Calcium-43 Trapped-Ion Qubit

Figure

Description

The calcium-43 trapped-ion qubit stores quantum information in the hyperfine ground-state manifold of ${}^{43}{\text{Ca}}^{+}$, whose nuclear spin $I=7/2$ splits the $4S_{1/2}$ level into $F=3$ and $F=4$ manifolds separated by 3.226 GHz at zero field, for a total of 16 ground-state sublevels. In the Oxford high-coherence implementation, the qubit is encoded in the field-independent hyperfine transition $|4,0⟩\leftrightarrow |3,+1⟩$ at a bias field of about 146 G, giving a qubit frequency near 3.200 GHz and strong suppression of first-order magnetic-field noise. This operating point underlies the long memory times reported by Harty et al. (2014) and the record single-qubit fidelities reported by Smith et al. (2025).

The ${}^{43}{\text{Ca}}^{+}$ ion is the workhorse qubit of the Oxford trapped-ion program because it pairs atomic-clock-grade coherence with microwave control in a microfabricated surface-electrode trap. Chip-integrated microwave resonators and waveguides drive single-qubit rotations electronically, avoiding laser-based single-qubit addressing. Lasers are still required for Doppler cooling (397 nm), repumping (866 nm), shelving/reset (393/850/854 nm), and fluorescence-based state preparation and readout, but coherent single-qubit control itself is all-electronic.

Different ${}^{43}{\text{Ca}}^{+}$ experiments use different hyperfine encodings depending on the task. The best single-qubit results use the 146 G clock transition above, while Ballance et al. (2016) used a low-field Raman-driven hyperfine basis for the 99.9(1)% two-qubit gate result. Together these experiments make ${}^{43}{\text{Ca}}^{+}$ a benchmark species for long-lived, high-fidelity trapped-ion qubits.

Hamiltonian

For the $4S_{1/2}$ ground manifold of ${}^{43}{\text{Ca}}^{+}$ ($I=7/2$, $J=1/2$), the relevant Hamiltonian is the hyperfine-plus-Zeeman Hamiltonian

$H=A_{\mathrm{hfs}}\mathbf{I}\cdot \mathbf{J}+{\mu }_B{g}_J{J}_{z}B-{\mu }_N{g}_I{I}_{z}B.$

Because $J=1/2$, there is no electric-quadrupole hyperfine term in this manifold. At zero field this produces the familiar $F=4$ and $F=3$ hyperfine manifolds with splitting $\Delta E_{\mathrm{hfs}}=h\times 3.226\mathrm{GHz}$. At finite magnetic field the eigenenergies follow the Breit-Rabi structure, and the Oxford clock qubit is chosen at the field-independent point near $B_0\approx 146\mathrm{G}$ on the transition

$|0⟩\equiv |4,0⟩,|1⟩\equiv |3,+1⟩,$

with qubit frequency ${\omega }_{01}/2\pi \approx 3.200\mathrm{GHz}$. Near that operating point,

${\omega }_{01}(B)\approx {\omega }_{01}(B_0)+\frac{1}{2}\kappa (B-B_0{)}^2,$

so the first-order magnetic sensitivity vanishes. Separate low-field gate experiments on the same species use other hyperfine basis states; for example, Ballance et al. (2016) operated a Raman-driven two-qubit gate in a low-field regime rather than at the 146 G clock point.

Motivation

The ${}^{43}{\text{Ca}}^{+}$ system demonstrates that microwave-driven trapped-ion qubit gates can reach error rates deep into the fault-tolerant regime while preserving the long coherence times expected from hyperfine clock transitions. Microwave control offers fundamental scaling advantages: fields can be generated on-chip via lithographically defined resonators, avoiding the alignment, pointing stability, and individual beam-delivery burden of laser-based single-qubit addressing. The record $10^{-7}$-level single-qubit errors place 1Q operations effectively in the “error-free” regime for practical architectures, shifting the real bottleneck to entangling gates, optical infrastructure, and system integration.

Experimental Status

High-fidelity qubit operations — Harty et al. (2014):

• First comprehensive characterization of ${}^{43}{\text{Ca}}^{+}$ as a microwave-driven hyperfine qubit.
• Qubit encoded in the field-independent $|4,0⟩\leftrightarrow |3,+1⟩$ transition near $B_0\approx 146\text{G}$.
• Single-qubit gate fidelity: 99.9999% (error $10^{-6}$).
• State preparation and readout fidelity: 99.93%.
• Coherence time: $T_2^{\ast }=50\text{s}$ in room-temperature surface-electrode trap without magnetic shielding.
• Demonstrated all coherent single-qubit operations using chip-integrated microwave resonators.

Two-qubit gate record — Ballance et al. (2016):

• Two-qubit gate fidelity: 99.9(1)% using a Raman-driven geometric phase (${\sigma }_z\otimes {\sigma }_z$) gate.
• Demonstrated in ${}^{43}{\text{Ca}}^{+}$ hyperfine qubits at Oxford in a low-field operating regime ($B=0.196\text{mT}$), distinct from the 146 G clock-state operating point used for the 1Q record.

Single-qubit gate record — Smith et al. (2025):

• Clifford gate error: $1.5(4)\times 10^{-7}$ (fidelity 99.999985%).
• All-time record across all physical qubit platforms.
• Fastest gates (4.4 μs) achieved error of $2.9(5)\times 10^{-7}$.
• Calibration errors suppressed below $10^{-8}$.
• Dominant errors: qubit decoherence ($T_2\approx 70\text{s}$), leakage ($<10^{-9}$ per gate), and measurement.
• Performed on a microfabricated surface-electrode Paul trap at room temperature without magnetic shielding.

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_1$	>10,000 s	Hyperfine ground states; practically limited by ion lifetime, not radiative decay	Harty et al. 2014
$T_2^{\ast }$	~50–70 s	146 G field-independent clock transition $	4,0\rangle \leftrightarrow
Zero-field hyperfine splitting	3.226 GHz	Separation between the $F=4$ and $F=3$ ground manifolds	Harty et al. 2014
Operating qubit frequency	3.200 GHz	Oxford clock-state operating point near $B_0\approx 146\text{G}$	Harty et al. 2014
1Q gate fidelity	99.999985%	Microwave randomized benchmarking, error $1.5\times 10^{-7}$; all-time record across physical qubits	Smith et al. 2025
2Q gate fidelity	99.9%	Raman-driven geometric phase gate in a low-field hyperfine basis, not the 146 G clock-state basis	Ballance et al. 2016
SPAM fidelity	99.93%	Single-shot fluorescence readout with shelving-based state discrimination	Harty et al. 2014
Gate time (1Q)	4.4–35 μs	Speed-fidelity tradeoff in microwave control	Smith et al. 2025
Nuclear spin	$I=7/2$	16 ground-state sublevels in the $4S_{1/2}$ manifold	Harty et al. 2014
Operating temperature	Room temp (trap)	Ion motion cooled to mK scale; no cryogenic hardware required	Harty et al. 2014; Smith et al. 2025

Scaling Considerations

• Microwave scalability: chip-integrated microwave resonators enable on-chip qubit control without individual laser addressing, a major advantage for scaling to large arrays.
• Isotope abundance: ${}^{43}\text{Ca}$ has only 0.14% natural abundance, requiring enriched sources and isotope-selective photoionization for efficient loading.
• Large ground-state manifold: the 16 ground-state sublevels from $I=7/2$ increase the risk of leakage to non-computational states, though Smith et al. measured leakage at $<10^{-9}$ per gate.
• Two-qubit gates: the current 99.9% two-qubit fidelity lags behind the single-qubit record by orders of magnitude; closing this gap is the main challenge.
• Operating-point mismatch: the best 1Q and 2Q results were obtained in different hyperfine bases and magnetic-field regimes, so integrating record-level single- and two-qubit performance in one architecture remains an open systems challenge.
• Optical transitions: cooling and readout still require UV/visible lasers (397 nm, 866 nm, plus shelving/reset wavelengths), limiting the “all-electronic” advantage to qubit manipulation only.

References

High-fidelity operations

• T. P. Harty et al., “High-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit,” Phys. Rev. Lett. 113, 220501 (2014) | arXiv:1403.1524

Two-qubit gates

• C. J. Ballance et al., “High-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine Qubits,” Phys. Rev. Lett. 117, 060504 (2016) | arXiv:1512.04600

Single-qubit gate record

• M. C. Smith et al., “Single-Qubit Gates with Errors at the $10^{-7}$ Level,” Phys. Rev. Lett. 134, 230601 (2025) | arXiv:2412.04421

Linked Papers

• harty-2014-high-fidelity-preparation
• ballance-2016-ion-gate-fidelity
• smith-2024-single-qubit-gate-errors

Related Entries

• trapped-ion-qubit — parent platform
• ytterbium-hyperfine-qubit — alternative hyperfine species, dominant in commercial systems
• strontium-88-ion-qubit — optical-clock trapped-ion species with a different control stack
• cirac-zoller-gate — foundational trapped-ion gate proposal