Submicrosecond entangling gate between trapped ions via Rydberg interaction

Generating quantum entanglement in large systems on timescales much shorter than the coherence time is key to powerful quantum simulation and computation. Trapped ions are among the most accurately controlled and best isolated quantum systems1 with low-error entanglement gates operated within tens of microseconds using the vibrational motion of few-ion crystals2,3. To exceed the level of complexity tractable by classical computers the main challenge is to realize fast entanglement operations in crystals made up of many ions (large ion crystals)4. The strong dipole–dipole interactions in polar molecule5 and Rydberg atom6,7 systems allow much faster entangling gates, yet stable state-independent confinement comparable with trapped ions needs to be demonstrated in these systems8. Here we combine the benefits of these approaches: we report a two-ion entangling gate with 700-nanosecond gate time that uses the strong dipolar interaction between trapped Rydberg ions, which we use to produce a Bell state with 78 per cent fidelity. The sources of gate error are identified and a total error of less than 0.2 per cent is predicted for experimentally achievable parameters. Furthermore, we predict that residual coupling to motional modes contributes an approximate gate error of 10−4 in a large ion crystal of 100 ions. This provides a way to speed up and scale up trapped-ion quantum computers and simulators substantially. A quantum system combining the accuracy of trapped ions and the speed of Rydberg atoms is reported; the implemented fast gate is a step towards a scalable quantum computer.

Generating quantum entanglement in large systems on timescales much shorter than the coherence time is key to powerful quantum simulation and computation. Trapped ions are among the most accurately controlled and best isolated quantum systems 1 with low-error entanglement gates operated within tens of microseconds using the vibrational motion of few-ion crystals 2,3 . To exceed the level of complexity tractable by classical computers the main challenge is to realize fast entanglement operations in crystals made up of many ions (large ion crystals) 4 . The strong dipole-dipole interactions in polar molecule 5 and Rydberg atom 6,7 systems allow much faster entangling gates, yet stable state-independent confinement comparable with trapped ions needs to be demonstrated in these systems 8 . Here we combine the benefits of these approaches: we report a two-ion entangling gate with 700-nanosecond gate time that uses the strong dipolar interaction between trapped Rydberg ions, which we use to produce a Bell state with 78 per cent fidelity. The sources of gate error are identified and a total error of less than 0.2 per cent is predicted for experimentally achievable parameters. Furthermore, we predict that residual coupling to motional modes contributes an approximate gate error of 10 −4 in a large ion crystal of 100 ions. This provides a way to speed up and scale up trapped-ion quantum computers and simulators substantially.
Trapped atomic ions are one of the most promising architectures for realizing a universal quantum computer 1 . The fundamental single-and two-qubit quantum gates have been demonstrated 2,3 with errors less than 0.1%, sufficiently low for fault-tolerant quantum error-correction schemes. A single-qubit coherence time of 10 min (ref. 9 ) as well as the proof-of-principle demonstration of error correction 10,11 have been realized. Nevertheless, a scalable quantum computer requires a large number of qubits and a large number of gate operations to be conducted within the coherence time. Most established gate schemes using a common motional mode are slow (typical gate times are between 40 μs and 100 μs) and difficult to scale up since the motional spectrum becomes more dense with increasing ion number. Many schemes have been proposed and implemented 12,13 , with the fastest experimentally achieved gate in a two-ion crystal being 1.6 μs (99.8% fidelity) and 480 ns (60% fidelity) 13 , realized by driving multiple motional modes simultaneously. Although the gate speed is not limited by the trap frequencies, the gate protocol requires the phase-space trajectories of all modes to close simultaneously at the end of the pulse sequence 13 . In long ion strings with a large number of vibrational modes, it becomes increasingly challenging to find and implement laser pulse parameters that execute this gate with a low error. Thus, a slow-down of gate speed appears inevitable.
Two-qubit entangling gates in Rydberg atom systems 7 are substantially faster, owing to strong dipole-dipole interactions. The gate fidelities in recent experiments using neutral atoms are fairly high 14,15 . However, the atom traps need to be turned off during Rydberg excitation. This can cause unwanted coupling between qubits and atom motion as well as atom loss 8 . By employing blue-detuned optical tweezers at a magic wavelength, one may achieve the trapping of Rydberg states, although the predicted residual change in trapping frequency of about 50% (ref. 16 ) will still result in entanglement between qubits and motional states and thus a reduction in gate fidelity. In addition, switching between different trapping potentials or switching the trap off and on will exponentially heat up the atoms 17 and reduce the gate fidelity (the unwanted motional effects are stronger at higher temperatures). Because cooling without destroying the qubit information is challenging (direct cooling by lasers will destroy the qubit, whereas sympathetic cooling will cause entanglement between qubit and coolant atoms unless their interaction is state-independent 18 ), this may limit the number of gate operations despite the long coherence times of these systems.
In solid-state platforms, such as superconducting circuits 19 and silicon-based qubits 20 , the interactions are also strong, enabling fast two-qubit gates, and tremendous progress has been made recently. However, the number of entanglement operations that can be executed in the coherence time using these systems is typically about 10 3 (with superconducting circuits gate times of approximately 50 ns and coherence times of approximately 100 μs), which is orders of magnitude Article less than about 10 6 in atomic systems (with trapped-ion gate times of around 100 μs and coherence times of around 100 s, Rydberg gates may improve this to about 10 8 ).
Combining the benefits of trapped-ion qubits and Rydberg interactions is a promising approach for scalable quantum computation 21 : Rydberg interactions may enable fast, motion-independent gates between trapped ions. Additionally, because ions are trapped via their electric charges and interact via the state-independent Coulomb interaction, Rydberg ions do not suffer from most of the limitations of neutral Rydberg atom systems. It has been shown that ions in Rydberg states can be confined 22,23 and coherence between Rydberg states and low-lying states can be maintained 24 in radio-frequency traps. However, strong interactions between Rydberg ions and their use in fastentangling gates had not been previously demonstrated.
In our experiment 88 Sr + ions are confined in a linear Paul trap. Two low-lying electronic states ( 〉 |0 and 〉 |1 ) are used to store a qubit, and 〉 |0 is coupled to Rydberg state 〉 r | via a two-photon laser field. The relevant level scheme is shown in Fig. 1a, and more details can be found in ref. 24 .
Two ions excited to Rydberg states interact through the dipoledipole interaction: where i μ μ is the electric dipole moment of ion i (i = 1,2), r = r 2 − r 1 is the relative ion position and n r r = /| |. Trapped ions in atomic eigenstates have zero dipole moments and V dd has no first-order effect. The secondorder effect (van der Waals interaction) can be sufficiently strong to cause Rydberg blockade in neutral atom systems with principal quantum number n ≈ 50 within a few micrometres 25 . However, this interaction is much weaker in Rydberg-ion systems; it scales with net core charge as Z −6 (for Sr + with one valence electron Z = 2). Instead, we achieve a strong first-order interaction by inducing rotating electric dipole moments via a microwave field.
When two Rydberg states 〉 s | and 〉 p | are coupled by a microwave field with Rabi frequency Ω MW and detuning Δ MW (Fig. 1b), the eigenstates become: where C is the normalization constant 26 . In our system the electric dipole moments of the dressed states 〉 |± rotate with the microwave field in the plane perpendicular to the magnetic field, as shown in Fig. 1c, d. For two ions, each in state 〉 〉 r | ≡ | + , the dipole-dipole interaction given by Eq. (1) yields an energy shift: . For the measurements described here, we use Rydberg states with principal quantum number n = 46 and ion separation 4.2 μm, resulting in V max ≈ 2π × 1.9 MHz. By tuning the ratio between Ω MW and Δ MW the interaction strength can be varied between approximately zero and V max . Higher-order terms in V dd can be neglected because the energy between dressed states Ω MW ≫ V max .
We probe the interaction between Rydberg ions as shown in the Rabi oscillations between 〉 |0 and 〉 r | in Fig. 2. Either one or two ions are trapped and initialized in 〉 |0 . A two-photon laser field then couples 〉 〉 r |0 ↔ | , the pulse length is varied and the population that is excited to 〉 r | and subsequently decays to 5S 1/2 is measured by fluorescence detection (for details of the detection scheme see ref. 24 ). From Fig. 2b to Fig. 2d, the dipole-dipole interaction strength is increased: this has no effect on the Rabi oscillations when a single ion is trapped (green data points), while two-ion oscillations (pink data points) are suppressed as the interaction shifts the pair state 〉 rr | out of resonance from are used to store a qubit 〉 〉 {|1 , |0 }. The qubit transition is driven by a 674-nm laser field. Projective measurements in any basis are carried out by qubit rotations and fluorescence detection at 422 nm. Rydberg excitation is driven by a two-photon laser field; the 243-nm laser field and the 306-nm laser field couples . c, The Rydberg electron density for the microwave-dressed state 〉 〉 s p (1/ 2)(| + | ) yields a permanent dipole that rotates in antiphase with the microwave electric field about the magnetic field direction. d, The two microwave-dressed Rydberg ions interact via the dipole-dipole interaction. They are confined on the trap axis. the laser excitation. When 〉 rr | is far from resonance the population oscillates between two states: 〉 |00 and the Bell state 〉 〉 r r (1/ 2)(|0 +| 0 ). This is the blockade regime 27,28 , which is corroborated by the 2 enhancement of the two-ion Rabi oscillation frequency (light blue data points) over the single-ion oscillation frequency (green data points) in Fig. 2d.
We then use the maximum interaction strength (V = V max ) to implement a 700-ns controlled phase gate between two ions. The experimental sequence is described in Fig. 3. First the two ions are initialized in state 〉 〉 〉 〉 (|00 + |01 + |10 + |11 ), 1 2 and then the gate operation is applied-the population is transferred from 〉 〉 〉 r |0 → | →|0 , and the Rydberg interaction causes component 〉 |00 to acquire phase ϕ. Finally, qubit rotations and projective measurements are used to determine the final two-ion state.
The gate operation consists of a double stimulated Raman adiabatic passage (STIRAP) pulse sequence 24,29 . The three levels 〉 |0 , 〉 e | and 〉 r | are coupled by two laser fields with coupling strengths Ω 1 and Ω 2 . 〉 |0 and 〉 r | are resonantly coupled while 〉 e | is detuned by Δ (see Fig. 1). Ω 1 and Ω 2 are gradually changed such that an ion initially in 〉 |0 adiabatically follows an eigenstate to 〉 r | and back to 〉 |0 . An ion initially in 〉 |1 is unaffected. For the initial pair states 〉 |11 , 〉 |10 and 〉 |01 the eigenenergies remain zero and no phase is accumulated. From the initial state |00〉, the population can be excited to 〉 rr | (provided Δ ≳ V max , more details can be found in the Supplementary Information or ref. 29 ), the energy of which is shifted due to the Rydberg interaction. Thus 〉 |00 acquires the phase , with the two-ion density operator ρ t( ) and pulse length T. We achieve ϕ ≈ π using sinusoidal profiles for Ω 1 (t), Ω 2 (t) and T V = 8π/3 ≈700 ns max . In the ideal case the final target state 〉 〉 〉 〉 (− |00 + |01 + |10 + |11 ) 1 2 is maximally entangled. The correlation of the final state is measured as follows: the projection of ion 1 on 〉 〉 {|0 , |1 } 1 1 and the phase between 〉 |0 2 and 〉 |1 2 of ion 2 are measured simultaneously; results are shown in Fig. 4a. The phase between 〉 |0 2 and 〉 |1 2 of ion 2 is 0 (π) when ion 1 is projected onto 〉 |1 1 ( 〉 |0 1 )-this indicates ϕ = π. Entanglement is characterized by parityoscillation measurements 30 after rotating the target state to a Bell state (Fig. 4b). The coherence and population of the Bell state are measured to be C = 0.72± 0.04 and The contributions of different gate error sources are estimated by numerical simulation (Table 1). They can account for the observed gate infidelity. The largest error contributions are technical and can be diminished by improving the microwave power stability, increasing the laser intensities and decreasing the laser linewidths. Further, several error contributions depend on the gate time, which can be reduced by using higher Rydberg states which interact more strongly, together with higher laser intensities. In our gate implementation we use sideband cooling to mitigate the mechanical effects of Rydberg ion polarizability 23,31 , which will become unnecessary when we use microwave-dressed Rydberg states with zero polarizability 26 . In turn, this may allow implementation of the gate in higher-dimensional ion , and the energy of state 〉 rr | is shifted by the tunable Rydberg interaction. b-d, Rabi oscillations between 〉 〉 r |0 ↔ | for one and two ions. From panel b to panel d, the interaction strength V is increased from 0 to V max by tuning the microwave field amplitude and detuning. The y axes show the population, which is excited to 〉 r | and subsequently decays to 5S 1/2 . The green data points for the one-ion excitation 〉 〉 r |0 ↔ | are similar in b-d. Pink data points show the two-ion excitation probability for 〉 〉 rr |00 ↔| , which is similar to the square of the oneion (green) data (b). From panel b through panel c to panel d, excitation to 〉 rr | is increasingly suppressed as 〉 rr | is shifted out of resonance from the laser excitation, with Rydberg blockade in d. The light blue data points show the twoion excitation probability for 〉 〉 〉 r r |00 ↔(1/ 2)(|0 +| 0 ); a 2-enhanced Rabi frequency compared with the one-ion case (green) is observed in the blockade regime (d). Error bars indicate one standard deviation due to quantum projection noise. Dephasing is mainly caused by finite Rydberg lifetime and laser linewidth. Solid lines show results of numerical simulations of the Lindblad Master equation including all known decoherence sources and using no free parameters. More details of the numerical simulation can be found in the Supplementary Information.

Article
crystals. Importantly, we observe no heating effects of the ion motion after the gate operation in the two-ion crystal. Numerical simulation indicates that the gate error induced by mechanical forces between interacting Rydberg ions, which can excite ion motion, is about 10 −4 in a 100-ion crystal using a zero-polarizability microwave-dressed state. This simulation and more details about gate errors caused by mechanical effects in large ion crystals can be found in the Supplementary Information (see Supplementary Figs. 1, 2). The gate error induced by black-body radiation through spontaneous decay and double-ionization may be further reduced in a cryogenic environment (see Supplementary Fig. 3 for more details).
In summary, we have demonstrated a strong Rydberg dipole-dipole interaction (2π × 1.9 MHz) and a 700-ns controlled phase gate in a trapped ion experiment. This provides a promising way to increase both the number of entangling operations within the coherence time and the number of qubits in a trapped-ion quantum computer or simulator. Although gates based on Rydberg interactions have already been demonstrated for neutral atoms, their availability in trapped ion systems offers many opportunities and advantages.
Ions are trapped by their electric charges, which enables deep, tight and state-independent confinement. The spread of the motional wavefunction is typically about 10 nm, much smaller than the effective excitation laser wavelengths, and as a result the system is in the Lamb-Dicke regime and motion is largely decoupled from electronic transitions.
The state-independent Coulomb interaction enables many important techniques in trapped-ion systems, such as sympathetic cooling 32 and multi-element logic gates 33 , which are compatible with the Rydberg gate. The ion crystal can be cooled via the common motional modes without affecting the qubits or the Rydberg gate operation (since the gate does not rely on the motional modes), so the crystal is protected from heating by the trap or other noise sources. The advantages of different ion species can be combined; for example, qubit information can be transferred to ions with long coherence times for storage and to ions with strong interactions for gate operations.
The insensitivity of the Rydberg gate to ion temperature and motional modes enables its application in scalable architectures, such as large or even higher-dimensional ion crystals and ion shuttling systems 34 . A Rydberg gate may be applied using a fixed pulse sequence to achieve a robust gate fidelity in large crystals or shuttling systems despite the difficulty in reaching low ion temperatures, and even though the number of ions and the motional modes vary. Here we have explored a short-range Rydberg interaction, but inside a microwave cavity the interaction can be made all-to-all without decreasing the interaction with distance.
Furthermore, the combination of the Rydberg dipole-dipole interaction and the Coulomb interaction may enable coupling between electronic and vibrational degrees of freedom, and thus interesting  1 2 , followed by the gate operation involving Rydberg excitation using a double-STIRAP sequence, the qubit rotation is applied for parity-oscillation measurements, and finally the two-ion state is measured using qubit rotation and fluorescence detection. b, Gate operation: the 〉 〉 e |0 ↔ | and 〉 〉 e r | ↔| coupling strengths (Ω 1 and Ω 2 / 2) are varied sinusoidally over 700 ns. c, Simulation of one-and two-ion population dynamics during gate operation; the controlled phase accumulated by 〉 |00 is proportional to area enclosed by the 〉 rr | population curve. Other states ( 〉 |01 , 〉 |10 and 〉 |11 ) do not accumulate phases because their eigenenergies remain zero.
Rotation axis (π) a, A Ramsey-type experiment measures the relative phase between 〉 |0 2 and 〉 |1 2 (a π/2 pulse of varied phase applied on ion 2 followed by projection measurement on 〉 〉 {|0 ,|1 } 2 2 ) conditional on the state of ion 1; the π-phase difference between the cases 〉 〉 {|0 ,|1 } 1 1 is consistent with the target entangled state The mismatch between simulation and data is mainly because of imperfect single-ion addressing of the qubit laser beam. b, (78 ± 3)% entanglement fidelity is determined by population (green) and parity-oscillation (blue) measurements after rotating the target state to the Bell state (1/ 2)(|00⟩ + i|11⟩). The small phase mismatch between simulation and experiment (owing to the minor dynamical phase) does not affect the entanglement fidelity. Error bars in a and b indicate one standard deviation due to quantum projection noise. quantum simulations, such as tunable multi-body interactions 35 or excitation transport through motion in biomolecules 36 .

Online content
Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-020-2152-9. Coupling to the quadrupole field of the radio-frequency trap is not considered; the effect is negligible for the n = 46 state. For higher n, it may give a non-negligible energy shift oscillating with the quadrupole field that need to be compensated by modulating the laser and microwave frequencies to achieve low error.

Data availability
The datasets generated during and analysed during the current study are available from the corresponding authors on reasonable request.