= Solution
{c}
Yes. Replace the target <qubit> by a register of $k$ <qubits> and implement the <controlled unitary gate>
$$
C(U)=|0\rangle\langle0|\otimes I+|1\rangle\langle1|\otimes U.
$$
The algebra in <5/c> uses only normalization and <unitarity>, so for an arbitrary pure register <quantum state> $|\phi\rangle$ it still gives $\langle Z\rangle=\operatorname{Re}\langle\phi|U|\phi\rangle$. For a mixed register <quantum state> $\rho$, linearity gives $\operatorname{Re}\operatorname{Tr}(\rho U)$. With $\rho=I/2^k$,
$$
\boxed{\operatorname{Re}\operatorname{Tr}U=2^k(P_0-P_1).}
$$
Only the control <qubit> needs to be pure. A uniformly randomized <computational basis> preparation supplies the target <maximally mixed state>. This is the principle of <trace estimation by a maximally mixed Hadamard test>. The extension is algebraic; efficiently implementing controlled-$U$ still requires a known suitable gate construction. Moreover, estimating the unnormalized <trace> to fixed additive precision requires increasingly precise estimation of the normalized signal as $k$ grows. There is no automatic efficient reconstruction of all entries of $U$.
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