  We study multi-parameter sensing of 2D and 3D vector fields within the
Bayesian framework for $SU(2)$ quantum interferometry. We establish a method to
determine the optimal quantum sensor, which establishes the fundamental limit
on the precision of simultaneously estimating multiple parameters with an
$N$-atom sensor. Keeping current experimental platforms in mind, we present
sensors that have limited entanglement capabilities, and yet, significantly
outperform sensors that operate without entanglement and approach the optimal
quantum sensor in terms of performance. Furthermore, we show how these sensors
can be implemented on current programmable quantum sensors with variational
quantum circuits by minimizing a metrological cost function. The resulting
circuits prepare tailored entangled states and perform measurements in an
appropriate entangled basis to realize the best possible quantum sensor given
the native entangling resources available on a given sensor platform. Notable
examples include a 2D and 3D quantum ``compass'' and a 2D sensor that provides
a scalable improvement over unentangled sensors. Our results on optimal and
variational multi-parameter quantum metrology are useful for advancing
precision measurements in fundamental science and ensuring the stability of
quantum computers, which can be achieved through the incorporation of optimal
quantum sensors in a quantum feedback loop.
