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

Natalie C. Brown, Kenneth R. Brown

Published:

Many systems used for quantum computing possess additional states beyond
those defining the qubit. Leakage out of the qubit subspace must be considered
when designing quantum error correction codes. Here we consider trapped ion
qubits manipulated by Raman transitions. Zeeman qubits do not suffer from
leakage errors but are sensitive to magnetic fields to first-order. Hyperfine
qubits can be encoded in clock states that are insensitive to magnetic fields
to first-order, but spontaneous scattering during the Raman transition can lead
to leakage. Here we compare a Zeeman qubit ($^{174}$Yb$^+$) to a hyperfine
qubit ($^{171}$Yb$^+$) in the context of the surface code. We find that the
number of physical qubits required to reach a specific logical qubit error can
be reduced by using $^{174}$Yb$^+$ if the magnetic field can be stabilized with
fluctuations smaller than $10$ $\mu$G.