 Solution

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.