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Many systems used for quantum computing possess additional states beyondthose defining the qubit. Leakage out of the qubit subspace must be consideredwhen designing quantum error correction codes. Here we consider trapped ionqubits manipulated by Raman transitions. Zeeman qubits do not suffer fromleakage errors but are sensitive to magnetic fields to first-order. Hyperfinequbits can be encoded in clock states that are insensitive to magnetic fieldsto first-order, but spontaneous scattering during the Raman transition can leadto leakage. Here we compare a Zeeman qubit ($^{174}$Yb$^+$) to a hyperfinequbit ($^{171}$Yb$^+$) in the context of the surface code. We find that thenumber of physical qubits required to reach a specific logical qubit error canbe reduced by using $^{174}$Yb$^+$ if the magnetic field can be stabilized withfluctuations smaller than $10$ $\mu$G.

Comparing Zeeman qubits to hyperfine qubits in the context of the surface code: $^{174}$Yb$^{+}$ and $^{171}$Yb$^{+}$