The Clifford group encompasses a set of quantum operations that map the set of n-fold Pauli group products into itself. It is most famously studied for its use in quantum error correction.
Definition
The Pauli matrices,
: <math>\sigma_0=I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \sigma_1=X=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_2=Y=\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \text{ and } \sigma_3=Z=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}</math>
provide a basis for the density operators of a single qubit, as well as for the unitaries that can be applied to them. For the <math>n</math>-qubit case, one can construct a group, known as the Pauli group, according to
: <math>\mathbf{P}_n=\left\{ e^{i\theta\pi/2} \sigma_{j_1} \otimes \cdots \otimes \sigma_{j_n} \mid \theta = 0,1,2,3,j_k = 0,1,2,3 \right\}.</math>
The Clifford group is defined as the group of unitaries that normalize the Pauli group: <math>\mathbf{C}_n=\{V\in U_{2^n}\mid V\mathbf{P}_nV^\dagger = \mathbf{P}_n\}.</math> Under this definition, <math>\mathbf{C}_n</math> is infinite, since it contains all unitaries of the form <math>e^{i \theta}I</math> for a real number <math>\theta</math> and the identity matrix <math>I</math>. Any unitary in <math>\mathbf{C}_n</math> is equivalent (up to a global phase factor) to a circuit generated using Hadamard, Phase, and CNOT gates, so the Clifford group is sometimes defined as the (finite) group of unitaries generated using Hadamard, Phase, and CNOT gates. The n-qubit Clifford group <math>\mathbf{C}_n</math> defined in this manner contains <math>2^{n^2+2n+3}\prod_{j=1}^{n}(4^j-1)</math> elements.
Some authors choose to define the Clifford group as the quotient group <math>\mathbf{C}_n/U(1)</math>, which counts elements in <math>\mathbf{C}_n</math> that differ only by an overall global phase factor as the same element. The smallest global phase is <math>\frac{1+i}{\sqrt{2</math>, the eighth complex root of the number 1, arising from the circuit identity <math>HSHSHS=\frac{1+i}{\sqrt{2I</math>, where <math>H</math> is the Hadamard gate and <math>S</math> is the Phase gate. For <math>n=</math> 1, 2, and 3, this group contains 24, 11,520, and 92,897,280 elements, respectively. The number of elements in <math>\mathbf{C}_n/U(1)</math> is <math>2^{n^2+2n}\prod_{j=1}^{n}(4^j-1)</math>.
Another possible definition of the Clifford group can be obtained from the above by further factoring out the Pauli group <math>\{I,X,Y,Z\}</math> on each qubit. The leftover group is isomorphic to the group of <math>2n\times 2n</math> symplectic matrices over the field <math>\mathbb{F}_2</math> of two elements. Here, reference It has <math>2^{n\log(n)+O(n)}</math> elements.
- Borel group, a maximal solvable subgroup, which is generated by the product of the lower triangular invertible Boolean matrices (CNOT circuits with controls on top qubits and targets on the bottom qubits) with diagonal subgroup elements (circuits with Phase and CZ gates).