= Solution
A <universal quantum gate set> can approximate any unitary operation on any finite register, to arbitrary requested accuracy, using a finite sequence of its gates on the register. Unobservable overall <global phase> is ignored. Approximation matters: a fixed finite alphabet has only countably many finite circuits, whereas there are uncountably many exact <unitary matrices>.
The <Hadamard gate> $H$ and <T gate> generate a dense set of single-qubit operations, but cannot themselves entangle different <qubits>. Adding the <controlled-NOT gate> gives a universal multi-qubit set. More generally, an available two-qubit gate must be genuinely entangling: it must take at least one product input to an entangled output. A product of local unitaries, or a <SWAP gate> composed with local unitaries, cannot provide this resource. Together with arbitrary single-qubit control, an entangling two-qubit gate on a connected interaction network supplies universal register control.
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