We consider the task of performing quantum state tomography on a $d$-state spin qudit, using only measurements of spin projection onto different quantization axes. By an exact mapping onto the classical problem of signal recovery on the sphere, we prove that full reconstruction of arbitrary qudit states requires a minimal number of measurement axes, $r_d^{mathrm{min}}$, that is bounded by $2d-1le r_d^{mathrm{min}}le d^2$. We conjecture that $r_d^{mathrm{min}}=2d-1$, which we verify numerically for all $dle200$. 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 $rapprox3d$.
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