Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 3 / a

For normalized vector representatives in the same Hilbert space, the vector-norm convention is $\Vert |\psi ⟩-|\varphi ⟩{\Vert }_2\le \epsilon$. Since physical pure states identify global phases, a phase-insensitive version isThe equality follows by choosing the phase to make the overlap real and nonnegative. This states an explicit pure-state norm distance convention; fidelity-based or trace-distance conventions use different numerical error parameters. We use the phase-insensitive convention for the universality claim below.

Under fixed phase representatives, two unitary operators on the same space are $\epsilon$-close when $\Vert U-V{\Vert }_{\mathrm{op}}\le \epsilon$, where the operator norm isThus every input vector is approximated uniformly, not merely one selected input. Under the phase-insensitive convention, replace this bound by ${\min }_{\theta }\Vert U-e^{i\theta }V{\Vert }_{\mathrm{op}}\le \epsilon$. The minimization uses one phase for the whole operator, not a separately chosen phase inside the supremum for each input. This is operator-norm distance between unitary gates with the stated global-phase identification.

A finite universal quantum gate set can approximate every unitary on any fixed finite number of qubits to any prescribed positive accuracy, using a finite circuit of its gates on allowed wires. Explicitly, for every target $U$ and $\epsilon >0$ there is such a circuit $C$ withThe global-phase convention is the one just specified. Universality means arbitrary approximation, not exact representation of an uncountable family by finitely described circuits. Products of approximate gates can be controlled using the quantum circuit gate-error telescoping bound.