High fidelity manipulation of a qubit built from a synthetic nucleus

The recently demonstrated trapping and laser cooling of $^{133}$Ba$^+$ 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 $^2$P$_{3/2}$ and $^2$D$_{5/2}$ hyperfine splittings, as well as the $^2$P$_{3/2}$$\leftrightarrow$ $^2$S$_{1/2}$and $^2$P$_{3/2}$ $\leftrightarrow$ $^2$D$_{5/2}$ transition frequencies. Using these transitions, we demonstrate high fidelity $^{133}$Ba$^+$ 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 $\mathcal{F} = 0.99971(6)$, a factor of $\approx$ 2 improvement over the best reported performance of any qubit.

For these NISQ devices, state preparation and measurement (SPAM) infidelity ( s ) causes a reduction in computational fidelity that is exponential in qubit number, F s = (1 − s ) Nq . The requirement to perform faithful SPAM therefore limits the number of qubits to N q < ln(2)/ 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 therefore important to develop new means to improve 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 m F = 0 hyperfine and optical "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 D 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. Using a simple threshold discrimination and modest fluorescence collection optics (0.28 NA), we measure an average single-shot SPAM fidelity of F = 0.99971(6), 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.
We trap and laser cool 133 Ba + ions as described in [26]. Briefly, 133 Ba + ions are loaded into a linear Paul trap (ω ≈ 2π × 100 kHz) by laser ablating an enriched BaCl 2 salt deposited on a platinum substrate. Laser cooling is accomplished using external cavity diode lasers (ECDLs) near 493 nm and 650 nm tuned to frequencies ν c 493 and ν c 650 , with fiber electro-optic modulators (EOMs) used to add repumping sidebands ν sb 493 , ν op 493 , and ν sb 650 (Fig. 1). The qubit is defined on the pair of m F = 0 "clock" states in the ground state 2 S 1/2 manifold as |0 ≡ |F=0 and |1 ≡ |F = 1; m F = 0 . This hyperfine qubit is initialized to the |0 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 cos(φ)x + sin(φ)ŷ 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. 2 found in 200 trials, measured with a technique described later, is plotted versus the application duration of microwave radiation at Rabi frequency Ω R ≈ 2π × 57 kHz. The |1 state can be prepared after initialization into |0 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 R(π, π 6 )R(π, 0)R(π, π 2 )R(π, 0)R(π, π 6 ). As shown in Figs. 2(b-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, state readout is accomplished via hyperfine-selective optical cycling (ν c 493 and ν c 650 in Fig. 1) and collecting any resulting fluorescence. Projection into the |0 or |1 state is then determined by e.g. a threshold discrimination on the number of collected photons, as an atom in the |1 state scatters many photons, while an atom in the |0 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 state was determined to be |1 , |0 = 3.03(4) × 10 −2 , and the fraction of experiments in which an ion prepared in the |1 state was determined to be |0 , |1 = 8.65(9) × 10 −2 . The average SPAM fidelity is defined as F = 1− = 1− 1 2 ( |0 + |1 ) = 0.941 (1). The fidelity of this technique is limited by offresonant excitation to the | 2 P 1/2 , F = 1 manifold during the readout phase, which can decay to either |0 or |1 , thereby causing misidentification of the original qubit state [28].
For high fidelity SPAM, 133 Ba + offers another, unique path to state detection. The |1 qubit state can be shelved [33] 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 or |1 state is then determined by e.g. a threshold discrimination on the number of collected photons, as an atom in the |0 state scatters many photons, while an atom in the 2 D 5/2 state, indicating |1 , does not. Offresonant scatter is negligible in this case as the Doppler cooling lasers are detuned by many THz from any 2 D 5/2 state transitions.
Shelving of the |1 qubit state via 2 D 5/2 ↔ 2 S 1/2 transition requires application of a laser near 1762 nm (ν 1762 ). Similarly, shelving of the |1 state via optical pumping requires application of the frequencies ν 455 , ν 585 , and ν c 650 (and ν 614 for deshelving). However, 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 455 138,133 and δν 614 138,133 ) and hyperfine splittings ∆ 3 and ∆ 5 (Fig.  1). To measure ∆ 3 and δν 455 138,133 , the atom is prepared in the | 2 S 1/2 ;F = 1 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 and its frequency scanned. 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 0.74, 0.23, and 0.03 to the 2 S 1/2 , 2 D 5/2 , and 2 D 3/2 respectively [34] 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 via 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 is repeated 200 times per laser frequency, and the average population is shown Fig. 3(a) as a function of frequency. From these data, we find ∆ 3 = 623(30) MHz, and δν 455 138,133 = +358(30) MHz relative to 138 Ba + .
For SPAM of the |0 state, initialization with optical pumping proceeds as described above. After preparation, the |1 state is shelved as previously described, and the state is read out via Doppler cooling. During |1 state shelving, off-resonant excitation to the | 2 P 3/2 ; F=1 followed by spontaneous emission can shelve the ion to the 2 D 5/2 . This results in an expected SPAM fidelity of F = 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 optically-pumped shelving scheme. 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 is chosen based on the average number of counts from the bright state to discriminate between |0 and |1 . The fraction of events in which an attempt to prepare the |0 state was measured to be |1 is |0 = 1.9(4) × 10 −4 , while the fraction of experiments in which an attempt to prepare the |1 state was measured to be |0 is |1 = 3.8(5) × 10 −4 . The average SPAM fidelity is F = 1 − 1 2 ( |0 + |1 ) = 0.99971 (6). Table I 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 = 1 × 10 −4 , determined by measuring the |1 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 and |1 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 -exp( 4.5×10 −3

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) ≈ 1.5×10 −4 . This could be corrected with maximum likelihood methods [35,36] or higher efficiency light collection [37]. Finally, the readout of the 2 S 1/2 manifold is limited by background gas collisions, which we characterize 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 = 0.99997 (2).
It should be possible to further improve this fidelity to F > 0.9999 by use of a laser near 1762 nm (Fig. 1) in two ways. First, optical-frequency qubit manipulations have been demonstrated (in other species) with a π-pulse fidelity of F = 0.99995 [4]. Second, even without the narrow-band laser used for optical qubit manipulations, a 1762 nm laser could be used to saturate the transition and transfer 0.875 of the population into the 2 D 5/2 state. If this is followed with the optically-pumped shelving scheme, we expect an infidelity in state preparation of = 4 × 10 −5 .
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 s = 2.9(6)×10 −4 via threshold discrimination. This represents a ≈ 2× reduction of SPAM error for any qubit [7], and is sufficient for single-shot, projective readout of a register of ≈ 2000 individually resolved qubits.  4. We prepare and read out one of the two qubit states in blocks of 200 consecutive trials, alternating between qubit states for a total of 156,581 |0 state trials and 157,211 |1 state trials. Detection of the |0 (bright) states returns an average of 39 collected photons, while detection of the |1 (dark) state returns an average of 1 collected photon. Using standard spin-1/2 techniques for |0 state preparation [28,29], a five π-pulse composite pulse sequence [32] to prepare the |1 state, and electron shelving for high fidelity readout, we measure an average SPAM error of = 2.9(6) x 10 −4 .

Process
Average Office under award W911NF-18-1-0097. We thank Anthony Ransford, Christian Schneider and Conrad Roman for helpful discussions. We thank Peter Yu for technical assistance.