Abstract
The ability to perform entangling quantum operations with low error rates in a scalable fashion is a central element of useful quantum information processing^{1}. Neutralatom arrays have recently emerged as a promising quantum computing platform, featuring coherent control over hundreds of qubits^{2,3} and anytoany gate connectivity in a flexible, dynamically reconfigurable architecture^{4}. The main outstanding challenge has been to reduce errors in entangling operations mediated through Rydberg interactions^{5}. Here we report the realization of twoqubit entangling gates with 99.5% fidelity on up to 60 atoms in parallel, surpassing the surfacecode threshold for error correction^{6,7}. Our method uses fast, singlepulse gates based on optimal control^{8}, atomic dark states to reduce scattering^{9} and improvements to Rydberg excitation and atom cooling. We benchmark fidelity using several methods based on repeated gate applications^{10,11}, characterize the physical error sources and outline future improvements. Finally, we generalize our method to design entangling gates involving a higher number of qubits, which we demonstrate by realizing lowerror threequbit gates^{12,13}. By enabling highfidelity operation in a scalable, highly connected system, these advances lay the groundwork for largescale implementation of quantum algorithms^{14}, errorcorrected circuits^{7} and digital simulations^{15}.
Main
Errors limit the computational capabilities of current quantum devices and must be made sufficiently low to permit efficient quantum error correction. In particular, twoqubitgate error rates below 1% (that is, fidelities above 99%) are required to surpass quantum errorcorrecting thresholds^{7}. Moreover, maintaining a combination of such low error rates, highly parallel control and a high degree of connectivity while scaling system size is crucial to realizing largescale quantum computers. Although highfidelity entangling operations were realized on isolated qubit pairs early on^{16,17,18,19}, only recently have these techniques been extended to larger systems. Stateoftheart examples include 99.4% fidelity on a 72qubit superconducting chip^{20} and 99.4–99.6% fidelity on a 31ion chain^{21}. Scaling these systems to even larger numbers of qubits while maintaining low error and efficient control is an exciting frontier^{22,23}, yet it also presents substantial platformspecific scientific and engineering challenges.
Recently, arrays of neutral atoms have emerged as a promising quantum processing platform capable of coherent control of hundreds of qubits^{2,3} for analogue quantum simulations. This platform also features a flexible, dynamically reconfigurable architecture^{4}, whereby entangling operations can be performed between neutralatom qubits with arbitrary connectivity and in a highly parallel manner. Although these capabilities open unique opportunities for both largescale digital simulations^{1} and computation with errorcorrected qubits^{7}, the main outstanding challenge in the field has been to improve the twoqubit gate fidelity substantially above the previously demonstrated value of approximately 97.5% (refs. ^{5,24}). In this article, we experimentally realize twoqubit controlled phase (CZ) gates with 99.5% fidelity while operating on up to 60 neutralatom qubits in parallel, closing the gatefidelity gap to other stateoftheart platforms^{20,21,23,25}. This advance is achieved by using a family of optimal gate schemes^{8,26} relying on the Rydbergblockade mechanism that are robust to experimental imperfections and spontaneous scattering, alongside the implementation of several experimental tools to overcome previously dominant error sources. To characterize the twoqubit gates, we use several complementary benchmarking methods using repeated gate applications, each giving consistent results. Finally, these techniques are generalized to entangling operations involving a higher number of qubits, allowing us to experimentally realize parallel, highfidelity, threequbit entangling gates.
Neutralatom entangling gates
In our approach, quantum information is encoded in longlived m_{F} = 0 hyperfine qubits^{27}, in which highfidelity (>99.97%) coherent singlequbit rotations are driven by Raman laser pulses. Entangling operations are performed in parallel by positioning the atoms, trapped in individual optical tweezers, into designated gate sites, followed by stateselective excitation into highly excited atomic Rydberg states using a twophoton transition (Fig. 1a). Errors in such quantum operations can occur owing to spontaneous emission from the intermediate atomic state e⟩, atomic temperature effects, Rydbergstate decay during the gate (Fig. 1a inset), as well as miscalibrations and experimental imperfections, such as laser noise or inhomogeneity.
We address these errors through the combination of gate schemes relying on optimal control and experimental improvements. Our method for gate implementation is inspired by the recently proposed timeoptimal gate by Jandura and Pupillo^{8}, which uses a numerically optimized continuous phase profile^{8,24,28} for a single laser pulse (as opposed to a discrete phase jump between two laser pulses^{5}). We generalize this gate scheme to a family of singlepulse gates with a small set of tunable gate parameters, including a version of the timeoptimal gate consisting of a parameterized sinusoidal phase modulation, as well as a second, smoothamplitude gate (Fig. 1c; see Methods for details).
We calibrate these gates by tuning several global parameters (Fig. 1d), which lends robustness to experimental imperfections: an optimal set of gate parameters can be found even in the presence of systematic offsets, such as finite Rydberg laser pulse rise time (Extended Data Fig. 4a). We further optimize our control pulses to suppress scattering from the shortlived intermediate state e⟩ by minimizing population in the ‘bright’ dressed state (\(B\rangle \propto 1\rangle +\sqrt{\frac{2\Omega }{\Delta }}e\rangle +r\rangle \)) containing e⟩ and maximizing population in the ‘dark’ state (D⟩ ∝ −1⟩ + r⟩) not containing e⟩ (in which Ω is the twophoton Rabi frequency and Δ is the intermediatestate detuning)^{9,29}. This optimization is achieved through the appropriate selection of the relative signs of the intermediate and twophoton detunings (Fig. 1b), as well as through smooth pulse shaping for the smoothamplitude gate (Extended Data Fig. 2).
Our experimental realization makes use of the apparatus described previously in refs. ^{2,4}, with which we rearrange ^{87}Rb atoms into programmable, defectfree arrays. Two main experimental upgrades facilitate highfidelity entanglinggate operation. First, we suppress scattering by substantially increasing intermediatestate detuning while maintaining a high twophoton Rabi frequency (Ω/2π = 4.6 MHz), enabled by excitation to a lowerlying (n = 53) Rydberg state with a tenfold higher power laser (Extended Data Fig. 1). Second, to suppress decoherence from atomic velocity and position fluctuations, we implement Λenhanced grey molasses cooling and an improved optical pumping technique (Methods) to achieve colder temperatures (radial phonon occupation \(\bar{n}\approx 12\)).
Entanglinggate characterization
To characterize the CZ gates realized with this approach, we create arrays of ten Bell pairs by arranging qubit pairs into separated gate sites (Fig. 2a) and pulsing global Rydberg and Raman lasers. Using the parameterized timeoptimal gate from Fig. 1c, we create a Bell state \(\left{\Phi }^{+}\right\rangle =\frac{1}{\sqrt{2}}\left(\left00\right\rangle +\left11\right\rangle \right)\) (ref. ^{5}), which is then characterized by measuring the populations of 00⟩ and 11⟩ and the oscillation amplitude of the twoatom parity \(\langle {\sigma }_{1}^{z}{\sigma }_{2}^{z}\rangle \) on applying a global singlequbit π/2 pulse of variable phase (Fig. 2b). We extract a raw Bellstate fidelity of 98.0(2)%, exceeding previous work by about 2% (ref. ^{5}), already suggesting a greatly improved gate fidelity. Because this Bellstate fidelity seems to be dominated by state preparation and measurement (SPAM) errors (Methods), to characterize the fidelity of the entangling gate more systematically, we apply an oddnumbered train of CZ gates to repeatedly entangle and disentangle the pairs and then characterize the fidelity of the final resulting Bell state^{25,30} (Fig. 2c). We fit the decreasing fidelity to an exponential decay to extract a CZ gate fidelity F_{CZ} = 99.52(4)% (Fig. 2d).
As a separate characterization of the gate fidelity, we apply random global singlequbit rotations between sequences of CZ entangling gates (Fig. 3a). This method averages over different states involved in the entangling operation similar to randomized benchmarking (see Extended Data Fig. 3 for numerical comparisons)^{10,11,31,32}. In the absence of errors during the gate sequence, a precisely calculated final singlequbit operation returns the qubit pair to their initial 00⟩ state. Applying a sequence of up to 20 CZ gates with random singlequbit rotations in between, we fit the decaying state fidelity as a function of CZ gate number and extract F_{CZ} = 99.54(2)% (Fig. 3b), consistent with the Bellstate method of Fig. 2d. Not only do these methods agree quantitatively but, in practice, we optimize the gate with this global randomized benchmarking method (Extended Data Fig. 5) and find that the exact same parameters are optimal for generating Bell states. The qubits also acquire a singleparticle phase during the CZ gate, which this benchmarking approach eliminates by using X gates in between CZ gate pairs (Fig. 3a). Therefore, we use a second method of global randomized benchmarking without these X gates, which allows for calibration of the singleparticle phase (used for calibrating the Bellstate measurement in Fig. 2b) and also benchmarks a gate fidelity of F_{CZ} = 99.48(2)% (Extended Data Fig. 6).
We next demonstrate that our gate methods are versatile, for which various pulse profiles can all realize a highfidelity CZ gate. Specifically, in Fig. 3b, we also realize and benchmark the smoothamplitude gate (Fig. 1c) and achieve a similar fidelity of F_{CZ} = 99.55(3)%. Different gate implementations can be tailored to specific use cases; for example, the smoothamplitude gate strongly suppresses scattering even with a closerdetuned excitation, which can help achieve high gate fidelities in situations in which laser intensity is limited.
Scaling up
We next explore the scalability of our approach to larger system sizes. Despite the fact that all calibration and control is done globally and not for individual gate sites, we find that the fidelity of the timeoptimal gate is constant across the ten individual gate sites within statistical error (Fig. 3c). This observation of homogeneity across the array highlights the inherent potential for scalability: more gate sites do not increase the calibration overhead. Motivated by this observation, in Fig. 3d, we extend to a 60qubit system by using larger Rydberg beams (while maintaining the same intensity) and achieve a gate fidelity of F_{CZ} = 99.48(2)% with good homogeneity across the array (Fig. 3e).
To understand the requirements for continued scaling and realizing highfidelity operation in even larger system sizes, we analyse the physical error sources in the system. In particular, we compare observed gate fidelities to detailed modelling, which uses two eightlevel atomic systems (Extended Data Fig. 3a) with quantitative decoherence rates informed by experimental measurements. This modelling accounts for the remaining CZ gate infidelity and reveals four main error sources (Extended Data Table 1): Rydberg decay, coupling to the other Rydberg m_{J} level, intermediatestate scattering and our measured groundRydberg \({T}_{2}^{* }=3\,{\rm{\mu s}}\), which is dominated by laserlightshift fluctuations and finite atomic temperature. We further analyse error sources by studying gate–site correlations in the experimental data. We observe that highweight correlated errors are largely absent from our data, which suggests the feasibility of stable, largescale operation (Extended Data Fig. 7). Careful analysis reveals small growth in the covariance between neighbouring gate sites (Extended Data Fig. 8), which can result from either correlated detuning fluctuations (corresponding to our \({T}_{2}^{* }\)) or longrange interactions caused by a Rydberg atom decaying into an adjacent Rydberg state (corresponding to our finite Rydberg lifetime; see further discussion in Methods).
Informed by these microscopic error sources, we conclude that the dominant challenge in maintaining high fidelity at an even larger number of parallel gate sites is to continually scale laser power and maintain beam homogeneity, as the other decoherence mechanisms seem to be independent of system size. However, we emphasize that, even with the present laser parameters, system sizes can be directly increased to hundreds of qubits by shuttling atoms in and out of the entangling zone during a quantum circuit^{4,33} or by redirecting the beam to dynamically redefine the position of the entangling zone.
Fast multiqubit gates
Finally, we explore the generalization of these methods to multiqubit gates. Using optimal control methods, we find a timeoptimal CCZ gate and corresponding ansatz phase profile (Fig. 4c) that realizes a native CCZ gate between three qubits^{5,12,13} in a time only 44% longer than the timeoptimal CZ and faster than other known CCZ profiles^{8}. This CCZ gate is realized experimentally by rearranging triplets of atoms into triangular gate sites (Fig. 4a) and applying the CCZ pulse profile with our global laser pulse. We characterize our CCZ gate using the sequence described in Fig. 4b to repeatedly entangle and disentangle a threequbit Greenberger–Horne–Zeilinger (GHZ) state and subsequently measure the final GHZstate fidelity (see Extended Data Fig. 9d for example GHZ states). Although this approach does not constitute a rigorous benchmarking of the CCZ gate fidelity (which can be done using randomized benchmarking^{13} for example), our data indicate highperforming threequbit entangling gates across 21 qubits in parallel, consistent with a fidelity F = 97.9(2)% (Fig. 4d). These optimal control methods extend to higherqubitnumber controlledZ gates. We numerically search for and find fast gates for up to six qubits (Fig. 4e), with gate times markedly shorter than those required to decompose an Nqubit controlledZ gate into 2N CZ gates and various singlequbit gates^{34}. Generically, with these global pulses and Rydberg blockade, one can natively realize symmetric, diagonal gates^{8} (for example, CPHASE gates as illustrated in Extended Data Fig. 4b), which are important for efficient realization of digital quantum simulation algorithms^{15}.
Discussion and outlook
Our results enable a new era of highfidelity digital circuits with neutral atoms. On the basis of the detailed microscopic understanding of error sources, we anticipate various paths to further improve gate fidelity in future work. For example, performing the gate at three times higher Rabi frequency and two times further detuning would theoretically result in a gate fidelity of 99.9%. This would require suppressing the coupling to the adjacent Rydberg state, optimization of pulse rise times and managing high laser intensity^{35} (Methods). The understanding of microscopic error sources can also be used to analyse the type of error, that is, the decomposition into different Pauli channels, atom loss and leakage^{36}, as described in Extended Data Table 1.
These observations open the door for explorations of largescale quantum error correction with efficient parallel control of logical qubits^{4,7}. The remaining ingredients associated with midcircuit readout can be implemented by moving atoms^{33} away from the entangling zone to a readout zone, using a second atomic species as ancilla qubits^{37}, shelving data qubits in auxiliary atomic levels^{38}, nondestructive readout with optical cavities^{39,40} or ancillary atomic ensembles^{41}. Alkalineearth atoms also present further opportunities, including singlephoton Rydberg excitation and nuclear spin control^{42,43,44}, as well as using erasure conversion for efficient quantum error correction^{45,46,47}. Furthermore, highfidelity multiqubit gates enable many possible scientific directions based on digital quantum simulation^{27} of models including nonAbelian topological physics^{15}, quantum gravity^{48,49} and quantum chemistry^{14}. Combining the analogue capabilities of the neutralatom platform with digital circuits^{4} opens the door for hybrid analogue–digital quantum simulation, including techniques such as shadow tomography^{50}. Finally, the highfidelity gate can be used as a tool for other applications with neutral atoms, for example, creating a wide variety of entangled states for use in metrology^{51,52}, and enabling new optical lattice simulators^{53,54,55,56}.
Methods
Experimental system
We stochastically load hundreds of ^{87}Rb atoms into a programmable array of 850nm optical tweezers generated by a spatial light modulator. A second set of 810nm optical tweezers generated by a crossed pair of acoustooptic deflectors is then used to rearrange into defectfree arrangements^{2}. Subsequently, the atoms are cooled first with polarization gradient cooling on the 780nm D2 line and then with Λenhanced grey molasses cooling on the 795nm D1 line^{57,58,59,60}. We implement Λenhanced grey molasses cooling owing to experimental simplicity (simply combined with our existing polarization gradient cooling path) as well as the potential for enhanced loading^{58} (which we do not use in this work as it reduces our cycle rate). We estimate an average radial motional quantum number of \(\bar{n}\approx 12\) from fitting a droprecapture curve^{61} to about 10 μK in approximately 1mKdeep traps.
Following rearrangement and cooling, we prepare atoms in the qubit basis, formed from clock states 0⟩ = F = 1, m_{F} = 0⟩ and 1⟩ = F = 2, m_{F} = 0⟩. In our previous work^{4,5}, we pump into F = 1, m_{F} = 0⟩ using Ramanassisted optical pumping, in which we repeatedly apply π pulses on the F = 1, m_{F} = −1⟩ → F = 2, m_{F} = −1⟩ transition and the F = 1, m_{F} = +1⟩ → F = 2, m_{F} = +1⟩ transition, followed by resonant depumping of the F = 2⟩ manifold. The main disadvantage of this scheme is that it can require scattering many photons (we performed 40–70 pumping cycles in refs. ^{4,5}) to end up in F = 1, m_{F} = 0⟩ and, empirically, we find that this causes enough heating to negate the benefits of the colder atoms following the Λenhanced grey molasses cooling step we use in this work.
To address this challenge, here instead we first optically pump into the F = 2, m_{F} = +2⟩ stretched state with σ^{+}polarized 780nm light, which scatters only several photons^{38,62}. Then we rotate the magnetic field by 90° such that the Raman laser propagation axis has an orthogonal component and can thus drive σ^{±}polarized transitions in the hyperfine manifold (see ref. ^{63}). We apply two separate Raman pulses that transfer population first from F = 2, m_{F} = +2⟩ to F = 1, m_{F} = +1⟩ and then to F = 2, m_{F} = 0⟩. We use Knill composite π pulses^{64} for these transfer steps and suppress unwanted transitions to adjacent m_{F} states by using Gaussianshaped optical pulses^{64}. We note that, during this transfer process, the magnetic field direction is along the axis of the tweezer, which quadratically suppresses vector light shifts from the tweezer polarization gradient^{65} that would otherwise limit coherence of the nonclock m_{F} states. Finally, we rotate the magnetic field back to its original configuration before performing quantum circuits.
The measurements in Fig. 2b and Fig. 4d have better SPAM performance than other measurements in this paper owing to suppression of previously undetected resonant leaked light that we discovered at the end of our measurements, as well as adding a final eight rounds of our previous Ramanassisted optical pumping (also using Gaussianshaped optical pulses to suppress offresonant excitation) to further improve the state preparation fidelity. With leaked light managed and with these two steps of optical pumping, we achieve both low temperatures and an estimated pumping fidelity of about 99.7–99.8%, probably dominated by residual leaked light.
To measure in the computational basis, we illuminate with a strong resonant light coupling F = 2 to the F′ = 3 on the D2 transition, which heats up and expels all atoms in the 1⟩ state; the remaining atoms are imaged in 0⟩. We estimate the combined fidelity of pushout and imaging to be about 99.83%. Finally, we also have a background atomic loss of roughly 0.25% before the circuit begins and about 0.1% after the circuit ends, dominated by a 10s vacuum lifetime.
To drive arbitrary singlequbit rotations, we use a Raman laser system^{63}, which globally illuminates the atoms with a Raman Rabi frequency of 1 MHz. Singlequbit rotations are implemented using robust BB1 pulses^{64,66}, whereas Z rotations are implemented by adjusting the phase in the control software. By applying sequences of random singlequbit rotations, we estimate a fidelity of 99.97% when sampling over a Haarrandom ensemble of singlequbit rotations, implemented by a combination of two Z rotations and an arbitrary BB1 pulse. This fidelity of 99.97% is consistent with the Raman scattering limit at our 180GHz detuning and therefore fidelity can be improved by detuning the Raman laser further.
Rydberg excitation
Extended Data Fig. 1 presents an overview of the atomiclevel structure used and an example pulse sequence for running a quantum circuit, largely the same as our previous work^{4}. The atoms are excited from the F = 2, m_{F} = 0⟩ state to the 53S_{1/2} Rydberg state in a twophoton scheme with a 420nm σ^{+}polarized and a 1,013nm σ^{−}polarized light. We are able to increase the intermediatestate detuning from our previous works, while maintaining similar or higher twophoton Rabi frequencies, in two ways. First, by operating at n = 53, we benefit from a 50% increase in the Rabi frequency compared with the previously used n = 70. Furthermore, we upgraded our 10W 1,013nm fibre amplifier laser to a 100W 1,013nm laser (IPG Photonics), which we operate at 20–50 W with a duty cycle <1%. Combined, this allows us to operate at the intermediatestate detuning of Δ/2π = 7.8 GHz with a Rabi frequency of Ω/2π = 4.6 MHz. For the data in Fig. 3d,e, we use the same Rabi frequency but an intermediatestate detuning of 6.3 GHz, which marginally increases scattering but allows us to work with lower Rydberg beam intensity.
The 1,013nm laser seed originates from a TOPTICA DL pro externalcavity diode laser, which is then locked to and filtered through a Stable Laser Systems ultralowexpansion cavity (finesse of 50,000 at 1,013 nm) that then injection locks another laser diode, which then seeds our 100W laser. Our 420nm laser is an 840nm TiSapph (M Squared), which is locked (but not filtered) to the same cavity (finesse of 30,000 at 840 nm) and then frequency doubled with an M Squared ECDX. For realizing the pulses and waveform shaping for quantum gates, we use an arbitrary waveform generator (Spectrum M4i.6631x8) that allows for arbitrary amplitude, frequency and phase control, which simultaneously drives two 420nm acoustooptic modulators (AOMs) (MQ240B40A0,2UV, AA Opto Electronic) in a tandem configuration and maps the RF waveform onto the 420nm light. The 1,013nm AOM (M1377aQ80L1 coated for 1,013 nm, Isomet) is pulsed on for several hundred microseconds and intensity stabilized during the duration of the entire circuit.
The two Rydberg beams are shaped into flat intensity tophat profiles by spatial light modulators to maximize intensity while maintaining homogeneity across the gate region^{2}. For the 20atom, onerow data, we aim for a flat 10 μm × 10 μm beam crosssection (to suppress sensitivity to drift) and for the 60atom, tworow data, we aim for a flat 20 μm × 10 μm region so that both rows are homogeneous. We tune beam parameters, including X and Y positions, focus and aberrations, to optimize homogeneity as measured by the differential light shift on the hyperfine qubit states. We stabilize the beam positions using a reference camera and motorized mirror mounts. To compensate for relative drift between the beam position and the atom array, we recalibrate the position often (several times per day) by stepping the beam positions to maximize the intensity at the atoms as measured by the differential light shift on the qubit transition, which takes about 5 min. We find that keeping the beams well centred on the atoms is important to ensure homogeneity and reduce sensitivity to relative beam drifts, and further find that gate parameters are highly reproducible (consistently reproducing fidelities of 99.5%) as long as the beams are properly positioned.
We extract the singlephoton Rabi frequencies of the Rydberg excitation using measurements of the twophoton Rabi frequency and the 420nm light shift on the groundRydberg transition, taking note of the appropriate Clebsch–Gordan coefficients for the multiple intermediate states. We extract Ω_{420} = 2π × 237 MHz and Ω_{1013} = 2π × 303 MHz, in which we adopt a convention such that the twophoton Rabi frequency is given by Ω = Ω_{420}Ω_{1013}/2Δ. Note that, because of the presence of several intermediate states, the firstorder scattering estimate is proportional to \(\frac{4}{3}\times {({\Omega }_{420}/2\Delta )}^{2}\) and thus—from the scattering perspective—the effective singlephoton Rabi frequencies are well balanced (273 MHz versus 303 MHz).
Finally, we comment on the motivation for choosing the principal quantum number n = 53 for the Rydberg state. There are several effects that depend on n, including the finite Rydbergstate lifetime (∝n^{3} for radiative decay, ∝n^{2} for blackbody decay), matrix elements influencing the 1,013nm Rabi frequency (∝n^{−3/2}), interaction energy (∝n^{11}) and sensitivity to electric fields (∝n^{7}). Weighing these relative benefits, we work at n = 53, for which Rydberg lifetime effects begin to become more relevant but for which the matrix element is favourable for increasing intermediatestate detuning. A technical challenge relevant to this choice is that, because of the small blockade radius, we place atoms at 2μm separation to achieve strong interaction strength V_{Ryd}/2π ≈ 450 MHz. Operating at such close spacing is enabled by using highnumericalaperture objectives (NA = 0.65 from Special Optics) and allows us to pack many atoms into the entangling zone.
Parameterized entangling gates
Inspired by the methods used in ref. ^{8} to find the timeoptimal gates, we use optimal control to design gates with phase profiles given by simple analytical formulas. We find that this approach makes the gate experimentally robust and reproducible, as small systematic offsets (for example, rise time, atom–atom separation etc.) can often be compensated for and are captured by a slightly different set of optimal parameters.
In this work, we focus on two main gates: one with a fixed amplitude and a phase profile similar to the timeoptimal gate of ref. ^{8} and a second one in which the amplitude is also varied. We note that, in the past, many schemes were proposed to implement twoqubit gates with Rydberg interactions^{5,8,29,35,67,68,69,70,71,72,73,74,75}, to engineer robustness to errors^{28,76,77,78,79,80,81,82,83,84} and to perform multiqubit gates^{85,86,87,88,89,90,91}. The fixedamplitude gate, which we refer to as ‘timeoptimal’ because it resembles and is only 0.2% slower than the Jandura–Pupillo gate^{8}, has the phase profile given by
This profile is plotted in the left column of Fig. 1c for the parameters
which constitutes an exact gate with time (ΩT/2π) = 1.215, that is, slightly longer than a resonant singleatom 2π pulse. Note that this set of parameters is not unique and other parameter values can also realize an exact gate, for example, at nonzero detuning δ_{0} as used in Extended Data Fig. 4a.
The smoothamplitude gate has a varying phase and a varying Rabi frequency of the 420nm laser. We used optimal control methods in a threelevel atomic system to find a gate that optimally suppresses scattering, for a fixed intermediatestate detuning and 1,013nm Rabi frequency. The scattering from the intermediate state was incorporated through a nonHermitian Hamiltonian proportional to the scattering rate (−iγ_{e}e⟩⟨e). Finally, we inferred an analytical form of the phase and amplitude profiles, which are given by
in which τ = t −T/2; in principle, one can also add a relative phase offset similar to φ_{0} in equation (1) for further finetuning. This ansatz realizes an exact CZ gate for the gate parameters
which has a duration of (ΩT/2π) = 1.207. We set the reference point such that, for Ω_{420} = Ω_{1013}, the system is at twophoton resonance and has a twophoton Rabi frequency Ω. This smoothamplitude gate has advantages of stronger intermediatestate scattering suppression and reduced offresonant coupling to other states. On the other hand, owing to a larger peak Rabi frequency, this gate is more susceptible to finiteblockade effects. The error budget for both gates can be found in Extended Data Table 1.
Extending beyond twoqubit gates, we find that a slightly more general ansatz allows us to implement a nearly timeoptimal threequbit CCZ gate with the phase profile given by
in which the Rabi frequency is kept constant and the parameters are
which results in an exact gate with duration (ΩT/2π) = 1.751. Finally, we note that the same methods can be directly extended to the design of CPHASE(θ) and CCPHASE(θ) gates, which is important in the context of digital quantum simulation^{15}. In Extended Data Fig. 4b,c, we show how the twoqubit gate duration scales with the phase θ.
Dark states in twophoton Rydberg gates
In this section, we describe the physics associated with the threelevel system present in the twophoton transition to the Rydberg state and derive how the Rydberg population can be realized through either the dark or the bright states. The basic intuition can be developed at the singleparticle level, at which the system is described by the threelevel Hamiltonian in the {1⟩, e⟩, r⟩} basis,
in which we use symbols Ω_{b} ≔ Ω_{420} and Ω_{r} ≔ Ω_{1013} in this section for clarity of expressions. We also assume that the amplitude and phase of the red 1,013nm laser are kept constant at all times and the blue 420nm phase is captured by the timedependent twophoton detuning \(\delta := \delta (t)\propto \,{\phi }^{{\prime} }(t)\).
At large intermediate detunings (Δ ≫ δ, Ω_{b/r}), this system is conveniently described in the darkstate basis (as summarized in Extended Data Fig. 2a,b) formed by the eigenstates of equation (2) at the twophoton resonance (δ = 0), which to the leading order in Ω_{r}/Δ is,
in which α = Ω_{b}/Ω_{r}. Note that the ‘dark state’ D⟩ has no contribution from the intermediate state, the ‘bright state’ B⟩ populates the intermediate state ∝ 1/Δ^{2} and E⟩ is composed essentially entirely from e⟩.
For our purposes, the initial state is always 1⟩, which is subsequently dressed by the blue light to \(\left\widetilde{1}\right\rangle \). This is because the amplitude rise time of the blue laser to its initial value of Ω_{b}(0) is on the timescale of 10 ns, which is much slower than the adiabaticity limit set by Δ and much faster than the twophoton Rabi frequency relevant for populating the Rydberg state; thus, the initial state indeed corresponds to
which is well supported on the {D⟩, B⟩} states alone. Moreover, the excited state E⟩ is detuned from the other two by Δ and all direct couplings to it are on the order of Ω_{r}/Δ; thus any population transfer out of the {D⟩, B⟩} manifold will be suppressed by (Ω_{r}δ)^{2}/Δ^{4} and the subsequent evolution of state \(\left\widetilde{1}\right\rangle \) is described by an effective twolevel system (Extended Data Fig. 2b). In this picture, the energy splitting is set by the AC Stark shift (diagonal terms) and the effective offdiagonal coupling is given by a combination of the twophoton detuning and diabatic terms (offdiagonal terms). Crucially, the Rydbergstate population can be realized in many inequivalent ways; for example, for α = 1, states of the form \(\sqrt{1\beta }\leftB\right\rangle +\sqrt{\beta }\leftD\right\rangle \) have the same Rydberg population for β and β → 1 − β (in which β ∈ [0, 1]), despite very different intermediatestate contributions.
First, consider the case of the parameterized timeoptimal gate in which the blue Rabi frequency is kept constant throughout the duration of the gate (α(t) = 1), up to the finite rise and fall times. The initial state is simply \((\leftD\right\rangle \leftB\right\rangle )/\sqrt{2}\) and the Hamiltonian is equivalent to
in which the magnetic field sign is decided by Δ and the phase of the Rabi frequency is given by the sign of the twophoton detuning δ. The time evolution under this Hamiltonian (which corresponds to driving the twophoton transition) can be solved exactly for fixed δ and the population in the dark state is
which can go above or below 1/2, depending on the relative sign of the detunings. More precisely, the Rydberg population is realized predominantly by means of the dark state when δΔ < 0 (P_{D} > 1/2), that is, when the intermediatestate detuning Δ and the twophoton detuning δ have opposite signs; for a timedependent detuning, the relevant sign is the one at the beginning of the pulse. In Extended Data Fig. 2c, we plot the intermediatestate population and the Blochsphere trajectories for the timeoptimal gate at two different signs of the twophoton detuning δ. As expected, one of the trajectories realizes the Rydberg population through the dark state and, as a result, minimizes the intermediatestate population, leading to suppressed scattering.
The remaining scattering comes mostly from the large admixture of the bright state in the initial state and can be further reduced by using a smoothamplitude profile, which starts at low blue Rabi frequency (α ≪ 1) and only later increases to larger values, as is the case in gate schemes based on the adiabatic passage^{29}. Note that operating at a fixed lower α does not further reduce scattering because a larger admixture of the bright state is necessary to realize the same integrated Rydberg population as before. In Extended Data Fig. 2d, we show the intermediatestate population and the effective Blochsphere trajectory for the smoothamplitude gate introduced in the previous section. This gate occupies the instantaneous dark state for most of its execution time, admixing only as much bright state as is necessary to realize the required Rydberg population. We find numerically that the degree of scattering suppression depends on the speed relative to the timeoptimal gate (assuming fixed 1,013nm Rabi frequency): scattering is suppressed by a factor of 1.2 (relative to the timeoptimal gate) if the two gates take the same time and by a factor of roughly 2.5 if the smoothamplitude gate is twice as long as the timeoptimal gate. This tunability is useful for choosing a gate based on the dominant error source: a slower gate could be more beneficial when scattering dominates, whereas a faster gate can be used when \({T}_{2}^{* }\) or Rydberg decay is the main source of errors.
We note that, despite the presence of several intermediate states (Extended Data Figs. 1 and 3a), which are at slightly different detunings and couple with different Rabi frequencies, the darkstate picture remains valid. We find that this is the case numerically and note that the intermediatestate population plots in Extended Data Fig. 2c,d include contributions from all intermediate states in a numerical model realistic for ^{87}Rb.
Simulating twoqubit gate error sources
The level diagram in Extended Data Fig. 3a summarizes the atomiclevel structure used for numerical modelling, as well as the assumed scattering rates, lifetimes, branching ratios, Rabi frequencies and detunings. We model scattering and Rydberg lifetime by performing a full densitymatrix simulation of the two atoms with the eight levels depicted in Extended Data Fig. 3a (including three intermediate states). Our modelling also explicitly includes the coupling to the other (m_{J} = −1/2) Rydberg state, which is 24 MHz lower in energy and is driven with a Rabi frequency suppressed by a factor of three (owing to Clebsch–Gordan coefficients). For the gate, we assume approximately 20 ns minmax rise/fall times of our AOM pulse profile (in a Blackman profile), which has substantial implications for the offresonant coupling to the adjacent Rydberg state, in terms of whether it is adiabatically ‘dressed’ or diabatically occupied. Because the impact of the other Rydberg state on the gate fidelity depends on the details of the pulse power profile and degree of calibration, we report a range of values that is reasonable for the assumptions mentioned above.
Finite temperature is also assumed in our error modelling, in which—for our given temperature—we calculate the position spread of the atom in a trap and the corresponding fluctuation in interaction strength from the distancedependent blockade interaction V_{Ryd} ∝ r^{−6}. We note that finite temperature can also contribute to singlequbit dephasing through both velocity spread and photon recoil^{92}, but these effects are already encompassed by our singlequbit groundRydberg \({T}_{2}^{* }\) measurements. The \({T}_{2}^{* }\) can also have contributions from other phenomena, such as electricfield fluctuations and fluctuations in the 1,013nm light shift (which has a differential light shift of about 20 MHz on the groundRydberg transition), and so, for our simulations, we simply use the measured \({T}_{2}^{* }=3\,{\rm{\mu }}s\) assuming a Gaussian distribution of detunings.
Projecting path to 99.9% and error breakdown
We can also use our detailed microscopic error modelling, which reproduces similar fidelities as the measured 99.5%, to project future performance. To reach 99.9% fidelity, the sum of the errors in Extended Data Table 1 needs to be suppressed to below 0.1%, which can be achieved by, for example, going two times further detuned, having a three times longer Rydberg lifetime (for instance, exciting to a higher n state), two times longer \({T}_{2}^{* }\) (note that dephasing error scales as \(\propto \,1/{(\Omega {T}_{2}^{* })}^{2}\)) and suppressing coupling to the other m_{J} state. This suppression can be achieved by applying a larger magnetic field, using the smoothamplitude gate or eliminating coupling altogether by exciting from a stretched state or through the 6P_{1/2} excited state. An alternative approach to reaching 99.9% fidelity could be going to three times higher Rabi frequency (again while suppressing coupling to the other m_{J} state) and two times larger detuning. Other unique opportunities towards future improvements include singlephoton excitation to the Rydberg state, which circumvents intermediatestate scattering but has higher Doppler and recoil sensitivity^{79,82,92}, and has been explored in a variety of contexts with both alkali^{93,94} and alkalineearth(like) atoms^{46,47,51}.
Separately, this microscopic error analysis can also be used to analyse the type of error produced, that is, whether it is a Pauli (X, Y, Z) error, atom loss or leakage to other m_{F} states. Such an understanding is particularly important for quantum error correction^{36,45}, for which neutral atoms have various unique opportunities^{95,96}, as knowing the noise structure can be used to enhance the performance of errorcorrecting schemes. Our present modelling suggests that most errors are Ztype and loss/leakagetype errors, as previously highlighted in ref. ^{36}. If atom loss is directly detected, these errors would constitute a socalled erasure error^{45} and, moreover, atom loss in this case is in fact a biased erasure error because almost all of it originates from state 1⟩, as pointed out and developed in ref. ^{96}. Alkalineearth(like) atoms are particularly well suited to erasure conversion, owing to their metastable qubit structure^{45,46,47}. In Extended Data Table 1, we summarize how each error source breaks down into the five error types mentioned above and find that only the scattering and Rydberg decay errors can lead to X and Y Pauli errors. For simplicity, we estimate the effective singleparticle error channel; that is, we compute the process matrix for the twoqubit gate and then trace out one of the qubits. The full process matrix can be used to study more complicated properties of this Pauli + loss/leakage noise model, such as correlations.
Gate calibration and benchmarking
We calibrate the gate using the global randomized benchmarking method shown in Fig. 3. In Extended Data Fig. 5, we show an example sequential optimization of CZ gate parameters for the timeoptimal gate and the smoothamplitude gate, leading up to the measurements in Fig. 3b. Once found, these gate parameters are empirically optimal for all the other benchmarking methods, such as the Bellstate measurements in Fig. 2, and are consistent from day to day.
The qubits also pick up a global singleparticle phase during the gate, which we cancel here using global X gates between pairs of CZ gates for simplicity. For many applications, such as quantum error correction, our gates are naturally used in this configuration (as was done for the quantum circuits implemented in ref. ^{4}). We also further calibrate and benchmark the CZ gate without the X gate, by performing randomized benchmarking composed of repeated application of CZ gates and random singlequbit rotations, as shown in Extended Data Fig. 6a. Here the final several CZ gates and random rotations are calculated to return the qubit pair from the resulting entangled state back to the initial product state. We perform a singlequbit Z rotation after each CZ gate to compensate for the accumulated singleparticle phase, which we calibrate as simply another parameter of the gate to scan and optimize. As well as calibrating this singlequbit Z rotation, this circuit also benchmarks a gate fidelity of 99.48(2)% on 20 atoms. To measure the raw Bellstate data in Fig. 2b, we used both the CZ gate calibration by means of the first method of global randomized benchmarking and this singleparticle phase calibration, as two independent calibration stages.
We note that all of our randomized benchmarking methods use only global rotations for simplicity. The symmetry introduced by global rotations makes us less sensitive to certain types of error^{97}, SWAP being an extreme example, which we expect to be negligible. Nonetheless, because the atoms are identical and placed very close together, there is a large degree of symmetry between the two qubits and we expect this global benchmarking procedure to faithfully capture our gate fidelity, which we confirm with numerical simulations. This is further confirmed by the fact that the experimentally extracted fidelity is consistent between this method and the Bellstate measurement. Quantitatively, in Extended Data Fig. 3e, we simulate all the benchmarking methods, including the full randomized benchmarking protocol, using the microscopic error model developed in this work. We find that all methods give consistent results, with the Bellstate fidelity lowerbounding the other curves.
Here we describe some experimental procedures used while taking data. First, for the benchmarking curves involving varying numbers of CZ gates, we take data in a cyclic manner to avoid systematic biases that could be introduced by experimental drift (for example, alternating in the sequence of 20, 0, 16, 4, 12 and 8 CZ gates for the data taken in Fig. 3). We perform several rounds of this cyclic sequence in one continuous stretch of time (over roughly a few hours for each plot in Figs. 2 and 3). For each gate number, we average over 300 sets of random rotations.
To extract a gate fidelity, we fit our data to exponential decays. We note that, as we are mostly in the linear regime of the exponential curve, adding an offset to the fit (and then rescaling the fitted exponent, as done in some randomized benchmarking works) has a negligible effect on the extracted fidelity and so we fit to an exponential decay without an offset.
Bellstate fidelity
Here we outline the method used for the Bellstate data in this work. We measure the Bellstate fidelity as the average of coherences and populations^{5}. The coherence is extracted by measuring the amplitude of parity oscillations, using the circuit in Fig. 2c. The populations are calculated as the sum of the 00⟩ and 11⟩ states, which we correct for further atom loss as described below. The Bellstate populations can be overestimated owing to atom loss contributing to the perceived detection of state 11⟩ (ref. ^{5}), because loss shows up identically as 1⟩ in our statedetection procedure. To account for this, here we measure the atom loss probability when applying the sequence of gates (by turning off the pushout of state 1⟩ for state discrimination), to find the extra contribution of atom loss to the Bellstate population. To perform this loss subtraction, we subtract the observed 11⟩ (that is, observed loss of both atoms) population during the loss measurement directly from the measured populations, as well as measuring the lossperatompergate, which can also contribute to state 11⟩ by converting 01⟩ and 10⟩ to 11⟩. This loss subtraction is performed for Fig. 2d. We emphasize that this correction strictly lowers the measured gate fidelity (without applying this loss subtraction, the measured CZ gate fidelity on the raw Bellstate fidelity data is extracted to be 99.57(4)%).
We next evaluate a SPAMcorrected Bellstate fidelity from the measured raw Bellstate fidelity of 98.0(2)% after a single gate in Fig. 2b. To extract a SPAMcorrected Bellstate fidelity, we first measure relevant SPAM errors. In particular, we measure a population of 99.6(1)% in state 0⟩ when we try to prepare into 0⟩ and, likewise, a population of 99.4(1)% in state 1⟩ after state preparation into 1⟩. These measurements include further effects from loss and imaging/pushout fidelity. Specifically, there is a 0.35(5)% probability that an atom is lost during the sequence and the gate itself causes 0.17% further loss on top of this baseline loss. Our pushout fidelity of 99.83(1)% affects the measurement fidelity of 1⟩, for which we also correct. From these measurements, we estimate the amount of population leaked into other m_{F} levels during state preparation, as well as the probability of atom loss both before and after the circuit. From these values, we follow the method described in ref. ^{5} to extract a SPAMcorrected Bellstate fidelity of 99.4(4)%.
Analysis of correlations between gate sites
Here we further analyse our data to characterize whether gate errors across the array are correlated. We study the 20atom and 60atom global randomized benchmarking data from Fig. 3 and consider the distribution of the number of errors that occurs in each experimental shot, in which an error is defined as whenever a qubit pair does not return to the initial 00⟩ product state. In Extended Data Fig. 7a,c, we plot the number of errors occurring in each shot as a function of the number of CZ gates applied, in which the mean number of errors grows owing to the 0.5% error per CZ gate. We compare our data (bottom) with a model consisting of a Poissonian distribution of errors centred at the experimental mean (top). We find that the Poisson distribution model approximates our data and largescale correlated errors are not common in our system.
To analyse more quantitatively in a single plot, we average the data for all numbers of gates and plot the resulting distribution in Extended Data Fig. 7b,d. We find that higher weight errors for both the 20 atoms and 60 atoms data are greatly suppressed. More quantitatively, we compare our data to the average of the Poissonian distributions plotted in Extended Data Fig. 7a,c and find small deviation from the Poisson distribution. We find that these data are better described by a model in which the CZ gate fidelity is sampled from a Gaussian distribution in each shot (which would arise from, for example, global shottoshot fluctuations in detuning, captured by our \({T}_{2}^{* }\)).
In Extended Data Fig. 8a,c, we plot the covariance matrix between gate sites for the return to the initial state P_{00⟩}, after 0 CZ gates and after 20 CZ gates, qualitatively observing evidence of small positive covariance between nearby gate sites. In particular, Extended Data Fig. 8b,d shows the growth of covariance between neighbouring gate sites as a function of the number of CZ gates applied. These observations are consistent with known physical effects related to our error budget, namely, Rydberg lifetime and \({T}_{2}^{* }\). For example, decay of Rydberg atoms to nearby P states during the gate can cause detuning shifts owing to strong longrange interactions between S and P Rydberg states (decaying as 1/R^{3}), as well as hopping of the P state to nearby gate sites. Furthermore, fluctuations in Rydberg 1,013nm beam intensity can give rise to shottoshot fluctuations in gate fidelity with a spatial dependence. Classical Monte Carlo simulations of these two phenomena (not plotted here) reveal that both error sources can result in nonzero covariance, which seems to be described by quadratic growth for a small number of gates but linear asymptotically. We note that the crosstalk between gate sites, on the scale of 10 kHz, should be negligible.
This measurement of covariance after 20 gates is a highly sensitive probe to small correlations between gate sites building up over the course of the circuit. After one applied gate, this covariance seems to be smaller than other main error sources, so these correlations will have little effect for quantum circuits in which atoms are involved in gates not just at a single gate site but across the entire entangling zone. Moreover, when running quantum circuits using atom transport^{4}, the approximately 100μs delay between subsequent gates would result in leftover Rydberg states being expelled from the array, completely suppressing the effect from Rydberg decay described above.
CCZ gate design
To find timeoptimal gates for the multiqubit controlled phase gates, such as the CCZ gate, we use the optimal control methods similar to ref. ^{8}. The gates in ref. ^{8} are found by looking for twoqubit diagonal gates up to global singlequbit Z rotations. However, for more than two qubits, there are several distinct ways of realizing the controlledZ gates that are not connected by a Z rotation but rather by a general singlequbit rotation, and these various gate realizations can be different. We use the approach in which the controlled phase flip is applied to the 000⟩ state to find multiqubit controlled Z gates for a larger number of qubits; we present the obtained times in Fig. 4e and Extended Data Fig. 9a. In Extended Data Fig. 9b, we show the timeoptimal pulse profiles of multiqubit CZ gates up to the sixqubit CCCCCZ. Finally, we note that an analytical ansatz (defined in the section dedicated to parameterized gates) similar to that used for the CZ gate allows us to parameterize the threequbit controlledZ gate with only a marginal decrease in speed (ΩT/2π = 1.75). In fact, we find that this ansatz can also realize the CCCZ gate and we expect simple generalizations to be capable of realizing these gates for even larger numbers of qubits.
GHZ states
To characterize the CCZ gate experimentally, we create a GHZ state using the circuit shown in Extended Data Fig. 9c. In Extended Data Fig. 9d, we generate GHZ states after application of two CCZ gates, with populations of 92.9(3)% and parity contrast of 89(1)%, giving a raw GHZstate fidelity of 90.9(6)% (without any loss subtraction). For Fig. 4, we calibrate the gate by repeatedly applying the CCZπCCZ part of the circuit such that after six and ten CCZ gates, we generate GHZ states with reduced fidelity and we observe a 2.1(2)% reduction in the raw GHZ fidelity as a function of the number of CCZ gates applied. For the data in Fig. 4d, we operate at 7.8GHz intermediatestate detuning and 3.9MHz Rabi frequency.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank S. Hollerith for discussions on error contributions from motional states and M. Cain for insights on atomic dark states. We further thank M. Abobeih, H. Bernien, N.C. Chiu, S. Geier, J. Guo, A. Keesling, H. Pichler and P. Stroganov for useful discussions, technical support and careful reading of the manuscript. We also thank QuEra Computing and IPG Photonics and in particular N. Gemelke, M.G. Hu, M. Kwon and A. Lukin for support in the development and testing of the highpower 1,013nm Rydberg laser. We acknowledge financial support from the U.S. Department of Energy (DOE Quantum Systems Accelerator Center, contract numbers 7568717 and DESC0021013), the Center for Ultracold Atoms, the National Science Foundation, the Army Research Office MURI (grant number W911NF2010082) and the DARPA ONISQ programme (grant number W911NF2010021). S.J.E. acknowledges support from the National Defense Science and Engineering Graduate (NDSEG) fellowship. D.B. acknowledges support from the NSF Graduate Research Fellowship Program (grant DGE1745303) and the Fannie and John Hertz Foundation. T.M. acknowledges support from the Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering. N.M. acknowledges support by the Department of Energy Computational Science Graduate Fellowship under award number DESC0021110.
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S.J.E., D.B., M.K., S.E., T.M., H.Z., S.H.L. and A.A.G. contributed to the building of the experimental setup, performed the measurements and analysed the data. M.K. developed the twoqubit gate schemes and performed theoretical analysis. N.M. and M.K. developed the multiqubit gate schemes. T.T.W., H.L. and G.S. contributed to initial developments and insights into gate error sources. All work was supervised by M.G., V.V. and M.D.L. All authors discussed the results and contributed to the manuscript.
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M.G., V.V. and M.D.L. are cofounders and shareholders and H.Z. is an employee of QuEra Computing.
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Extended data figures and tables
Extended Data Fig. 1 Atomiclevel diagram and pulse sequence.
a, Level diagram showing key levels of ^{87}Rb used in our quantum circuits. The clock states, 0⟩ and 1⟩, are the qubit states used in this work. Excitation to the Rydberg state between 1⟩ and r⟩ is carried out by a twophoton transition driven by 420nm and 1,013nm lasers. Singlequbit rotations are realized with an amplitudemodulated 795nm laser that drives Raman rotations between the m_{F} = 0 clock states. A DC magnetic field of 8.5 G is applied throughout this work. The Rydberg detuning signs and polarization signs are carefully selected for various optimizations: for suppressing 420induced differential light shift between 0⟩ and 1⟩, we reddetuned the 6P_{3/2} transition; for using darkstate physics (nominally the phase profile corresponds to a certain sign of twophoton detuning), we thereby choose positive twophoton detuning, which—in turn—then suppresses coupling to m_{J} = − 1/2 by being primarily on the upper side of m_{J} = +1/2; and, finally, the 1,013nm light shift is lower (by about 30%) at this singlephoton detuning sign, as there is a magic wavelength of about 1 GHz reddetuned of 6P_{3/2} for the 1⟩ → 53S_{1/2} transition^{98}. Two downsides of this detuning choice are that this choice of 420nm polarization and detuning causes a vector light shift in the hyperfine groundstate manifold that causes the m_{F} levels to be pushed closer together, as opposed to further apart, which could exacerbate effects arising from 420induced vector light shifts coupling adjacent m_{F} states in the groundstate manifold (although negligible here), and the other downside is that the m_{J} = −1/2 Rydberg pair states are closer detuned to the twophoton excitation and so we require a larger interaction strength to suppress their excitation, although the matrix element to these states is smaller. b, Example pulse sequence, here for making a \(\left{\Phi }^{+}\right\rangle \) Bell state between two qubits. Traps are pulsed off for a few hundred ns during the Rydberg gate to avoid both antitrapping of the Rydberg state and inhomogeneous light shifts that broaden the transition, and the groundstate atoms are then recaptured for roughly 3 μs between consecutive gate applications.
Extended Data Fig. 2 Darkstate physics in twophoton Rydberg gates.
a, In the fardetuned limit, the threelevel system can be effectively described as a twolevel system with a ‘dark’ state D⟩ and a ‘bright’ state B⟩. The population in the excited state E⟩ is suppressed by a factor ∝ Δ^{−4} and does not participate in system dynamics. b, The Hamiltonian of the bare atomic system and the effective twolevel system, in which α is the (timedependent) ratio between the blue and red Rabi frequencies and \(\mathop{\alpha }\limits^{.}\) is its time derivative. c, Intermediatestate population during the parameterized timeoptimal gate in the dark and bright configurations together with their Blochsphere trajectories in the {D⟩, B⟩} basis. d, Intermediatestate population during the smoothamplitude gate and its Blochsphere trajectory in the instantaneous {D⟩, B⟩} basis. The simulation parameters correspond to those mentioned in Extended Data Fig. 3a and Extended Data Table 1.
Extended Data Fig. 3 Atomic physics errorlevel diagram and numerical comparison of benchmarking methods.
a, Level diagram shows the eight states assumed in the simulation. We assume a 88μs Rydbergstate lifetime (based on measured T_{1} with 1,013nm scattering lifetime subtracted) and a 110ns lifetime for the intermediate states. We assume the following branching ratios for the intermediate states^{99}: η_{e→L} = 0.6142, η_{e→1} = 0.2504, η_{e→0} = 0.1354, and the following ones for the Rydberg states^{100}: η_{r→L} = 0.894, η_{r→1} = 0.053, η_{r→0} = 0.053. We use the branching ratios between different channels of intermediatestate scattering as reported in ref. ^{99} and we also assume a simplified model in which all indirect paths (through 4D and 6S) populate the groundstate manifold uniformly. The Rydberg lifetime has both radiative decay (170 μs) and blackbody decay (128 μs) components, which we obtain by rescaling the values in ref. ^{100} to n = 53. The microwave component results purely in atom loss and we assume that the radiative decay populates the groundstate manifold uniformly. We note that a more accurate treatment of the decay channels^{36} could increase error modelling precision in future work. b, Benchmarking of the CZπCZ sequence with global random rotations, which is insensitive to the singleparticle phase. c, Benchmarking a standalone CZ gate with global random rotations, which enables separate calibration of the singleparticle phase. d, The usual interleaved randomized benchmarking method using random twoqubit Clifford gates (not performed in this work). e, Numerical simulation of the presented benchmarking methods and the Bellstatepreparation method, using the realistic error model developed in this work. All approaches give consistent results, with the Bellstate fidelity measurement lowerbounding the other curves.
Extended Data Fig. 4 Robust and versatile twoqubit gates.
a, Robustness to experimental offsets. For some systematic experimental offsets, such as finite rise time of the 420nm Rydberg laser pulse, an exact CZ gate can still be found. The relative values of gate parameters for the timeoptimal gate are plotted as a function of pulse rise time. For no rise time, the parameter values used here are: A/2π = 0.0988, ω/Ω = 1.3629, φ_{0} = −2.6082 and δ_{0}/Ω = −0.0187. b, Duration of a controlledphase gate CPHASE(θ). The CZ gate (θ = π) is the longest in this gate family. Because faster gates are expected to have higher fidelity, an average CPHASE gate should perform even better than the CZ gate reported in this work, which is an exciting perspective for nearterm digital simulation. c, Plotting on a log–log plot, we see that, for small angles θ, the gate time decreases with an approximate power law ΩT(θ)/2π = 1.6 × (θ/2π)^{0.23}. This suggests that applying very small phases can be costly and should be taken into account when designing digital simulation schemes. Although these smallangle gates are timeoptimal when applying a single, fixedamplitude pulse, different approaches could perform better. Exploring other exact and approximate gate schemes, such as Rydberg dressing and a detuned 2π pulse, in the smallangle regime is an interesting direction for future work.
Extended Data Fig. 5 Empirical optimization of twoqubit gate fidelity.
a, Calibrating gate parameters for the timeoptimal gate, indicating the chronological sequence of calibrations performed before measurement in Fig. 3b. We scan individual gate parameters and measure the probability of return to the initial state after application of 20 entangling gates. b, Analogous calibration of gate parameters for the smoothamplitude gate, which we performed in the sequence shown before the smoothamplitude measurement in Fig. 3b. Additional calibration of other gate parameters was performed before these measurements.
Extended Data Fig. 6 CZ gate singleparticle phase calibration and benchmarking.
a, Digital circuit for global randomized benchmarking method used to calibrate singleparticle phase, in which a Z rotation can be performed after each CZ gate to compensate for the acquired phase. R_{rand} are singlequbit rotations sampled from a Haarrandom distribution and the five rotations R_{f} are computed to return the atom pair to the initial product state in the absence of gate errors. For the 0 CZ gates point, 20 random rotations are applied, as well as a final rotation precomputed to return population to the initial state. b, Experimental data used for calibrating the singleparticle phase by optimizing the return probability P_{00}_{⟩} after 20 CZ gates (inset). For the optimal choice of ϕ, we extract a 99.48(2)% CZ gate fidelity, fitting to an exponential decay.
Extended Data Fig. 7 Correlated errors in experimental shots.
a, Distribution of errors in each experimental shot as a function of the number of CZ gates applied for the 20atom data in Fig. 3, showing qualitative agreement with a Poisson distribution centred at the experimental mean for the number of errors in a shot. b, Histogram of the number of errors in a shot, averaging over all numbers of gates for the 20atom data. We compare to one model assuming a Poissonian distribution of errors about the mean, finding some deviation from our data. In a second model, we consider that, in each shot, there is a slightly different gate fidelity, sampled from a Gaussian distribution with a mean of 99.54% and standard deviation of 0.3%. This second model seems to better capture our data. c, Repeating the analysis for the 60atom data in Fig. 3, we notably find no shot (out of the 5,053 total repetitions) with 18 or more errors out of the 30 gate sites. Again, the data are qualitatively similar to a Poisson distribution model. d, Averaging over the 60atom data for all numbers of gates, we find again a small quantitative deviation between the data and a model with a Poisson distribution of errors in each shot. The data seem to be better described by a model in which, in each shot, we sample fidelities from a Gaussian distribution with a mean of 99.5% and a standard deviation of 0.3%.
Extended Data Fig. 8 Correlations between gate sites.
a, Covariance matrices for the 20atom data in Fig. 3b after 0 gates and 20 gates, in which local correlations appear after 20 gates. b, Covariance averaged over all neighbours, next nearest neighbours and next next nearest neighbours, as a function of the number of CZ gates applied. As a guide to the eye, data are fit to quadratic curves. c, Covariance matrices for 60atom data in Fig. 3d for 0 gates and 20 gates. d, Plotting covariance for neighbouring sites for the 60atom data. Once again, the covariance between nearby sites exhibits small growth throughout the 20 CZ gates applied, in particular for the nearestneighbour sites.
Extended Data Fig. 9 Timeoptimal pulses for multiqubit controlled phase gates and GHZstate data.
a, The execution time of a C_{N}Z blockade gate as a function of the number of qubits. The Nqubit gates are realized by applying a phase flip to the \({\left0\right\rangle }^{\otimes N}\) state, which is not equivalent to the method of ref. ^{8} for N > 2. For the CCZ gate, applying a π phase to the 111⟩ state (111⟩ → −111⟩), while leaving all other basis states invariant, is related by a global bitflip to applying a relative π phase to the 000⟩ state; however, the two implementations are not equivalent up to a global Z rotation, contrary to the twoqubit case. The timeoptimal CCZ gate using the second approach realizes the CCZ gate about 34% faster with (ΩT/2π) = 1.72, as compared with (ΩT/2π) = 2.61 from the first approach. The two approaches are different because the relative phase of π is accumulated between different basis states, which have different rates of phase accumulation. In the case of applying 111⟩ → −111⟩, the states with the slowest relative rate are 111⟩ and 011⟩, which are driven with the Rabi frequencies of \(\sqrt{3}\Omega \) and \(\sqrt{2}\Omega \), respectively, resulting in the phase accumulation rate proportional to \((\sqrt{3}\sqrt{2})\Omega \approx 0.32\Omega \). By contrast, when the relative phase is applied on the state 000⟩, the smallest accumulation rate is given by the 001⟩ state, which is driven with the Rabi frequency Ω. In general, an arbitrary global singlequbit rotation at the end of the gate can be included to incorporate all of the above approaches in the optimization procedure. b, Timeoptimal phase profiles (without analytic parameterization) for the C_{N}Z gates up to six qubits realized by applying a phase flip to the \({\left0\right\rangle }^{\otimes N}\) state. c, Circuit used to generate the GHZ state \(( 000\rangle + 111\rangle )/\sqrt{(2})\) after two CCZ gates. d, GHZ states measured experimentally on applying this circuit to seven threequbit groups in parallel, with populations in 000⟩ and 111⟩ of 92.9(3)% and a parity contrast of 89(1)%, giving a raw GHZstate fidelity of 90.9(6)%.
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Evered, S.J., Bluvstein, D., Kalinowski, M. et al. Highfidelity parallel entangling gates on a neutralatom quantum computer. Nature 622, 268–272 (2023). https://doi.org/10.1038/s4158602306481y
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DOI: https://doi.org/10.1038/s4158602306481y
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