  We extend recent work on a leakage-protected, adiabatic entangling gate for
exchange-only spin qubits [Doherty and Wardrop, PRL 111, 050503 (2013)] by
adapting to a setting where single spins are not assumed to be polarized on
preparation. Previous gate constructions do not function correctly when "gauge
spins" are uninitialized, because the entangling gate has different,
non-trivial action in different gauge subspaces. Our construction inherits many
of the desirable features of the previous work while addressing the
gauge-dependent behavior. Using numerical simulation, we show that the
resulting gate implements the same logical operation in both gauge subspaces to
first order in perturbation theory, and second-order terms introduce an error
that decreases quadratically in the duration of the gate. We add $1/f$ charge
noise to voltages modulating exchange in this model, which introduces errors
that increase with gate time, to show that there is an optimal gate duration
for a given set of device parameters.
