 Solution

Apply exact quantum phase estimation to $U(t)$ with the supplied eigenstate $|\Psi ⟩$. Sincethe promise that the phase has an $N$-bit representation makes an $N$-qubit control register recover $\varphi$ exactly. Multiplying by $-2\pi$ modulo $2\pi$ gives $\lambda t\mathrm{mod}2\pi$. Equivalently, phase estimation may be run on $U(t{)}^{†}$, whose eigenphase is $\lambda t/(2\pi )$ modulo one.