Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 58 / 4 / b

For a unitary operator $W$, $W{W}^{†}=I$. Thus for every nonnegative integer $j$,Insert this into the convergent matrix exponential series in the finite-dimensional qubit setting:Consequently unitary conjugation commutes with the exponential:For an unbounded self-adjoint Hamiltonian, the same identity follows from the spectral theorem for normal operators, with the domain transported by $W$; no unbounded power-series manipulation is needed.

Use one target ancilla qubit initially in $|0⟩$. Apply a CNOT gate from each data qubit $x_j$ to that same target, for $j=1,\dots ,n$. Each CNOT gate adds its control bit modulo two without changing the control. After all $n$ gates the target contains the parity bit $f(x)=x_1\oplus \cdots \oplus x_n$. Thus the required circuit is the $n$-gate parity fan-in:This is parity computation by CNOT gates. The circuit also satisfies $W|x⟩|y⟩=|x⟩|y\oplus f(x)⟩$ for either target value, and $W^{-1}=W$ because all these shared-target CNOT gates commute and individually square to the identity. The action on general superpositions follows by linearity.

The product of the data Pauli Z gates has computational-basis eigenvalueUse the parity circuit $W$ from part (ii), apply the target unitary gate $R=e^{itZ}=\mathrm{diag}(e^{it},e^{-it})$, and then uncompute the parity with $W^{†}$. For every basis input,This equals $e^{it{Z}^{\otimes n}}$ on the data, with the ancilla qubit returned to zero. The compute-phase-uncompute construction therefore gives an exact linear-size circuit:This is a Pauli-string phase by parity computation, an instance of simulation of a computable diagonal Hamiltonian. Part (i) also explains it by $W^{†}(I\otimes Z)W=Z^{\otimes n}\otimes Z$, whose exponential restricts correctly to the target-$|0⟩$ subspace. If no extra line is desired, accumulate parity into the last data qubit, apply $e^{itZ}$ there, and undo the $n-1$ CNOT gates, for $2n-1$ gates. Both constructions implement the global phase as well as the relative phases exactly.