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# Trapped ion qubits

**Trapped ion qubits** use individual ions, commonly ytterbium or calcium, confined by oscillating electric fields in a linear Paul trap and encode qubit states in stable internal atomic levels such as hyperfine ground states. Because the ions are held purely by electric fields and repel each other via Coulomb interaction, they form a rigid, self-assembling chain with no fabrication-induced disorder, giving this platform some of the longest coherence times and highest gate fidelities of any qubit technology.

Single-qubit gates are driven directly by laser or microwave fields resonant with the qubit transition. Two-qubit gates work differently: they use the ions' shared vibrational motion (a phonon mode of the whole chain) as a data bus. A laser conditionally displaces this shared motional mode depending on the internal state of one ion, and a second ion picks up a state-dependent phase from that displacement, entangling the two ions without them ever touching directly.

$$H = \frac{\hbar\omega_0}{2}\sigma_z + \hbar\omega_m a^\dagger a + \hbar\eta\Omega\left(a e^{-i\delta t} + a^\dagger e^{i\delta t}\right)\sigma_x$$

$\omega_0$ is the qubit's internal transition frequency, $\omega_m$ is the frequency of the shared motional mode with creation and annihilation operators $a^\dagger, a$, $\eta$ is the Lamb-Dicke parameter coupling internal and motional states, $\Omega$ is the laser Rabi frequency, and $\delta$ is the detuning from a motional sideband. Driving near a sideband ($\delta \approx \pm\omega_m$) is what lets the laser transfer information between the qubit and the shared bus mode, the basis of gates like the Mølmer-Sørensen gate.

## Practical characteristics

Trapped ions offer all-to-all connectivity within a chain, since every ion couples to the same shared motional modes, unlike platforms with only nearest-neighbor coupling. The tradeoff is gate speed: because gates depend on the comparatively slow motional dynamics of the chain, they are typically microseconds to milliseconds, much slower than superconducting qubit gates. Scaling beyond a few dozen ions in a single trap also runs into motional mode crowding, which is why larger systems use shuttling between multiple trap zones or photonic links between separate traps.