= Solution
A <universal quantum gate set> can implement any <unitary operator> on any finite number of <qubits>, exactly when arbitrary continuous gate parameters are allowed, or to arbitrary accuracy with a suitable finite gate alphabet. More precisely, for every target $V$ and $\epsilon>0$ there is a finite <quantum circuit> $W$ with $\min_\gamma\|W-e^{i\gamma}V\|<\epsilon$ in <operator norm>. This ignores only an overall <global phase> of the complete operation. Universality lets a fixed collection of controllable elementary operations run arbitrary <quantum algorithms> rather than requiring a different physical interaction for each algorithm.
All single-<qubit> <unitary gates>, together with the <controlled-NOT gate> between selectable pairs, form an exact <universal quantum gate set>. One way to see sufficiency is to eliminate entries of a finite <unitary matrix> by two-level rotations between <computational basis> vectors; these rotations decompose into controlled single-<qubit> rotations, which in turn decompose into single-<qubit> gates and <controlled-NOT gates>. Single-<qubit> gates alone preserve <product states> and cannot supply general <quantum entanglement>. A finite approximate universal example is the <Hadamard gate>, the <T gate> $T=\operatorname{diag}(1,e^{i\pi/4})$, and the <controlled-NOT gate>; finite exact words over a finite alphabet cannot represent every continuously parameterized <unitary operator> exactly.
Distinguish the <Hamiltonian operator> $H_{\rm int}$ from the <Hadamard gate> $H_{\rm Had}$. For the printed <collisional phase gate>,
$$
H_{\rm int}=\Omega|01\rangle\langle01|.
$$
As written, $\Omega$ has units of <energy>. The <Schrödinger equation> gives
$$
\begin{array}{c|c}
\text{initial computational basis state}&\text{state after time }t\\\hline
|00\rangle&|00\rangle\\
|01\rangle&e^{-i\Omega t/\hbar}|01\rangle\\
|10\rangle&|10\rangle\\
|11\rangle&|11\rangle
\end{array}
$$
because only $|01\rangle$ has a nonzero <energy eigenvalue>. In the ordered <computational basis> $(00,01,10,11)$, define
$$
\boxed{U_\phi=e^{-iH_{\rm int}\tau/\hbar}
=I+(e^{-i\phi}-1)|01\rangle\langle01|
=\operatorname{diag}(1,e^{-i\phi},1,1),\qquad \phi=\frac{\Omega\tau}{\hbar}.}
$$
If an interaction strength is instead quoted as an <angular frequency> $\Omega_{\rm f}$, the <Hamiltonian operator> is $\hbar\Omega_{\rm f}|01\rangle\langle01|$ and $\phi=\Omega_{\rm f}\tau$. An alternative definition with $e^{+i\phi}$ would have $\phi=-\Omega\tau/\hbar$. The minus sign above follows the usual <time-evolution operator> convention. At $\phi=\pi$ these conventions give the same $U_\pi$.
For the <quantum circuit>, use $X$ on the first <qubit> to move the selective minus sign from $01$ to $11$:
$$
(X\otimes I)U_\pi(X\otimes I)=\operatorname{diag}(1,1,1,-1)=\mathrm{CZ}.
$$
Conjugating the target of this <Controlled-Z gate> by the <Hadamard gate> gives the required <controlled-NOT gate>:
$$
\boxed{\mathrm{CNOT}_{1\to2}
=(I\otimes H_{\rm Had})(X\otimes I)U_\pi(X\otimes I)(I\otimes H_{\rm Had})
=(X\otimes H_{\rm Had})U_\pi(X\otimes H_{\rm Had}).}
$$
The rightmost <matrix> acts first. Thus the network applies $X_1,H_2$, then $U_\pi$, then $X_1,H_2$, and maps $|a,b\rangle$ to $|a,b\oplus a\rangle$, where $\oplus$ is <exclusive or>.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2005/7-and-8-cnot-network.png]
{title=Controlled-NOT from the printed collisional phase gate}
{description=Time runs from left to right. Pauli X gates act on the first wire and Hadamard gates on the second, before and after the selective phase gate. The phase gate changes the sign of 01 only.}
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