipole-dipole interaction strength J / Ωoat the optimal time T*, illustrating how the optimal detuning value changes as the interaction strength increases. Notably, the optimization results demonstrate that as J / Ωoincreases, the optimal gate time decreases from approximately T≈6.3 / Ωofor J / Ωo= 10 to T≈6.1 / Ωofor J / Ωo= 70, providing faster quantum gates. An important technical advantage illustrated in this figure is that the laser field requires only amplitude modulation (which can be implemented as a simple square pulse) and operates at a fixed detuning Δowith no phase modulation, while all temporal phase modulation is performed only on the microwave field. This significantly simplifies hardware requirements since phase modulation at microwave frequencies (typically in the GHz range) is substantially easier to implement with high precision than phase modulation at optical frequencies (hundreds of THz).

[0063] FIG. 18 demonstrates the comprehensive robustness optimization of quantum gates against interatomic distance fluctuations for both rubidium for both cesium (left column, panels a-d) and rubidium (right column, panels e-h) atoms with a dipole-dipole interaction strength of J / Ωo= 10. Panel (a) shows the Bell state infidelity as a function of relative fluctuation in interatomic distance 8R / R for different stabilization protocols with varying allowed increases in gate time 8T*. The dark grey lines represent time-optimal exact gates without robustness optimization, while colored lines show gates optimized for robustness with ΩoδT* = 0 (blue), 0.1 (orange), and 0.2 (green). Panels (b), (c), and (d) for rubidium and panels (f), (g), and (h) for cesium show the corresponding optimal microwave phase cpmw, microwave amplitude Ωmw / Ωo, and laser detuning Δo / Ωoas functions of normalized time t / T. Critically, across all optimization scenarios, the laser field is implemented with simple amplitude modulation (e.g., approximating a square pulse) and fixed detuning without any phase modulation, while all complex temporal modulation is applied only to the microwave field's phase and amplitude. This represents a significant practical advantage for implementation in quantum computing hardware, as phase modulation at microwave frequencies is technically much more straightforward and precise than optical phase modulation. The horizontal red dashed line in panel (a) and (e) indicates the 99.9% fidelity threshold, demonstrating that properly optimized robust protocols can maintain high fidelity across significant interatomic distance variations of up to 3% without requiring complex laser phase modulation. Exemplary laser and microwave modulation functions are shown in FIG. 19.

[0064] FIG. 19 shows various pulse sequence implementations for the quantum gate protocols disclosed herein using dipole-dipole interactions, highlighting the flexibility and robustness in experimental realization. All three implementations (a-c) maintain constant detunings for both laser and microwave fields. Implementation (a) and (b) use square pulse shapes for both laser and microwave fields with different relative timing, while featuring complex phase modulation of the microwave field. As shown in line (c) of Fig. 19, an example gate protocol is shown where the laser field Ωois only amplitude modulated with an essentially square pulse shape at constant detuning Ao, while the microwave field employs both amplitude modulation with three distinct peaks and smooth phase modulation. While square pulses are shown in Fig. 19 as actual squares, in an actual implementation the edges of such pulse will be smooth due to bandwidth limitations of electronics and laser modulators (e.g., limited by the amplitude modulation bandwidth of acousto-optical modulators and similar equipment).

[0065] FIG. 20 illustrates a comprehensive conceptual overview of the dipoledipole interaction quantum gate implementation as disclosed herein. The left portion shows a pair of atomic particles represented by orange and yellow circles, trapped within optical beams (green cones) similar to the optical traps described in connection with the quantum register 620 of FIG. 6. The dipole-dipole interaction strength J between the two atomic particles is visualized by the curved red arrows, representing the physical mechanism enabling the quantum gate operation. The right upper portion depicts a control system similar to the control unit 730 of FIG. 7, with a display showing the microwave amplitude Ωmwand phase φmwmodulation waveforms that are applied during gate execution. A microwave antenna, corresponding to the microwave antenna (720) of FIG. 7, is shown directing microwave radiation toward the atomic particles. The bottom right portion presents an energy level diagram showing the relevant atomic states involved in the quantum gate implementation, including the qubit states | o) and 11), with the laser field Ωo(blue arrow) coupling the qubit state 11) to a first Rydberg state, and the microwave field(green arrow) coupling between two Rydberg states, consistent with the level structure described in FIG. 1. The bottom left portion illustrates a quantum circuit representation of the CZ (controlled-Z) gate that is implemented by the dipole-dipole interaction protocol, demonstrating how thephysical mechanism is utilized within a quantum computing algorithm as described in the method of FIG. 4.

[0066] FIG. 21 illustrates optimal gate protocol for the rubidium Rydberg states in Table 3 of supplement A. The optimal microwave phase (a), microwave amplitude (b), and Rydberg detuning (c) are obtained upon adiabatic elimination of the intermediate state |e). (d) Gate infidelities computed under realistic experimental conditions with intermediate state detuning A_e = 2. x 7.75 GHz, and single-photon Rabi frequencies Ω1= Ω2= 2π × 278 MHz, which result in an effective two-photon Rabi frequency Ωeff= Ω1Ω2 / 2Δe≈ 2π × 5 MHz. The simulation also includes intermediate-state scattering, finite Rydberg lifetimes, and atomic motion with an initial motional state at 2 pK and a trap frequency ωtrap= 100 kHz.

[0067] The foregoing disclosure provides illustration and description but is not intended to be exhaustive or to limit the aspects to the precise form disclosed.Modifications and variations may be made in light of the above disclosure or may be acquired from practice of the aspects. As used herein, the term component is intended to be broadly construed as hardware, firmware, or a combination of hardware and software. As used herein, a processor is implemented in hardware, firmware, or a combination of hardware and software.

[0068] It will be apparent that systems and / or methods described herein may be implemented in different forms of hardware, firmware, or a combination of hardware and software. The actual specialized control hardware or software code used to implement these systems and / or methods is not limiting of the aspects. Thus, the operation and behavior of the systems and / or methods were described herein without reference to specific software code— it being understood that software and hardware can be designed to implement the systems and / or methods based on the description herein.

[0069] Even though particular combinations of features are recited in the claims and / or disclosed in the specification, these combinations are not intended to limit the disclosure of various aspects. In fact, many of these features maybe combined in ways not specifically recited in the claims and / or dis-closed in the specification. Althougheach dependent claim listed below may directly depend on only one claim, the disclosure of various aspects includes each dependent claim in combination with every other claim in the claim set. A phrase referring to “at least one of’ a list of items refers to any combination of those items, including single members. As an example, “at least one of: a, b, or c” is intended to cover a, b, c, a-b, a-c, b-c, and a-b-c, as well as any combination with multiples of the same element (e.g., a-a, a-a-a, a-a-b, a-a-c, a-b-b, a-c-c, b-b, b-b-b, b-b-c, c-c, and c-c-c or any other ordering of a, b, and c).

[0070] No element, act, or instruction used herein should be construed as critical or essential unless explicitly described as such. Also, as used herein, the articles “a” and “an” are intended to include one or more items, and may be used interchangeably with “one or more.” Furthermore, as used herein, the terms “set” and “group” are intended to include one or more items (e.g., related items, unrelated items, a combination of related and unrelated items, and / or the like), and may be used interchange-ably with “one or more.” Where only one item is intended, the phrase “only one” or similar language is used. Also, as used herein, the terms “has,” “have,” “having,” and / or the like are intended to be open-ended terms.

[0071] As used herein, the phrase “based on” shall not be construed as a reference to a closed set of information, one or more conditions, one or more factors, or the like. In other words, the phrase “based on A” (where “A” may be information, a condition, a factor, or the like) shall be construed as “based at least on A” unless specifically recited differently.

[0072] As used herein, the term “or” is an inclusive “or” unless limiting language is used relative to the alternatives listed. For example, reference to “X being based on A or B” shall be construed as including within its scope X being based on A, X being based on B, and X being based on A and B. In this regard, reference to “X being based on A or B” refers to “at least one of A or B” or “one or more of A or B” due to “or” being inclusive. Similarly, reference to “X being based on A, B, or C” shall be construed as including within its scope X being based on A, X being based on B, X being based on C, X being based on A and B, X being based on A and C, X being based on B and C, and X being based on A, B, and C. In this regard, reference to “X being based on A, B, or C” refers to “at least one of A, B, or C” or “one or more of A, B, or C” due to “or” beinginclusive. As an example of limiting language, reference to “X being based on only one of A or B” shall be construed as including within its scope X being based on A as well as X being based on B, but not X being based on A and B.

[0073] Further, process diagrams such as Fig. 3, and Fig. 4 do not necessarily indicate a particular order or sequence of steps. For example, steps may also be performed in a different order or, if hardware capabilities allow it, simultaneously, without deviating from the scope of the present disclosure.

[0074] Further details of the aspects described above are in Supplement A making reference to Figs. 8 to 18 and 21 As will be apparent to the person skilled in the art, Supplement A is a scientific publication that is based on and / or makes use of several aspects of the present disclosure. Details and examples provided in Supplement A are included for ease of understanding and thus not to be understood to be limiting in any way to what was described above. For instance, aspects of the present disclosure discussed above may also be implemented using alkaline-earth atoms such as88Sr. Further, as will also be appreciated by the skilled person, terminology used in Supplement A might, in some cases, be different from the terminology used above.Supplement ASection 1: Introduction

[0075] The advent of digital neutral-atom quantum computers relies on the development of fast and robust protocols for high-fidelity quantum operations. In this work, we introduce a novel scheme for entangling gates using four atomic levels per atom: a ground-state qubit and two Rydberg states. A laser field couples the qubit to one of the two Rydberg states, while a microwave field drives transitions between the two Rydberg states, enabling a resonant dipole-dipole interaction between different atoms. We show that controlled-Z gates can be realized in this scheme without requiring optical phase modulation and relying solely on a microwave field with timedependent phase and amplitude. We demonstrate that such gates are faster and less sensitive to Rydberg decay than state-of-the-art Rydberg gates based on van der Waals interactions. Moreover, we systematically stabilize our protocol against interatomicdistance fluctuations and analyze its performance in realistic setups with rubidium or cesium atoms. Our results open up new avenues to the use of microwave-driven dipolar interactions for quantum computation with neutral atoms.

[0076] The last decade has witnessed the rapid development of quantum platforms based on Rydberg atom arrays. In addition to their remarkable success as analog quantum simulators, neutral atoms trapped via optical tweezers or lattices are emerging as one of the most promising architectures for digital quantum computing, thanks to their scalability, their long coherence times, and their reconfigurable geometry that enables arbitrary qubit connectivity. High-fidelity single-qubit and two-qubit quantum operations have also been demonstrated with several atomic species, including rubidium, cesium, strontium, and ytterbium. Yet, further improving their accuracy remains one of the outstanding challenges toward realizing a large-scale, fault-tolerant quantum computer with neutral atoms, making it crucial to develop novel schemes for Rydberg gates.

[0077] Typically, two-qubit gates with Rydberg atoms rely on the strongly repulsive van der Waals force arising when the two atoms are in the same Rydberg state. The state-of-the-art approach for these schemes involves state-selectively coupling the atomic qubit of each atom to one Rydberg level via a laser, whose amplitude and phase are modulated in time to yield a controlled-Z (CZ) gate up to a local phase. The phase and amplitude pulses required to achieve the desired two-qubit gate are not unique, providing a degree of freedom that can be exploited to minimize gate execution time or maximize gate robustness against fluctuations in specific parameters. Time optimality is particularly relevant as the finite Rydberg lifetime is among the major sources of decoherence. Such protocols have been tested in Rydberg arrays of rubidium, cesium, strontium, and ytterbium atoms, achieving gate fidelities above 99%.

[0078] Here, we present a different approach for realizing CZ Rydberg gates mediated by the dipole-dipole interaction between pairs of distinct Rydberg levels | n>, | r2). The scheme is depicted in Fig. 8: the atomic qubit state 11) is optically coupled to the Rydberg state | n>, and | n> is microwave coupled to nearby Rydberg level | r2>,enabling a flip-flop interaction of strength J between the two atoms. Throughout the supplement A, we refer to J as the resonant dipole-dipole interaction to distinguish it from the off-resonant, second-order van der Waals interaction commonly used in blockade-based gates. The laser field can be configured as either global, to enable parallel gate operations, or local, to target individual pairs of atoms. We use the microwave amplitude Ωmwand phase φmwas time-dependent control functions, and apply the Gradient Ascent Pulse Engineering (GRAPE) method as known in the art to obtain the time-optimal protocol for this scheme. We show that the resulting gate is up to 20% faster than the time-optimal "van der Waals protocol" while offering several other advantages such as lower sensitivity to Rydberg decay, increased interaction strengths for longer-range gates, and the possibility to combine optical addressability and global microwave control. Notably, our protocol relies solely on microwave phase and amplitude modulation and does not necessarily involve phase shaping of laser pulses, significantly simplifying hardware requirements and potentially reducing the impact of laser phase noise on gate fidelity.

[0079] We give a physical interpretation of the numerically obtained pulses in a regime where the resonant dipole-dipole interaction strength J is much larger than all other energy scales and find that the protocol can be formally interpreted as a reparameterization of the standard van der Waals protocol with time-dependent interaction strength. We then focus on realistic experimental conditions and employ a GRAPE-based method for further optimizing the pulse shapes to make the gate robust against fluctuations of the interaction strength J induced by the atomic motion. Finally, we perform numerical simulations that take atomic motion and finite Rydberg lifetimes into account for rubidium and cesium atoms. The results indicate that Bell state fidelities exceeding 99.9% are achievable with our scheme, highlighting its feasibility and effectiveness.

[0080] The structure of supplement A is as follows. In Section 2, we discuss two different protocols for realizing entangling gates with resonant dipole-dipole interactions between Rydberg atoms in the limit J / Ωo→ ∞. In particular, the first one consists of two resonant laser n-pulses with a fast microwave pulse at constant detuning in between; the second requires a time modulation of the microwave detuning at constant laser detuning and is mathematically equivalent to a van der Waals protocolwith time-dependent interaction strength. In Section 3, we focus on the second protocol and lay out the numerical procedure employed to obtain the pulses that implement the CZ gate in the ideal case where no van der Waals interaction occurs between the Rydberg pair states. In Section 4, we consider rubidium and cesium atoms where the Rydberg states | n> and |are, respectively, P and S states with principal quantum numbers from n=40 to n=70. We compute their resonant dipole-dipole interaction strength J and van der Waals interaction strengths Vn, Vi2, V22at various distances, and repeat the GRAPE optimization to show that a qualitatively similar gate protocol exists in the presence of van der Waals forces. In Section 5, we optimize the pulse robustness against fluctuations of the interatomic distance and find a stabilized protocol that significantly enhances gate fidelities when considering the coupling to the atomic motional degrees of freedom. We benchmark the performance of the robust pulses including both atomic motion and Rydberg decay, and demonstrate that our protocol competes with, and in some regimes even outperforms, state-of-the-art protocols. In Section 6, we draw our conclusions and outline potential directions for future work.Section 2: CZ gates from resonant dipole-dipole interaction

[0081] The Hamiltonian that describes the two four-level systems depicted in Fig. 8 reads:where A and B label the two atoms, Ω0and Δ0are the amplitude and detuning of the laser that couples the computational basis state |1〉 to the Rydberg state |r1〉, Ωmwand φmware the amplitude and phase of the microwave radiation that couples the two Rydberg states | n> and | m), and J > o is the strength of the dipolar exchange interaction. Such interaction is present in Rydberg states for which a dipole transition is allowed, e.g. an s state and a p state (cf. Section 4). The real Hamiltonian for the two atoms also includes van der Waals interaction terms of the form Vy | nq) (nq |, which wewill neglect in this section and discuss at length in Sections 4 and 5. The most general maximally-entangling two-qubit gate realizable within this scheme is a CZ gate up to a single-qubit phase shift, which can be parameterized by an angle 0 and can be written in the computational basis {| 00), |oi), |io), |n)}asCZ(0) = diag

[0082] We now outline two ways for obtaining such two-qubit gate from the dynamics (see above) when J » 2o, |Ao|, Ωmw, |Δmw|. We postpone the discussion of gate protocols at finite J to Section 3b. In what follows, we switch to a description in terms of the microwave detuning Δmw= dφmw / dt by applying the unitary transformation U = UA 0 UB, where UA= UB= diag(1, 1, 1, exp(i φmw)). The time evolution of |oi) (or 110)) and 111) is governed by the Hamiltonians HOiand Hu depicted in Fig. 9. The flip-flop interaction only enters in Hu and, for J » 2o, | Ao |, Ωmw, |Δmw|, renders the states (| rxr2> + |r2Ti)) / 2 and | r2r2> far off-resonant and effectively decoupled from the dynamics. As a result, the optical Rabi frequency Ω0sets the gate timescale, and will fix our time units throughout the work presented in the supplement A. A simple protocol for realizing a CZ(0) gate in this limit consists of three steps. First a resonant optical π-pulse transfers the single qubit population from |1〉 to |r1〉, such thatA second off-resonant microwave pulse U2at constant detuning Amwis then applied to the atoms. Since the laser is off during this pulse, the state |r1r1〉 does not evolve under U2(cf. Fig. 9). Hence, one can choose a pulse duration Tmw= 2π / sqrt(Δmw2+ Ωmw2) such that lor,) undergoes a complete Rabi oscillation U2|0r1〉 = exp(i φmw)|0r1〉, acquiring a phase π(1 + Δmw / sqrt(Δmw2+ Ωmw2)) that can be adjusted by tuning Amw. Setting Δmw= ∓Ωmw / sqrt(3) yields φmw= ±π / 2 and ΩmwTmw= sqrt(3) π. The combined result for these two pulses isFinally, a resonant optical n-pulse U3 is applied to bring back the population to the computational basis:By comparing with the earlier equation, one can see that this pulse sequence realizes a CZ(±3π / 2) gate in a time

[0083] As we discuss in ^Xppendix 1 of supplement when Ωmw≫ Ω0, the microwave pulse U2is much faster than the two laser pulses U1and U3and can be executed without turning off the laser with negligible effect on the gate fidelity. This condition is easily achievable with standard microwave sources. Moreover, a proper time-modulation of the microwave phase can accommodate a finite resonant dipole-dipole interaction strength J. However, we could not find a straightforward extension of this protocol that is stable upon the inclusion of van der Waals interactions Vy.

[0084] A natural question that emerges at this point is whether both optical and microwave couplings can be used at the same time to find an improved gate protocol. In the following, we present such a protocol and show that it can be systematically adapted to the case where van der Waals interactions are present and will be the focus of the rest of the main text. We describe below its simplest version when J ≫ Ω0, |Δ0|, Ωmw, |Δmw| and give its physical interpretation. We assume laser and microwave fields to be always on with constant amplitudes Ω0and Ωmw, and consider the microwave detuning Δmw≫ Ω0, Δ0, Ωmw. In this limit the dynamics of |oi) depicted in Fig. 9 can be further restricted to only two levels. To see this, we diagonalize HOiin the subspace {Ion), |or2)} at first order in λ = Ωmw / (2Δmw). H01expressed in the dressed basis {|01〉, |or1〉 + λ|or2〉, |or2〉 - λ|or1〉} reads

[0085] The AC Stark shift induced on the dressed states |or1〉 + λ|or2〉 and |or2〉 - λ|or1〉 can be made finite by setting Δmw= τ Ωmw2. If we now assume Ωmw≫ Ω0, Δ0the state |or2〉 decouples from the dynamics of 101), which is governed by the two-level Hamiltonian

[0086] The same argument can be applied to H11upon replacing {|or1〉, |or2〉} with {(|1r1〉 + |r11〉) / √2, (|1r2〉 + |r21〉) / √2}. In the basis {|11〉, (|1r1〉 + |r11〉) / √2, |r1r1〉} the resulting three-level Hamiltonian is

[0087] The dynamics described by the above is mathematically equivalent to the one of two three-level systems { | o), 11), | r)}, where the state |rr〉 interacts with van der Waals force V = -1 / (2τ), and where the detuning from the Rydberg transition is given by Δ = Δ0- 1 / (4τ), i.e. a two-qubit gate scheme. The Hamiltonian for this system is

[0088] In the prototypical van der Waals gate, the distance between the atoms and thus the van der Waals coefficient V are usually constant during the gate operation. In our scheme, instead, a time-dependent V simply corresponds to a time modulation of the microwave detuning Amw. We will show in the next section that this additional control function provides a substantial speedup with respect to constant-V van der Waals gates. In particular, we will demonstrate that the shortest gate time for a CZ operation with the above Hamiltonian is T ≈ 6.03 / Ω0, compared to the van der Waals gates execution time of T ≈ 7 / Ω0for V / Ω0≈ 1.3 (see Appendix 2 of supplement A) and T ≈ 7.6 / Ω0for V / Ω0= ∞. We will also show that the obtained protocol can be extended to finite Ωmw / Ω0and J / Ω0with only a slight increase of the gate execution time.Section 3: Gate speed optimization

[0089] In this section, we use GRAPE to find the time-optimal protocol that realizes a CZ(θ) gate within the scheme of Fig 8. This gate maps the product state |++) to the Bell statesuch that the Bell state fidelitywhere T exp is the time-ordered exponential.

[0090] This quantity has been often used to benchmark and optimize the CZ gate. Since such gate is diagonal, FBell= 1 is equivalent to F = 1, where F is the average gate fidelity. We take the Bell state infidelity 1 - FBellas a cost function for the optimization, discretize time with a time step dt, and numerically minimize the cost function for a fixed total time T. The number of variational parameters for dt = T / N is kN + 2, where k is the number of control functions and N is the number of time steps. Such parameters are the values fi = f(ti) of the unknown functions f computed on the timegrid ti= (i + 1 / 2) dt, i = 0,..., N-1, the detuning Ao from the optical transition, and the single-qubit rotation angle 0. We will always keep a constant laser amplitude Ω0andconstant detuning Δ0. We use φmw(t) as a control function for J / Ω0= ∞ and φmw(t), Ωmw(t) for finite J / Ω0. We repeat the optimization for increasing T until a time T* is found for which the cost function C vanishes. The numerical minimization is performed using the method "L-BFGS-B" as implemented in the Python library SciPy. To speed up this procedure we provide the SciPy routine with the gradient of the (time-discrete) cost function, which can be straightforwardly computed analytically. Moreover, when the microwave amplitude 2mw is also used as a variational control function, we set a lower bound Ωmwmin= 0 in the optimizer to ensure its positivity, and add a term to the cost function that enforces Ωmw(t0) = Ωmw(tN-1) = 0:

[0091] In the remainder of this section, we employ GRAPE to first show that the scheme of Fig. 8, in the double limit J / Ω0→ ∞ and Ωmw / Ω0→ ∞ with Δmw= τ Ωmw2, enables the realization of an exact CZ(θ) gate in a time T ≈ 6.03 / Ω0. We then apply a modified version of the same method to find a time-optimal CZ gate protocol attainable with a Hamiltonian as discussed above and a finite resonant dipole-dipole interaction strength J.Section 3.1 Infinite J / Ω0

[0092] In Section 2, we proved that the system of two dipole-dipole interacting four-level atoms depicted in Fig. 8 maps to a system of two three-level atoms interacting via a time-dependent van der Waals force V(t) and Rydberg-transition detuning A(t). This mapping is valid in the limits J / Ω0→ ∞ and Ωmw / Ω0→ ∞ with Δmwand the relations between the parameters of the two models are V(t) = -1 / (2 (t)) and A(t) = Ao - 1 / (4T(t)). We now perform a GRAPE optimization on the Hamiltonian above using 1 / τ(t), Δ0and the CZ gate angle θ as variational parameters. The results of this optimization for N=ioo time steps are the solid black lines in Fig. 10.

[0093] The infidelity sharply drops to zero (within numerical precision) at the minimum time T* for which an exact CZ gate is realized, demonstrating the existence, under ideal conditions, of an exact CZ(θ) gate in a time T ≈ 6.03 / Ω0. In Fig. 10c, weplot the time dependence of the control function 1 / τ(t) = -2V(t) for Ω0T = 5.95. Such function has a direct interpretation in the resonant dipole-dipole interacting gate scheme of Fig. 8 via 1 / τ(t) = Ωmw2 / dφmw / dt.

[0094] Although this correspondence only holds when Ωmw / Ω0= ∞, GRAPE remarkably finds a qualitatively similar solution for finite Ωmw / Ωo. To show this, we minimize the Bell state infidelity obtained by evolving |++〉 with the an Hamiltonian as discussed above, using the microwave phase φmw(t) and optical detuning Δ0as variational parameters. We enforce the constraint J / Ω0= ∞ by projecting out the states |r1r2〉, |r2r1〉, | r2r2) from the dynamics (cf. Fig. 9b). We plot in Fig. 10a the minimum infidelity as a function of Ω0T for different values of Ωmw / Ω0, and in Fig. 10c the function 1 / τ(t) = Ωmw-2 / dφmw / dt for Ω0T = 5.95, where d / dt denotes the numerical derivative taken for a time step T / N with N=3oo,6oo. We compare this result to the one obtained from the effective van der Waals Hamiltonian and observe a rapid convergence to the limit Ωmw / Ωo= 00.

[0095] The finite-Ωmwpulse depicted in Fig. 10d is discontinuous for t / T ≈ 0.06, 0.28, 0.72, 0.94. This is because the optimal V(t) oc i / T(t) for the model vanishes, causing divergences in the optimal φmw(t) for the model and slow convergence of the result with the number of time steps N (cf. Fig. 10b, 10d). Although this discontinuity can be made arbitrarily small by reducing the time step dt = T / N, for too large N the numerical optimization becomes unstable and eventually fails to produce a continuous solution. This issue is even more severe when the constraint J / Ω0= ∞ is relaxed. For this reason, for the GRAPE optimization at finite J / Go below we introduce a regularizer in the cost function that penalizes discontinuous solutions.Section 3.2 Finite J / Ω0

[0096] For the GRAPE optimization of the CZ gate protocol implemented via the Hamiltonian discussed in section 2 above (para.

[0075] ) we use the control functions φmw(t), Ωmw(t). As we show below, introducing the additional control function Ωmw(t) eliminates the divergences in the optimal microwave phase φmw(t) discussed in Section 3.1. Intuitively, this can be understood from the mapping of the infinite- J limit to the effective model of paragraph

[0081] where the diverging quantity is τ(t) = dφmw / dt / Ωmw2. Therefore, divergences in φmw(t) can be avoided by allowing Ωmw(t) to vanish. To steer the optimization towards smooth solutions with vanishing 2mw(t) rather than diverging cpmw(t), we add a regularizer to the cost function that enforces the smoothness of the control fieldswhere C is the cost function discussed above, s = t / T, f = φmw, Ωmwand η is a small constant that we adjust as the exact gate at time T* is approached. Specifically, we initially set r|0= 1O“6for To< T* and carry out the optimization until convergence. We then take the resulting optimal pulses as initial conditions for the optimization at T, = To+ dT, with dT = 0.002 / 20, and reset η1= ε C0, where C0is the Bell state infidelity obtained for the optimal solution at time T0. By iterating this procedure we systematically reduce η as T* is approached. We empirically find that the control functions obtained in this way are smooth and independent of the time step, as long as ε ≥ 10-3. Increasing ε results in a larger T*. Therefore, we tune ε to the minimum value for which the optimal pulses are independent of the discretization scale dt.

[0097] The resulting pulses are plotted in Fig. 17 (and also in Fig. 11) for several values of J / 2o and N=200. In Fig. 17a we show the Bell state infidelity as a function of the total time T. The minimum gate execution time T* is J-dependent and decreases with increasing J / 2o. As plotted in Fig. 12a, it ranges from T* ≈ 6.3 / Ω0for J / Ω0= 10 to T* ≈ 6.1 / Ω0for J / Ω0= 70. Fig. 12a also displays the relative speed-up with respect to the execution time T*V≈ 7.61 / Ω0of the time-optimal van der Waals gate, ranging from 17% up to 20% for J / Ω0= 10 and 70, respectively. We observe that some of the divergences in the microwave phase φmwreported in Fig. 10b become smooth peaks in Fig. 17c at times t / T ≈ 0.3, 0.7 due to the regularizer.

[0098] Another important quantity to monitor is the time spent in the Rydberg manifold during the protocol, as it upper-bounds the infidelity due to the finite lifetime of the Rydberg states. It is given bywhere Π = |r1〉〈r1| + |r2〉〈r2| is the projector on the Rydberg subspace of one atom, the sum runs on all the computational basis states |q) = {|oo), |oi), |io), 111)}, and |q(t)) is the time evolution of these states under the optimal protocol. We plot Ω0TRvs J / Ω0in Fig. 12b, demonstrating another substantial improvement ranging from 20% to 26% for J / Ω0= 10 and 70 with respect to the van der Waals gate TRV≈ 2.95 / Ω0.Section 4: Implementation with alkali atoms

[0099] So far, we neglected the van der Waals interactions arising when the two atoms are in a Rydberg state. Such interactions have the formand have to be included in the Hamiltonian of paragraph

[0071] . We do not expect the gate protocols depicted in Fig. 10 to be sensitive to the value of the interaction strengths V12(=V21), V22. In fact, J ≫ Ω0and |V12|, |V22| ≫ Ω0have the same effect on the state |11〉, decoupling the states |r1r2〉, |r2r2〉 from its dynamics. On the contrary, |V11| has to be much smaller than Ω0to avoid the decoupling of |r1r1〉, which plays an active role in our scheme, as we discussed in Section 3. Finally, we need to have J ≫ Ω0since the execution time of our protocol decreases with increasing J / Ω0(cf. Fig. 12a). Combining these requirements we have the condition J ≫ Ω0≫ |V11|.

[0100] The interaction strengths J and Vn depend on the interatomic distance R as J(R) ~ C3 / R3and V11(R) ~ C6 / R6, respectively. Hence, we can adjust J / V11~ (C3 / C6) R3by tuning R. However, 20has to be as large as possible since the real gate time T ∝ 1 / Ω0. The optical Rabi frequencies achievable in typical experimental conditions Ω0 / 2π≈ 1–10 MHz thus set the range of the required resonant dipole-dipole and van der Waals interactions to J / 2π ≈ 10–100 MHz and V / 2π ≈ 0.1–1 MHz.

[0101] The C3and C6coefficients depend on the atomic species and the quantum numbers of the Rydberg states. Their scaling with the principal quantum number n is C3~ n4and C6~ n11. Therefore, J / |V11| decreases with n for fixed R. Given the Rydberg lifetime scaling τ ~ n2at room temperature, we need to find a tradeoff between large J / |V11| and long τ. Below, we focus on rubidium with n=40, 50, 60 and cesium atoms with n=40, 50, 60, 70, which are among the most used atomic species in Rydberg atom experiments. In particular, we choose |r1〉 = |nP3 / 2, mJ= 3 / 2〉 and |r2〉 = |nS1 / 2, mJ= 1 / 2〉.

[0102] This choice is motivated by the fact that, in what follows, we will consider single-photon transitions from the computational qubit state 11) - typically an S state -to the Rydberg state |r1〉, and a direct transition to an S Rydberg state is forbidden by selection rules. While for n ≥ 60 the microwave transition between |r1〉 and |r2〉 falls in the 10–20 GHz range and can be directly modulated using routinely employed microwave techniques, for n = 40 and n=50 it lies in the more demanding 30-60 GHz range, which can be addressed using frequency modulation at a lower frequency combined with frequency multiplication or upconversion. An alternative implementation based on a two-photon transition to |r1〉 is discussed in Appendix 3.

[0103] We calculate the interaction strengths J, Vijusing the Python package PairInteraction. Selected results that will be employed in what follows are reported in Table 1. We note that Vu < o for cesium, a feature that makes the gate more robust against fluctuations in R, as we will show in Section 5 (cf. Fig. 18a; Fig. 13a).40 2.51 50 0.007 0.016 0.079 40 2.25 70 0.010 0.022 0.110 50 3.45 50 0.015 0.037 0.182 50 3.08 70 0.020 0.052 0.255 60 4.45 50 0.027 0.076 0.354 60 3.98 70 0.037 0.107 0.49640 2.43 50 -0.011 0.007 0.064 40 2.17 70 -0.015 0.010 0.089 50 3.36 50 -0.021 0.020 0.150 50 3.00 70 -0.030 0.028 0.210 60 4.35 50 -0.038 0.045 0.295 60 3.89 70 -0.053 0.062 0.413 70 5.41 50 -0.062 0.087 0.514 70 4.84 70 -0.086 0.121 0.720 Table I. Resonant dipole-dipole and van der Waals interaction strengths J and Vij between pairs of Rydberg states |r₁⟩ = |nP₃ / ₂, mJ = 3 / 2⟩ and |r₂⟩ = |nS₁ / ₂, mJ = 1 / 2⟩ with n = 40, 50, 60 for rubidium and n = 40, 50, 60, 70 for cesium atoms at a distance R.

[0104] We carried out the GRAPE optimization outlined in Section 3.2 including the van der Waals interaction into the Hamiltonian, with microwave phase and amplitude as control functions. We set 8 < J / flo < 14 and Vy as listed in Table 1. These interaction strengths enable Rabi frequencies 2o / 2n ® 3.6-6.3 MHz for J / 271 = 50 MHz and flo / 2.n ~ 5-8.75 MHz for J / 2.n = 70 MHz. The time-optimal pulses resulting from this procedure are similar to the ones depicted in Fig. 17 and yield almost the same gate time. One example obtained for J / Qo = 10 and the interaction strengths in the first row of Table 1 for rubidium and cesium Rydberg states are the pulses depicted in dark grey in Fig. 18b, 18c, 18d and Fig. 18f, 18g, 18h, respectively. The gate total time and Rydberg subspace occupation time during the gate are also comparable to thevalues shown in Fig. 12, ranging from Ω₀T ≈ 6.2 and Ω₀T^R ≈ 2.2 for J / Ω₀ = 14, to Ω₀T ≈ 6.35 and Ω₀T^R ≈ 2.4 for J / Ω₀ = 8.

[0105] Sub-Doppler Raman side-band cooling brings individual atoms to2 pK within 25 ms, after which optical tweezers are ramped to a depth of > 0.7 mK, giving trap frequencies cotrap / 271 in the 70-300 kHz range. These exemplary parameters ensure that the motional ground-state wave-packet width xosc= V[h / (2 m cotraP)] and the associated Lamb-Dicke factors q 0, η_mw remain within the bounds assumed in the numerical fidelity analysis presented herein.Section 5: Gate robustness optimization

[0106] The main drawback of the experimentally realizable gate protocols of Section 4 lies in the fact that the finite value of J / 2o and the inclusion of small but non-negligible van der Waals forces make them sensitive to fluctuations 8R of the interatomic distance, which induce variations of the interaction strengths 8J / J = -3 8R / R and 8V / V = -68R / R. The same issue arises with standard van der Waals protocols away from the blockade regime. To overcome this limitation we employ a simple cost function for GRAPE that targets pulse shapes more stable against small changes of R. Upon defining x = 8R / R, such cost function has the form:where XM = 8RM / R is the maximum fluctuation and FBeii(x) is the Bell state fidelity corresponding to a Hamiltonian with J(i - 3x) and Vy (1 - 6x).

[0107] For the numerical optimization, we discretize the integral in the above equation over K points and include the regularizer in the previous cost function, with r| = 10“7to ensure the smoothness of the optimal control functions. We use the time-optimal pulses obtained for a given set of parameters J, Vy as the initial condition for the robustness optimization and allow for a slight increase in the gate time, T = T* + 8T*, where T* is the minimal time for realizing an exact CZ(0) gate (cf. Fig. 17; fig. 11). This choice accelerates convergence, as the gradient evaluation in the robust costfunction is more computationally intensive due to the averaging over atomic displacements 8R. We have verified that the final optimized pulses are robust with respect to perturbations in the initial guess and changes in the optimization hyperparameters K and XM, consistently converging to the same control functions. While we cannot exclude the existence of alternative robust protocols, possibly operating at shorter gate times, this appears unlikely given that T* typically sets a lower bound for the existence of exact or robust gates.

[0108] The result of the GRAPE optimization is depicted in Fig. 18, Fig. 13 for the set of interaction strengths J, Vy in the first row of Table 1 for rubidium (top) and cesium (bottom) Rydberg states. While the stabilized pulses (colored lines) are qualitatively similar to the time-optimal exact protocol (dark grey lines), their Bell state infidelity in Fig. 18a is one order of magnitude smaller over the whole interval |8R / R| < 0.033, excluding a small neighborhood around 8R / R = o. We also observe that the stabilized pulses for cesium yield lower infidelities over a broader range of 8R / R, indicating that the attractive van der Waals interaction contributes positively to gate stabilization (cf. bottom and top of Fig. 18a). The performance further improves when a small increase in the gate duration 8T* is allowed.Section 5.1: Gate performance with atomic motion and Rydberg decay

[0109] To benchmark the performance of the robust pulses such as those shown in Fig. 6, we carried out numerical simulations including the spontaneous Rydberg decay and the atomic motion induced by thermal fluctuations and photon recoil. Below, we consider single-photon transitions to the Rydberg state |r,). In such a setup, photon recoil and Rydberg decay account for most of the gate infidelity, provided laser phase and amplitude noise are negligible. We note that single-photon excitation from hyperfine qubit states can also lead to unwanted coupling to multiple Rydberg Zeeman sublevels, even with pure light polarization. Experimentally, this can be effectively mitigated by quickly transferring the qubit to a stretched hyperfine state prior to excitation, achievable with minimal infidelity, or by operating at large magnetic fields to spectroscopically isolate a single Rydberg transition. Another possibility is to use a two-photon transition to a different Rydberg state |r,), as discussed in detail in Appendix 3.[ono] We assume that the initial motional state is a thermal statewhere cotraPis the trap frequency and a (a+) are the annihilation (creation) operators of the relevant motional modes. The tweezer traps are turned off during the gate, such that the vibrational modes' Hamiltonian readswhereand m is the atomic mass.

[0111] The full model Hamiltonian is H = Hmotion + Hatom + Hint + Hdecay. The second term is the Hamiltonian of the atomic levels including the momentum transfer of the laser and micro wave fields:whereη_o = 2πx_osc / λ_o; η_mw = 2πx_osc / λ_mware the Lamb Dicke parameters for the optical and microwave transitions, with xosc= sqrt(h / (2mcotraP)) and Xo (Xmw) the optical (microwave) transition wavelength.

[0112] The third term is the interaction Hamiltonian where the resonant dipoledipole and van der Waals potentials are expanded at first order in the interatomic distance fluctuations

[0113] Finally, the last term models the finite lifetimes of the Rydberg states | r,), | r2) via a non-Hermitian Hamiltonian of the formwhere the lifetimes i / n are listed in Table 2.the numerical simulations presented in Fig. 7 for rubidium (left) and cesium (right)

[0053] .

[0114] For the numerical simulations, we set the initial temperature to 2 pK and vary the trap frequency between 30 and 300 kHz. We used 8 vibrational modes per atom and verified that this number is sufficient to obtain converged results at all the considered trap frequencies. The results are shown in Fig. 14. We plot the Bell state infidelity as a function of cotraPin Fig. 14a and 14b for rubidium and cesium, respectively. The values of J and Vij are taken from the respective first rows of Table 1. The dark grey lines represent the exact protocol, whose execution time is T* ® 6.30 / Q0, while the colored lines are the robust protocols with T = T* + 8T* (cf. Fig. 18; Fig. 13). The latter considerably reduce the gate infidelity, especially at low trap frequencies.

[0115] Remarkably, the gate infidelity exhibits a non-monotonic dependence on the trap frequency, with an optimal cotraPthat minimizes the error. This behavior arises from two competing effects that dominate in different regimes. At low cotraP, the dominant contribution to the infidelity arises from position fluctuations: in shallow traps, the large spatial extent of the atomic wavepacket makes the gate more sensitive to variations in interatomic distance. As cotraPincreases, the atoms become more tightly confined and this source of error is suppressed. However, for sufficiently large trap frequencies, the infidelity increases again due to photon recoil: tighter traps reduce the spatial overlap between the kicked and unperturbed motional states, leading to increased decoherence, even when the atom remains in the motional ground state. We note that cesium yields the smallest infidelities thanks to its larger mass, which reduces the oscillator length xosc~ 1 / Vm, its larger optical transition wavelength Xo, which reduces the photon recoil, and its attractive van der Waals force between P states (see the discussion in Section 6a).

[0116] In Fig. 14c and 14d, we plot the gate infidelities obtained from robust protocols with 2o 8T* = 0.1 for all sets of interaction strengths in Table 1 and different optical Rabi frequencies f o / 2.n ® 3.6 - 6.3 MHz (red) and f o / 2.n ® 5 - 8.75 MHz (purple). The total gate times and Rydberg subspace occupation times for the stabilized protocols range from T ® 6.45 / Q0 and TR® 2.45 / Q0 to T ~ 6.30 / Q0 and TR® 2.3 / Q0 for the largest and smallest Rabi frequencies, respectively.

[0117] We observe that cesium consistently yields lower gate infidelities— decreasing with increasing n— compared to rubidium across the range of parametersconsidered. This behavior can be attributed to the nature of the van der Waals interaction. For cesium, Vu / £2o is negative and increasingly attractive with n, which, similarly to large J / 2o, suppresses the population of the states ( | rir2> + |r2rd) / V2 and |r2r2), thereby enhancing robustness to interatomic distance fluctuations. At the same time, the Rydberg decay rate decreases with increasing n, leading to improved overall fidelity. In contrast, for rubidium the van der Waals interaction is repulsive and increases with n, leading to a growing positive Vn / 20. This interaction suppresses the population of the state |r, r,), which plays a key role in the dynamics underlying our gate protocol (see Section 2). As a result, the positive Vn / £2o competes with the beneficial effects of large J / Qo. This competition reduces the robustness against interatomic distance fluctuations at large n, eventually making this the dominant source of infidelity despite the improved Rydberg lifetime.Section 6: Conclusions

[0118] In the work described in supplement A it is demonstrated how resonant dipole-dipole interactions between Rydberg atoms can mediate two-qubit entangling operations. Our proposed CZ gate protocols stand out by requiring only constantamplitude laser pulses and time-modulated microwave fields, avoiding the need for complex control of optical phases and potentially suppressing the gate sensitivity to laser phase noise. Compared to standard gate schemes based on van der Waals blockade, our approach is faster and less sensitive to the finite lifetimes of Rydberg states. Furthermore, we generalized our protocols to realistic atomic setups using rubidium and cesium atoms, and employed systematic stabilization methods to counteract fluctuations in atomic positions. We showed that the stabilized protocols achieve Bell state fidelities on par with or exceeding current experimental realizations of neutral-atom entangling gates.

[0119] In this work, we only considered experimental implementations with heavy alkali atoms, for which reliable atomic physics calculations can be carried out. Yet, alkaline-earth species such as strontium and ytterbium constitute promising candidates for realizing our scheme, respectively due to the attractive van der Waals interaction between Rydberg states in the singlet series for strontium, and the small predicted C6 coefficients for states in the singlet series of ytterbium. Another potentialapplication of the gate protocol outlined in this paper is the realization of long-range gates, thanks to the slower decay with the distance of the resonant dipole-dipole potential with respect to the commonly used van der Waals interaction. Finally, an interesting extension of this work is the generalization of our scheme to multi-qubit gates, which could natively be implemented on future neutral atom quantum computers.Appendix 1: Piecewise protocol

[0120] As discussed in the main text, the piecewise protocol is composed of two optical n-pulses separated by a slightly-detuned micro wave, with Amw= +Gmw / 3, and it yields an exact CZ(±3.n / 2) gate in the limit of J / Go 00. The time required is Go T = 2.n + V3.n Go / Gmw, which, in the limit of large microwave driving (Gmw» Go) yields a shorter time than the optimal van der Waals protocol. The time spent in the Rydberg state, however, is slightly less favorable: GoTR= 71 + V3 TI Go / Gmw. Even in the large microwave driving limit, this is always larger than the van der Waals protocol. We also note that there is an additional solution for Amw~ J, which yields a slightly lower execution time of Go T = 271 + V271 Go / Gmw. Physically, this detuning brings the state | ri r2) + | r2 ri) into resonance with |ri ri), as opposed to the previous case where all dynamics from |ri ri) are trivial (cf. Fig. 9). Similarly to the previous case, no phase modulation is necessary.

[0121] While the limit J / Go 00 is useful for gaining an analytical understanding, it is not the ideal regime for realizing fast gates in a practical setup. For finite interaction strength, we resort to optimal control techniques. Similarly to the main text, we use GRAPE to find an optimal modulation of the detuning Amw(t) for the intermediate pulse. As shown in Fig. 15a, feasible solutions for each branch are found up to J / Gmw~ 2. The pulses consist of smooth oscillations around their asymptotic value (cf. Fig. 15b), with a frequency increasing with J / Q™. It is worth noting that for large microwave driving, one can still realize an approximate CZ gate without turning off the optical drive during the intermediate pulse. As shown in Fig. 15c, 15b, by using the pulses found previously and simply adjusting the total laser pulse time T, one can achieve the desired gate with a fidelity improving both with Gmw / Go as well as J / Go. Notably, the branch with Amw~ J seems to have the highest fidelities compared to theother two. If the regime J, 2mw » Go is experimentally accessible, such a piecewise protocol can be appealing for practical implementations as it does not require any phase modulation on the laser beam.Appendix 2: Finite-V van der Waals protocols

[0122] In this section, we briefly discuss gate protocols using only a finite van der Waals interaction. We consider a single Rydberg state |r) = | n> for each atom and the laser beam couples it to the computational state 11). The total Hamiltonian isThis Hamiltonian splits into two blocks, each one becoming a two-level system in the limit of V / 2o (the so-called blockade limit). This was exploited to perform a pulse optimization using GRAPE, achieving the optimal time of T ~ 7.61 / Qo. Performing a similar optimization for finite V, we obtain several sets of solutions based on the initial condition, as shown in Fig. 16. Remarkably, numerically exact solutions can be found down to the regime of V ~ G. We also note that, contrarily to the asymptotic case, the pulse time does not necessarily correlate with the total time in the Rydberg manifold TR.Appendix 3: Gate protocol with two-photon Rydberg transitionTable III. Resonant dipole-dipole and van der Waals interaction strengths J and 1 between pairs of rubidium Rydberg stateswith n = 40.50, 60, 70 at a distance R.Rb Rb40 55 118 60 196 42350 111 239 70 317 684Table IV. Lifetimes for the rubidium Rydberg states |r₁⟩ = |(n - l)D5 / 2, m. J = 5 / 2) and |r2) = lnP3 / 2, mj = 3 / 2) employed for the numerical simulations presented in

[0123] Above, we analyzed implementations of our gate protocol based on a single-photon transition to the Rydberg manifold { | ri>, | m)}. Here, we consider two-photon transitions in a setup with rubidium atoms. Specifically, the Rydberg transition from the hyperfine qubit state |i) proceeds via the intermediate state |e) = 16P3 / 2>. Since selection rules allow coupling to either S or D Rydberg states, we choose |ri) = (n-I)D5 / 2, mj= 5 / 2 and | r2> = nP3 / 2, mj=3 / 2. This choice is motivated by the significantly smaller ratio Vn / J for D states compared to S states (cf. Table 1 and Table 3). We optimize our gate protocol for rubidium Rydberg states with n=40, 50, 60, 70, using the interaction parameters listed in Table 3. We carry out the GRAPE optimization assuming the adiabatic elimination of the intermediate state |e). The resulting microwave phase cpmw(t), amplitude £2mw(t), and Rydberg detuning Ao are shown in Fig.21a, 21b, 21c. Notably, the optimized phase and amplitude profiles are almost independent of n.

[0124] To refine the gate for finite intermediate state detuning Ae, we reoptimize the single-qubit rotation angle 0 and gate duration T by maximizing the gate fidelity when |e) has a finite detuning Ae» £21,where £2i and 12 are the two single-photon Rabi frequencies. We emphasize that the resulting gate protocol is not exact even in the idealized case as long as Ae / £2i, Ae / £22 are finite. For Ae / £2i = Ae / £22 = 27.8, which we use in what follows, we obtain an ideal infidelity i-F ® 0.001. We then benchmark the gate under realistic conditions, including the finite intermediate state lifetime T_e = 110 ns, the finite Rydberg states lifetime listed in Table 4, and atomic motion. The Hamiltonian used for the numerical simulation is analogous to Section 6a, upon inclusion of the intermediate state | e) whose finite lifetime is modeled with an imaginary term. Similarly to Section 6a we take the motional degrees of freedom at an initial temperature of 2 pK with a trap frequency cotrap / 271 = 100 kHz and assume thetrap to be off during the gate. We set Ω1 / 2π = Ω2 / 2π = 278 MHz and Δe / 2π = 7.75 GHz, such that effective two-photon Rabi frequency is Ωeff / 2π ≈ 5 MHz. We find that choosing AeAo < o improves fidelities by reducing intermediate-state scattering. The resulting gate infidelity, plotted in Fig. 2id, decreases with increasing n from 0.7% for n=40 to 0.4% for n=7O, thus demonstrating competitive performance. In contrast to Section 5, we did not find it advantageous to employ robust protocols such as those shown in Fig. 18. We attribute this to the dominant role of intermediate-state decay and the intrinsic infidelity arising from finite Δe. A promising direction for further improvement would be to incorporate | e) explicitly into the GRAPE optimization, either to make the gate exact at finite Δe or to further suppress intermediate-state scattering.

[0125] Although specific embodiments have been described, the skilled person will recognize that numerous modifications are possible.

Claims

CLAIMS1. Method (300) for performing a quantum gate on a pair of atomic particles (100) serving as qubits for quantum computing, comprising:illuminating (310) the pair of atomic particles with a first electromagnetic radiation such that in each atomic particle a first qubit state 11) is coupled to a first Rydberg state |r1); andilluminating (320) the pair of atomic particles with a second electromagnetic radiation such that in each atomic particle the first Rydberg state |r1) is coupled to a second Rydberg state |r2), such that a dipole-dipole interaction is induced between the pair of atomic particles.

2. Method of claim 1, wherein the first electromagnetic radiation comprises laser radiation and the second electromagnetic radiation comprises microwave radiation.

3. Method of claim 1 or claim 2, wherein illuminating the pair of atomic particles with the first and the second electromagnetic radiation comprises:applying the first and the second electromagnetic radiation to the pair of atomic particles in form of at least two radiation pulses.

4. Method of claim 3, wherein the at least two radiation pulses are applied to the pair of atomic particles at least partially simultaneously.

5. Method of claims 1 to 4, wherein illuminating the pair of atomic particles with the first and the second electromagnetic radiation comprises:determining a pulse duration for the first and the second electromagnetic radiation and one or more time-dependent pulse parameters, including one or more of a pulse amplitude, a pulse phase, a detuning; andmodulating the first and the second electromagnetic radiation based on the determined one or more time-dependent pulse parameters during the pulse duration, such that the dipole-dipole interaction between the two atomic particles realizes the quantum gate for the pair of atomic particles.

6. Method of claim 5, wherein determining the time-dependent pulse parameters comprises:determining the time-dependent pulse parameters based on optimizing a fidelity of the quantum gate; and / ordetermining the time-dependent pulse parameters based on optimizing a robustness of the quantum gate against inter-particle distance fluctuations of the pair of atomic particles.

7. Method of claim 6, wherein optimizing the robustness of the quantum gate against the inter-particle distance fluctuations of the pair of atomic particles further comprises:optimizing the fidelity of the quantum gate in presence of fluctuations of a dipolar interaction strength and / or a van der Waals interaction strength for the pair of atomic particles induced by the inter-particle distance fluctuations of the pair of atomic particles.

8. Method of any of claims 5 to 7, wherein determining the time-dependent pulse parameters further comprises:determining the time-dependent pulse parameters based on performing an iterative numerical optimization method, preferably a gradient ascent pulse engineering, GRAPE, optimization method.

9. Method of any of claims 5 to 8, wherein modulating the first and the second electromagnetic radiation comprises:modulating the pulse amplitude Ω0(t) of the first electromagnetic radiation at essentially constant detuning Δo; andmodulating the pulse amplitude Ωmw(t) and the pulse phase φmw(t) of the second electromagnetic radiation.

10. Method of claim 9, wherein the pulse amplitude Ω0(t) of the first electromagnetic radiation is modulated as an essentially square-pulse function.

11. Method of claim 9 or 10, wherein the pulse amplitude Ωmw(t) of the second electromagnetic radiation is modulated as an essentially square-pulse.

12. Method of any of claims 9 to 11, further comprising:selecting the essentially constant detuning Aoto lie in the interval [-0.25 20max, o] where 20max is the maximal value of the pulse amplitude Ω0(t) of the first electromagnetic radiation.

13. Method of any of claims 1 to 12,wherein the quantum gate is a two-qubit entangling gate; and / or wherein the pair of atomic particles are part of a neutral atom quantum register formed by an array of optical traps for atomic particles inside a vacuum chamber.

14. Apparatus for performing a quantum gate on a pair of atomic particles (100) in a quantum register (512), comprising:a laser source (705);a microwave source (710);a set of optical elements (715) configured to direct laser radiation generated by the laser source onto the pair of atomic particles;a microwave antenna (720), coupled to the microwave source, and configured to illuminate the pair of atomic particles with microwave radiation generated by the microwave source, wherein the microwave source is configured to control a duration, an intensity, a detuning and / or a phase of the microwave radiation;a laser modulator (725) configured to control a duration, an intensity, a detuning and / or a phase of the laser radiation directed to the pair of atomic particles;a control unit (730) configured to control the laser modulator and the microwave source to illuminate the pair of atomic particles with modulated laser radiation and modulated microwave radiation to perform the quantum gate on the pair of atomic particles.

15. Apparatus of claim 14, wherein the control unit is further configured to control the laser modulator and the microwave source to perform the quantum gate by carrying out the steps of the method of any of claims 1 to 13.

16. Method (400) for quantum computing, comprising:trapping (420) a plurality of atomic particles in an array of optical traps forming a quantum register of trapped particle qubits inside a vacuum chamber;performing (430) a set of quantum gate operations of a quantum computing algorithm on a selected subset of the trapped particle qubits based on dipole-dipole interactions by performing the steps of the method of any of the claims 1 to 13; and determining (440) a result of the quantum computing algorithm by measuring a state of at least the selected subset of trapped particle qubits.

17. Method for quantum computing of claim 16, further comprising:obtaining (410) a set of instructions for performing the set of quantum gate operations of the quantum computing algorithm on the selected subset of the trapped particle qubits of the quantum register; andoutputting (450) data corresponding to the result of the quantum computing algorithm.

18. Method for quantum computing of claim 17,wherein obtaining the set of instructions for performing the set of quantum gate operations of the quantum computing algorithm comprises obtaining the set of instructions from a remote user device via a network; and / orwherein outputting the data corresponding to the result of the quantum computing algorithm comprises sending, to the remote user device via the network, the data corresponding to the result of the quantum computing algorithm.

19. Neutral atom quantum computer (500) configured to operate a neutral atom quantum register (512) inside a vacuum chamber (510) to perform a quantum computing algorithm and comprising means to carry out the steps of the method of any of claims 16 to 18.

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