Informally, a set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Technically, this is impossible since the number of possible quantum gates is uncountable, whereas the number of finite sequences from a finite set is countable. To solve this problem, we only require that any quantum operation can be approximated by a sequence of gates from this finite set. Moreover, for the specific case of single qubit unitaries the Solovay–Kitaev theorem guarantees that this can be done efficiently.
One simple set of two-qubit universal quantum gates is the Hadamard gate, the gate, and the controlled NOT gate.
A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate, which performs the transformation
The universal classical logic gate, the Toffoli gate, is reducible to the Deutsch gate, thus showing that all classical logic operations can be performed on a universal quantum computer.
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