= Solution
With access to the uncontrolled physical gate, use <single-qubit process tomography>. Prepare the positive <eigenstate> of each of $X,Y,Z$, apply $V$, and on independent copies measure each of $X,Y,Z$. Arbitrary single-qubit <rotation gates> let one prepare these inputs and rotate each measurement into the <computational basis>. The nine measured means are
$$
R_{ij}=\operatorname{Tr}\left[\sigma_iV\frac{I+\sigma_j}{2}V^\dagger\right]=\frac12\operatorname{Tr}(\sigma_iV\sigma_jV^\dagger),\qquad i,j\in\{x,y,z\}.
$$
They determine the real <rotation matrix> $R$ of the <Bloch vector>, and hence the action of $V$ on every <qubit> state. To see precisely what this identifies, suppose another unitary $U$ has the same action. Then $V^\dagger U$ commutes with every <Pauli matrix>. Commuting with $Z$ makes it diagonal; commuting with $X$ makes its diagonal entries equal. Therefore $U=e^{i\gamma}V$.
Consequently \b[tomography determines $V$, and its <eigenvalues>, up to a common phase]. More explicitly, write a representative as $V=e^{i\gamma}(\cos a\,I-i\sin a\,\boldsymbol n\cdot\boldsymbol\sigma)$. Its <eigenvalues> are $e^{i(\gamma-a)}$ and $e^{i(\gamma+a)}$, and its <Bloch vector> action is <rotation> through $2a$ about $\boldsymbol n$. The measured <rotation> identifies the unordered relative <eigenphase> information; for example $\operatorname{Tr}R=1+2\cos(2a)$. Once a representative with a specified overall phase is supplied, diagonalizing that representative gives the corresponding two <eigenvalues>.
The common phase itself cannot be recovered with the stated uncontrolled access. Indeed, $(e^{i\gamma}V)\rho(e^{i\gamma}V)^\dagger=V\rho V^\dagger$ for every input, including inputs entangled with ancillas. Replacing $V$ by $e^{i\gamma}V$ therefore leaves every such preparation-and-measurement experiment unchanged, whereas it multiplies both <eigenvalues> by $e^{i\gamma}$. For example, $I$ and $e^{i\gamma}I$ are indistinguishable physical gates under these operations. This proves that the request for absolute <eigenvalue> phases needs an additional phase reference or controlled-gate assumption: the <global phase of an uncontrolled quantum gate is unobservable>. The controlled powers in the earlier parts supply that extra resource, since $\operatorname{diag}(I,e^{i\gamma}V)$ changes a relative phase between the two control branches.
The practical advantages are that no unknown <eigenstate> preparation, controlled powers, entangling gates or inverse <quantum Fourier transform> are needed: a fixed set of single-qubit preparations and measurements suffices. The disadvantages are <statistical estimation> and the phase ambiguity just proved. Estimating the bounded <Pauli measurement> means to additive accuracy $\epsilon$ at fixed confidence requires order $\epsilon^{-2}$ independent repetitions, giving the usual sampling cost for this direct procedure. The earlier <quantum phase estimation> circuit uses $N$ phase <qubits> and high controlled powers to resolve scale $2^{-N}$ in one coherent experiment with a constant success bound; constructing those powers from controlled uses of $V$ costs $2^N-1$ gate calls, so its exponential-looking precision is not obtained with only $N$ uses of $V$. Repetitions can raise confidence. Finally, full process tomography grows exponentially with the number of system <qubits>, whereas the eigenstate-based phase procedure can estimate a selected <eigenphase> without reconstructing the entire operator.
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