The Clifford group is generated by three gates: Hadamard, phase gate S, and CNOT.[2][3][4] This set of gates is minimal in the sense that discarding any one gate results in the inability to implement some Clifford operations; removing the Hadamard gate disallows powers of in the unitary matrix representation, removing the phase gate S disallows in the unitary matrix, and removing the CNOT gate reduces the set of implementable operations from to . Since all Pauli matrices can be constructed from the phase and Hadamard gates, each Pauli gate is also trivially an element of the Clifford group.
The gate is equal to the product of and gates. To show that a unitary is a member of the Clifford group, it suffices to show that for all that consist only of the tensor products of and , we have .
The Clifford gates do not form a universal set of quantum gates as some gates outside the Clifford group cannot be arbitrarily approximated with a finite set of operations. An example is the phase shift gate (historically known as the gate):
.
The following shows that the gate does not map the Pauli- gate to another Pauli matrix:
However, the Clifford group, when augmented with the gate, forms a universal quantum gate set for quantum computation.[5] Moreover, exact, optimal circuit implementations of the single-qubit -angle rotations are known.[6][7]
^Forest, Simon; Gosset, David; Kliuchnikov, Vadym; McKinnon, David. "Exact Synthesis of Single-Qubit Unitaries Over Clifford-Cyclotomic Gate Sets". Journal of Mathematical Physics.
^Ross, Neil J.; Selinger, Peter (2014). "Optimal ancilla-free Clifford+ T approximation of z-rotations". arXiv:1403.2975.
^Kliuchnikov, Vadym; Maslov, Dmitri; Mosca, Michele (2013). "Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates". Quantum Information and Computation. 13 (7–8): 607–630. arXiv:1206.5236. doi:10.26421/QIC13.7-8-4. S2CID12885769.