Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 324 / 3 / b

Use a clean quantum ancilla and apply the quantum circuit $C$, then the Pauli Z gate on the ancilla, then the inverse quantum circuit $C^{†}$:This compute-phase-uncompute construction implements $A\otimes I$ on every input with the ancilla initially $|0⟩$, and by linearity on any superposition of those inputs. The action on ancilla-$|1⟩$ inputs need not coincide with $A\otimes I$; the question only requires the clean-ancilla subspace.

Since $A|x⟩=(-1{)}^{f(x)}|x⟩$, the spectral decomposition givesReplace the middle gate of the previous compute-phase-uncompute construction bya one-qubit rotation about the z-axis. ThenThe circuit uses the gates of $C$, their inverses, and one fixed rotation; it returns the quantum ancilla to $|0⟩$. The same construction implements $e^{itA}$ with middle gate $e^{itZ}$, an instance of simulation of a computable diagonal Hamiltonian.