Let be the group of unitary matrices. The Pauli group is generated with the following complex matrices.
The -qubit Pauli group is then given with the -fold tensor product . We define the Clifford group as the normalizer of in the group of unitaries.
In quantum computing, we do not care about the global phase, and implicitly work with , quotienting out the phase.
Generators
Info
The -qubit Clifford group is generated by the following gates.
This can be proven by decomposing the group of into these elements
Conjugation
Pauli
The conjugation action of on can be useful to analyze and simplify quantum circuits. The nonidentity elements of anti-commute, so conjugation adds a phase.
Hadamard
The Hadamard gate send and (up to normalization). Studying the action of on these basis states, we can see that . Therefore and . Intuitively, conjugating by swaps and .
Phase
The phase gate is a half gate, as . From this we deduce and commute. Direct computation shows that and . Similar to Hadamard, swaps and gates, but adds a phase.
Controlled X
Some algebra shows that acts on the Pauli group in the following manner
Controlled Z
The controlled gate is defined by . Combining the Hadamard and controlled conjugation actions, we obtain conjugation action of .
Note that only applies phase the basis vector , thus it is symmetric on control and target. This implies that is also a , but with control and target swapped.