Trace estimation by a maximally mixed Hadamard test ($\mathrm{Tr}U=N\mathrm{Tr}[(I/N)U]$)

Prepare the auxiliary register in the maximally mixed state $I/N$ and estimate both mixed-state Hadamard-test quadratures. Multiplying the normalized expectation by $N$ estimates the trace of $U$. A uniformly randomized preparation of any orthonormal basis produces $I/N$; the eigenvectors of $U$ need not be known.