High-fidelity manipulation of a qubit enabled by a manufactured nucleus

The recently demonstrated trapping and laser cooling of 133Ba+ has opened the door to the use of this nearly ideal atom for quantum information processing. However, before high-fidelity qubit operations can be performed, a number of unknown state energies are needed. Here, we report measurements of the 2P3/2 and 2D5/2 hyperfine splittings, as well as the 2P3/2 ↔ 2S1/2 and 2P3/2 ↔ 2D5/2 transition frequencies. Using these transitions, we demonstrate high-fidelity 133Ba+ hyperfine qubit manipulation with electron shelving detection to benchmark qubit state preparation and measurement (SPAM). Using single-shot, threshold discrimination, we measure an average SPAM fidelity of F=0.99971(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}=0.99971(3)$$\end{document}, a factor of ≈2 improvement over the best reported performance of any qubit.

For these NISQ devices, single-shot state preparation and measurement (SPAM) infidelity (ϵ s ) causes a reduction in computational fidelity that is exponential in qubit number, F s ¼ ð1 À ϵ s Þ Nq (uncorrelated errors). The requirement to perform faithful SPAM therefore limits the number of qubits to N q < ln ð2Þ=ϵ s . While state readout error correction techniques can effectively lower measurement infidelity, they generally require a number of measurements that grows exponentially with N q and single-shot readout infidelity 25 . For these reasons, and given the desire to increase N q to tackle problems beyond the reach of classical computers, it is important to develop new means to improve ϵ s .
The A = 133 isotope of barium provides a potential path to improving fidelities in atomic ion quantum computing, as this isotope combines the advantages of many different ion qubits into a single system 26 . 133 Ba + has nuclear spin I = 1/2, which as we show here, allows fast, robust state preparation and readout of the hyperfine qubit. It possesses both hyperfine and optical m F = 0 "clock" state qubits, which are relatively insensitive to magnetic fields (m F is the projection quantum number of the total angular momentum F) 27 . It also possesses metastable 2 D J states (τ~1 min), allowing high-fidelity readout, and long-wavelength transitions enabling the use of photonic technologies developed for the visible and near-infrared spectrum. However, before these advantages can be realized, a number of unknown hyperfine and electronic transition frequencies must be determined.
Here, we measure the previously unknown 2 P 3/2 and 2 D 5/2 hyperfine structure, as well as the 2 P 3/2 ↔ 2 S 1/2 and 2 P 3/2 ↔ 2 D 5/2 electronic transition frequencies. Using this knowledge, we demonstrate 133 Ba + hyperfine qubit manipulation and electron shelving detection. Employing a threshold discrimination and modest fluorescence collection optics (0.28 NA), we measure an average single-shot SPAM fidelity of F ¼ 0:99971ð3Þ, the highest reported for any qubit.
In what follows, we first present qubit SPAM using standard hyperfine-selective optical cycling 28,29 combined with arbitrary qubit rotations and a composite pulse sequence for high-fidelity state transfer. We then present measurement of the unknown hyperfine and electronic transition frequencies. Finally, we use this information to demonstrate high-fidelity SPAM using electron shelving.

Qubit manipulation and hyperfine-selective SPAM
The hyperfine qubit is defined on the pair of m F = 0 "clock" states in the 2 S 1/2 manifold as 0 This hyperfine qubit is initialized to the 0 j i state after Doppler cooling via optical pumping by applying frequencies ν c 493 , ν op 493 , ν c 650 , and ν sb 650 (Fig. 1). Rotations of the qubit Bloch vector about cosðϕÞx þ sinðϕÞŷ through angle θ, R(θ, ϕ), are accomplished by using microwave radiation near 9.925 GHz 30 controlled by a modular digital synthesis platform 31 . An example rotation of the form R(Ω R t, 0) is shown in Fig. 2a, where the average population in state 1 j i found in 200 trials, measured with a technique described later, is plotted versus the duration of microwave radiation with Rabi frequency Ω R = 2π × 57.03(3) kHz. The 1 j i state can be prepared after initialization into 0 j i by R(π, 0); however, we employ a composite pulse sequence, referred to as the CP Robust 180 sequence (attributed to E. Knill) 32 , consisting of the five πpulses Rðπ; π 6 ÞRðπ; 0ÞRðπ; π 2 ÞRðπ; 0ÞRðπ; π 6 Þ. As shown in Fig. 2b, c, the broad flat features in both curves near zero detuning and θ = π demonstrate resiliency to both pulse area and detuning errors as compared to single π-pulses, enabling robust day-to-day operation.
Typically, for nuclear spin-1/2 hyperfine qubits, single-shot state readout is accomplished via hyperfine-selective optical cycling (ν c 493 and ν c 650 in Fig. 1) and collection of any resulting fluorescence.
The j0i and 1 j i states are determined by threshold discrimination on the number of collected photons, as an atom in the 1 j i state scatters many photons, while an atom in the 0 j i state does not 28,29 . Using this hyperfine-selective optical cycling for SPAM, we measure the fraction of events in which an ion prepared in the 0 j i state was determined to be 1 j i, ϵ 0 j i ¼ 3:03ð4Þ 10 À2 , and the fraction of experiments in which an ion prepared in the 1 j i state was determined to be 0 j i, ϵ 1 j i ¼ 8:65ð9Þ 10 À2 . The average SPAM fidelity is defined as F ¼ 1 À ϵ ¼ 1 À 1 2 ðϵ j0i þ ϵ j1i Þ ¼ 0:9415ð5Þ. The fidelity of this technique is limited by offresonant excitation to the j 2 P 1=2 ; F ¼ 1i manifold during readout, which can decay to either 0 j i or 1 j i, thereby causing misidentification of the original qubit state 28 . This readout fidelity could be improved with increased light collection efficiency 12,33 .

Spectroscopy
For high-fidelity SPAM, 133 Ba + offers another path to state detection. The 1 j i qubit state can be shelved 34 to the long-lived (τ ≈ 30 s) metastable 2 D 5/2 state via the 2 D 5/2 ↔ 2 S 1/2 transition, or optically pumped via the 2 P 3/2 state (Fig. 1), followed by Doppler cooling for state readout. Projection into the 0 j i or 1 j i state is then determined by threshold discrimination on the number of collected photons, as an atom in the 0 j i state scatters many photons, while an atom in the 2 D 5/2 state, indicating 1 j i, does not. Off-resonant scatter is negligible in this case as the Doppler cooling lasers are detuned by many THz from any 2 D 5/2 state transitions.
In principle, shelving of the 1 j i qubit state is possible via the 2 D 5/2 ↔ 2 S 1/2 electric quadrupole transition near 1762 nm (ν 1762 , currently unknown). However, as we demonstrate below, fast, high-fidelity shelving of the 1 j i state can be achieved with optical pumping by application of the frequencies ν 455 , ν 585 , and ν c 650 (and ν 614 for deshelving). Of these, only ν c 650 has been previously measured 26 . To determine these unknown frequencies, we measure the 2 P 3/2 ↔ 2 S 1/2 and 2 P 3/2 ↔ 2 D 5/2 isotope shifts relative to 138 Ba + (δν 455 133;138 and δν 614 133;138 ) and hyperfine splittings Δ 3 and Δ 5 (Fig. 1). To measure Δ 3 and δν 455 133;138 , the atom is prepared in the 2 S 1=2 ; F ¼ 1 i manifold by optical pumping with ν c 650 and ν sb 650 after Doppler cooling. A tunable laser near 455 nm (ν 455 ) is applied for 50 μs. When the frequency is near one of the two allowed transitions, excitation followed by spontaneous emission from the 2 P 3/2 with branching ratios 35 0.74, 0.23, and 0.03 to the 2 S 1/2 , 2 D 5/2 , and 2 D 3/2 states, respectively, optically pumps the ion to the 2 D 5/2 state. The population remaining in the 2 S 1/2 and 2 D 3/2 states is then detected by collecting fluorescence while Doppler cooling and using threshold discrimination on the number of collected photons to decide if the atom was in the 2 D 5/2 state. This sequence (see Supplementary Information) is repeated 200 times per laser frequency, and the average population is shown Fig. 3a as a function of frequency. From these data, we find Δ 3 = 623(20) MHz, and δν 455 133;138 = +358(28) MHz relative to 138 Ba + . To measure Δ 5 and δν 614 133;138 , the atom is Doppler cooled and shelved to the 2 D 5/2 state via one of the 2 P 3/2 hyperfine manifolds. j i state, the five π-pulse CP Robust 180 sequence Rðπ; π 6 ÞRðπ; 0ÞRðπ; π 2 ÞRðπ; 0ÞRðπ; π 6 Þ transfers population from the initially prepared 0 j i state. b Probability of shelving 1 j i vs. microwave detuning using the CP Robust 180 sequence with Ω R = 2π × 35.4(1) kHz. Points are experimental data and solid line represents theoretical prediction for this composite pulse sequence with no fit parameters. c Pulse area (t ¼ θ ΩR ) scan at zero detuning using the CP Robust 180 sequence. Dashed dotted lines in b, c are theory for a single π-pulse, R(π, 0). Statistical error bars on individual data points are smaller than markers. the 2 D 5/2 are to the 2 D 5=2 ; F ¼ 3 i manifold. Next, a tunable laser near 614 nm is applied for 100 μs. When the frequency is near the 2 spontaneous emission from the 2 P 3/2 state quickly deshelves the ion to the 2 S 1=2 ; F ¼ 1 i and 2 D 3/2 states. This deshelved population is then detected via Doppler cooling. This sequence is repeated 200 times per laser frequency, and the average population is shown Fig. 3b as a function of frequency. From these data, we find the 2 D 5/2 hyperfine splitting Δ 5 = 83 (20) MHz, and isotope shift δν 614 133;138 = +216(28) MHz.
High-fidelity SPAM With the required spectroscopy known, we can calculate the expected fidelity of optically pumped electron shelving detection of the hyperfine qubit as follows. For SPAM of the 1 j i state, the initial state is prepared as described above, followed by illumination with a laser resonant with the 2 P 3=2 ; F ¼ 2 i $ 2 S 1=2 ; F ¼ 1 i transition (ν 455 ) at an intensity below saturation ( Fig. 1 and Methods). After excitation of the atom, the 2 P 3/2 state quickly (τ ≈ 10 ns) spontaneously decays to either the 2 S 1/2 , 2 D 5/2 , or 2 D 3/2 state. Dipole selection rules forbid decay to the 2 S 1=2 ; F ¼ 0 i ( 0 j i) state, resulting in F ¼ 0:88 shelving fidelity, limited by population stranded in the 2 D 3/2 states. To further increase the shelving fidelity, a 650 nm laser near resonant with the 2 P 1=2 ; , and a laser near 585 nm (ν 585 ) resonant with the 2 P 3=2 ; F ¼ 2 i $ 2 D 3=2 ; F ¼ 2 i transition can be applied at an intensity below saturation. The hyperfine structure of 133 Ba + allows for concurrent repumping of the 2 D 3/2 states (ν 585 and ν c 650 ) with all polarization components during the application of ν 455 , simplifying the shelving sequence (see Supplementary Information) compared with other species 7 . Dipole selection rules forbid spontaneous emission to the 2 S 1=2 ; F ¼ 0 i ( 0 j i) state resulting in a fidelity of F % 0:999. This scheme is limited by off-resonant scatter of ν 455 to the 2 P 3=2 ; F ¼ 1 i state, where 0.44 of decays to the 2 S 1/2 are to the 2 S 1=2 ; F ¼ 0 i. If ν 455 is linearly polarized parallel to the magnetic field direction (π-light), dipole selection rules forbid excitation from the 2 P 3=2 ; F ¼ 1; m F ¼ 0 i $ 2 S 1=2 ; F ¼ 1; m F ¼ 0 i for the first scattered photon, and the expected fidelity increases to F ¼ 0:9998.
For SPAM of the 0 j i state, initialization with optical pumping proceeds as described above. After preparation, the 1 j i state is shelved as previously described, and the state is read out via Doppler cooling. During 1 j i state shelving, off-resonant excitation to the 2 P 3=2 ; F ¼ 1 i followed by spontaneous emission can shelve the ion to the 2 D 5/2 state. This results in an expected SPAM fidelity of F ¼ 0:9998.
To experimentally test these predictions, state preparation of each qubit state is applied to a single trapped 133 Ba + ion and read out using the highest fidelity optically pumped shelving scheme (see Methods for experimental parameters). Before each SPAM attempt, the Doppler cooling fluorescence is monitored to determine if an SPAM attempt can be made. If the count rate does not reach a predetermined threshold of 2σ below the Doppler cooling mean count rate, chosen before the experiment begins and constant for all SPAM measurements, the subsequent SPAM attempt is not included and deshelving and Doppler cooling are repeated until the threshold is met. Each qubit state is attempted in blocks of 200 consecutive trials, followed by the other qubit state, for a combined total of 313,792 trials. The number of photons detected after each experiment is plotted in Fig. 4, and a threshold at n th ≤ 12 photons maximally discriminates between 0 j i and 1 j i . The fraction of events in which an attempt to prepare the 0 j i state was measured to be 1 j i is ϵ 0 j i ¼ 1:9ð4Þ 10 À4 , while the fraction of experiments in which an attempt to prepare the 1 j i state was measured to be 0 j i is ϵ 1 j i ¼ 3:8ð5Þ 10 À4 . The average SPAM fidelity is F ¼ Table 1 provides an error budget with estimates of the individual sources of error that comprise the observed infidelity. In addition to the previously discussed errors, we have experimentally determined several sources of infidelity. The CP Robust 180 sequence is found to have an error of ϵ = 9(1) × 10 −5 , determined by measuring the 1 j i state SPAM infidelity as a function of the number of concatenated CP Robust 180 sequences. The state readout duration is determined by the need to statistically separate the 0 j i and 1 j i state photon distributions. Our limited numerical aperture requires detection for 4.5 ms, leading to an error due to spontaneous emission from the 2 D 5/2 state of 1 À expð 4:5 10 À3 30 Þ % 1:5 10 À4 . This could be reduced with maximum likelihood methods 36,37 or higher efficiency light collection 33 . Finally, the readout of the 2 S 1/2 manifold is limited by background gas collisions, characterized by the preparation and readout fidelities of the 2 S 1/2 and 2 D 5/2 manifolds in 138 Ba + , for which we achieve F ¼ 0:99997ð1Þ. j i (dark) state returns an average of 1 collected photon. Using standard spin-1/2 techniques for 0 j i state preparation 28,29 , a five π-pulse composite pulse sequence 32 to prepare the 1 j i state, and electron shelving for high-fidelity readout, we measure an average SPAM error of ϵ = 2.9(3) × 10 −4 . Fig. 3 2 P 3/2 and 2 D 5/2 hyperfine splitting measurement in 133 Ba+. a Measurement of the 2 P 3/2 hyperfine splitting by tuning the 455nm laser. The left (right) peak corresponds to F=2 (F=1) in 2 P 3/2 . b Measurement of the 2 D 5/2 hyperfine splitting by tuning the 614nm laser. Triangles are data for the j 2 P 3=2 ; F ¼ 2i $ j 2 D 5=2 ; F ¼ 3i transition, circles are for the j 2 P 3=2 ; F ¼ 2i $ j 2 D 5=2 ; F ¼ 2i transition. Solid lines are Lorentzian fits. Statistical error bars on individual data points are smaller than markers.
It should be possible to further improve the fidelity to F > 0:9999. Errors due to 0 j i → 1 j i state transfer and spontaneous emission during readout could be reduced with higher fidelity population transfer and improved light collection efficiency 7,33 . The shelving fidelity could be improved using a pulsed shelving scheme 37 , or by addition of a 1762 nm transfer step before optical pumping (Fig. 1) in two ways. First, optical-frequency qubit manipulations have been demonstrated (in other species) with a π-pulse fidelity of F ¼ 0:99995 4 , suggesting that high-fidelity, unitary transfer to 2 D 5/2 may be possible. Second, even without the narrow-band laser used for coherent transfer on the electric quadrupole transition, a broad-band 1762 nm laser could be used to saturate the transition to achieve 50% population transfer. Performing these operations to each of the ten available Zeeman sublevels will transfer the majority of the population to the 2 D 5/2 state. If either method via 1762 nm is followed with the optically pumped shelving scheme, we expect a shelving infidelity below 10 −6 .

DISCUSSION
In summary, we report measurements in 133 Ba + of the 2 P 3/2 and 2 D 5/2 hyperfine splittings and 2 P 3/2 ↔ 2 S 1/2 and 2 P 3/2 ↔ 2 D 5/2 transition frequencies, which are required for high-fidelity state readout and optical qubit manipulations. Using these measurements, we have demonstrated operation of the 133 Ba + hyperfine qubit, including use of the CP Robust 180 composite pulse sequence, to realize an average single-shot SPAM error of ϵ s = 2.9 (3) × 10 −4 via threshold discrimination. This represents a ≈2× reduction of SPAM error for any qubit 7 , and is sufficient for singleshot, projective readout of a register of ≈2000 individually resolved qubits.

METHODS
We trap and laser cool 133 Ba + ions as described in ref. 26 (Fig. 1). An applied magnetic field (B ≈ 5G) applied along a radial direction of the ion trap, with laser beams linearly polarized ≈45°from the magnetic field direction, are used to destabilize dark states that result from coherent population trapping (CPT) 38 .
State preparation of the 0 j i state is accomplished by removing ν sb 493 and adding ν op 493 for 100 μs after Doppler cooling. The 1 j i state is prepared via the CP Robust 180 sequence with approximately 3 W of microwave power directed with a microwave horn for ≈15 μs.
Electron shelving is accomplished by simultaneously applying three lasers near 455, 585, and 650 nm for 300 μs. The ECDL laser near 455 nm tuned resonant with the 2 P 3=2 ; F ¼ 2 i $ 2 S 1=2 ; F ¼ 1 i transition (ν 455 ) is linearly polarized parallel to the magnetic field (π-light) with saturation parameter s ≈ 1 × 10 −3 . The ECDL near 1171 nm frequency doubled using a periodically poled lithium niobate (PPLN) waveguide is tuned resonant The laser is linearly polarized ≈45°from the magnetic field direction with saturation parameter s ≈ 1 × 10 −2 . The ECDL near 650 nm is tuned to the same parameters as Doppler cooling except for the reduction of saturation parameter to s ≈ 1. Deshelving of the 2 D 5/2 manifold back to the cooling cycle is accomplished with an ECDL near 1228 nm frequency doubled with a PPLN waveguide and linearly polarized ≈45°from the magnetic field direction. The frequency is red-detuned approximately 40 MHz from the 2 P 3=2 ; F ¼ 2 i $ 2 D 5=2 ; F ¼ 2 i transition and applied for 500 μs with saturation parameter s ≈ 1.
State detection is accomplished by collecting only 493 nm photons for 4.5 ms using a 0.28 NA commercial objective and photomultiplier tube with approximately 15% quantum efficiency. The 493 and 650 nm lasers have the same parameters as Doppler cooling during detection. The collection efficiency, laser parameters, 2 P 1/2 branching ratio of approximately 3:1, and CPT of the lambda cooling system result in a 493 nm photon count rate of approximately 10 kHz. Background counts of approximately 150 Hz are dominated by 493 nm laser scatter from the 493 nm Doppler cooling beam.
To measure the 2 P 3/2 hyperfine splitting and 2 P 3/2 ↔ 2 S 1/2 isotope shift, Doppler cooling followed by optical pumping with ν c 650 and ν sb 650 prepares the S 1=2 ; F ¼ 1 i manifold. A laser near 455 nm (ν 455 ) is applied for 50 μs with saturation parameter s ≈ 1 × 10 −3 . Doppler cooling then determines population in the 2 D 5/2 states via fluorescence detection, followed by deshelving to reset the ion to the cooling cycle. The sequence is repeated 200 times per frequency, and the frequency scanned over the 2 P 3/2 hyperfine splitting. All lasers are linearly polarized ≈45°from the magnetic field direction.
To measure the 2 D 5/2 hyperfine splitting and 2 P 3/2 ↔ 2 D 5/2 isotope shift, the 2 D 5/2 F = 3 or F = 2 manifold is prepared by Doppler cooling and applying ν 455 . A laser near 614 nm (ν 614 ) is applied for 100 μs with saturation parameter s ≈ 1. Doppler cooling then determines population in the 2 D 5/2 states via fluorescence detection, followed by deshelving to reset the ion to the cooling cycle. The sequence is repeated 200 times per frequency, and the frequency scanned between the P 3=2 ; F ¼ 2 i and 2 D 5/2 hyperfine splitting. All lasers are linearly polarized ≈45°from the magnetic field direction.
All lasers are stabilized via a software lock to a High Finesse WSU-2 wavemeter 39 . Reported hyperfine measurements (see Supplementary  Information) include a systematic uncertainty of 20 MHz due to unresolved Zeeman structure. For isotope shifts, the relevant 133 Ba + centroid frequency is determined from the hyperfine splitting measurements and then compared to measurements of the corresponding 138 Ba + transition. All 133 Ba + and 138 Ba + measurements use the same experimental hardware and wavemeter. Reported isotope shifts include a 28 MHz systematic uncertainty due to wavemeter drift and unresolved Zeeman structure.

DATA AVAILABILITY
Data are available upon request.
Received: 9 October 2019; Accepted: 2 March 2020; Errors are estimates based on theoretical models and auxiliary experiments. The 0 j i state SPAM is limited by off-resonant scatter from the laser used for electron shelving. The 1 j i state electron shelving is limited by the 2 P 3/2 hyperfine splitting, where off-resonant scatter can cause spontaneous emission to the 0 j i state. Spontaneous emission of the 2 D 5/2 state and preparation of the 1 j i state via microwaves are the next largest contribution to the 1 j i state SPAM error.