= Solution
Use the <computational basis> in the stated order and let the rightmost operation act first. The <controlled phase gate> is $|0\rangle\langle0|\otimes I+|1\rangle\langle1|\otimes Z$. From the rotation definition,
$$
R_y(\theta)=\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix},\qquad
R_y(\pi/2)ZR_y(-\pi/2)=X.
$$
The control-zero block remains the identity, and the control-one block becomes $X$. Thus
$$
\boxed{R_y^{(2)}(\pi/2)U_{\rm CP}R_y^{(2)}(-\pi/2)=U_{\rm CNOT}.}
$$
This proves the relation for arbitrary <superpositions>, rather than only for computational-basis probabilities.
For the second relation write $S=U_{\rm SW}^{1/2}$. Its central block is $\bigl(\begin{smallmatrix}a&b\\b&a\end{smallmatrix}\bigr)$, with $a=(1+i)/2$, $b=(1-i)/2$. Because $a^2=i/2$, $b^2=-i/2$ and $ab=1/2$, direct multiplication gives
$$
S R_z^{(1)}(\pi) S=\operatorname{diag}(-i,1,-1,i).
$$
Meanwhile $R_z^{(1)}(\pi/2)R_z^{(2)}(-\pi/2)=\operatorname{diag}(1,-i,i,1)$. Multiplying these diagonal matrices and the overall factor $i$ gives
$$
\boxed{iR_z^{(1)}(\pi/2)R_z^{(2)}(-\pi/2)S R_z^{(1)}(\pi)S
=\operatorname{diag}(1,1,1,-1)=U_{\rm CP}.}
$$
The supplied $S$ is indeed a <square root of the SWAP gate>: $S^2$ exchanges $|01\rangle$ and $|10\rangle$ while fixing the other basis states. It acts as one on the triplet subspace and as $i$ on the singlet subspace.
For an isotropic <exchange interaction> with <Hamiltonian> $H=J(t)\,\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2/4$, define $\varphi=\int J(t)dt/\hbar$. Triplet and singlet energies differ by $J$, so
$$
U(\varphi)=e^{-i\varphi/4}(P_T+e^{i\varphi}P_S).
$$
An exchange pulse with $\varphi=\pi/2$ therefore implements $e^{-i\pi/8}S$. Two such pulses, interspersed with the specified local rotations, implement the <controlled phase gate>; the initial and final target rotations then turn it into a <CNOT gate>. Their common overall phase has no effect on an isolated gate's outcome probabilities; a subsequently controlled version must keep track of conditional phases.
\Image[/past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2012/paper-1-exchange-cnot-circuit.png]
{title=CNOT synthesis from two square-root-of-SWAP exchange pulses and local rotations, with time running left to right}
The construction converts a physically available two-spin interaction into a standard entangling gate, which together with arbitrary one-qubit rotations supplies <universal quantum computation>. Practical operation needs calibrated exchange pulse area and timing, individually controllable rotations, and reproducible phase references. Charge noise can change $J$; magnetic-field gradients add unwanted single-spin evolution, and anisotropic exchange modifies the assumed <Hamiltonian>. Pulses must suppress leakage into unwanted orbital states while finishing before excessive <decoherence>. Readout, preparation, cross-talk and the extra operations in this decomposition all affect the achieved <quantum gate fidelity>. The algebraic identity alone does not establish a <fault-tolerant quantum computation> implementation.
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