unitary matrix universal gate set universal quantum computer Solovay-Kitaev theorem qubit matrix ion qubits bit computer quantum logic gates bits
Any quantum computation (which is, in the above formalism, any unitary matrix of size 




2

n


×

2

n




{\displaystyle 2^{n}\times 2^{n}}

 over 



n


{\displaystyle n}

 qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.
