  We consider the task of performing quantum state tomography on a $d$-level
spin qudit, using only measurements of spin projection onto different
quantization axes. After introducing a basis of operators closely related to
the spherical harmonics, which obey the rotational symmetries of spin qudits,
we map our quantum tomography task onto the classical problem of signal
recovery on the sphere. We then provide algorithms with $O(rd^3)$ serial
runtime, parallelizable down to $O(rd^2)$, for (i) computing a priori upper
bounds on the expected error with which spin projection measurements along $r$
given axes can reconstruct an unknown qudit state, and (ii) estimating a
posteriori the statistical error in a reconstructed state. Our algorithms
motivate a simple randomized tomography protocol, for which we find that using
more measurement axes can yield substantial benefits that plateau after
$r\approx3d$.

Quantum Physics