 Solution

Use the computational basis in the stated order and let the rightmost operation act first. The controlled phase gate is $|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes Z$. From the rotation definition,The control-zero block remains the identity, and the control-one block becomes $X$. ThusThis proves the relation for arbitrary superpositions, rather than only for computational-basis probabilities.

For the second relation write $S=U_{\mathrm{SW}}^{1/2}$. Its central block is $\left(\begin{array}{cc}a& b\\ b& a\end{array}\right)$, with $a=(1+i)/2$, $b=(1-i)/2$. Because $a^2=i/2$, $b^2=-i/2$ and $ab=1/2$, direct multiplication givesMeanwhile $R_z^{(1)}(\pi /2)R_z^{(2)}(-\pi /2)=\mathrm{diag}(1,-i,i,1)$. Multiplying these diagonal matrices and the overall factor $i$ givesThe supplied $S$ is indeed a square root of the SWAP gate: $S^2$ exchanges $|01⟩$ and $|10⟩$ 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 $H=J(t){\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2/4$, define $\phi =\int J(t)dt/\hslash$. Triplet and singlet energies differ by $J$, soAn exchange pulse with $\phi =\pi /2$ 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.

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.