Past exam of the computer science course of the University of Oxford / 2022 / Quantum information / 2 / a

For normalized vector representatives in the same Hilbert space, two pure states are $\epsilon$-close in pure-state norm distance whenHere $\Vert \cdot {\Vert }_2$ is the Euclidean norm of the coefficient vector in any orthonormal basis. Equivalently, $2-2\mathrm{Re}⟨\psi |\varphi ⟩\le {\epsilon }^2$. If the convention identifies vector representatives differing by a global phase, first minimize over that phase, giving ${\min }_{\gamma }\Vert |\psi ⟩-e^{i\gamma }|\varphi ⟩{\Vert }_2=\sqrt{2-2|⟨\psi |\varphi ⟩|}$; the vector-norm convention above specifies the representatives.

Two unitary transformations $U,V$ on the same Hilbert space are $\epsilon$-close in operator-norm distance between unitary gates whenThus their outputs are $\epsilon$-close in pure-state norm distance for every normalized input. This is an operator norm condition, stronger than checking only one chosen input. If global phase is disregarded for the whole operation, use ${\min }_{\gamma }\Vert U-e^{i\gamma }V{\Vert }_{\mathrm{op}}$ instead, with one phase chosen for the operator rather than separately for each input.

An entangling gate is a unitary operator on two quantum systems for which some pure product input is mapped to an entangled state:It need not create quantum entanglement from every product state. For example, a controlled-NOT gate entangles $|+⟩|0⟩$ but leaves $|00⟩$ a product state.