Clifford Gate Generator:

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. https://en.wikipedia.org/wiki/Clifford_group. Requested librairie to run the function:

• Numpy
• Qiskit

1 Qubit Clifford Gates

The 1 qubit set of Clifford gates is composed of the following gates:

Identity:

$$ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$

Gate $X$:

$$ X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$

Gate $Y$:

$$ Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$

Gate $Z$:

$$ Z = \begin{pmatrix} 1 & 0 \\0 & -1 \end{pmatrix} $$

Hadamard Gate:

$$ H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$

Gate $S$:

$$ S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} $$

Gate $V$:

$$ V = H S H S = \frac{1}{2} \begin{pmatrix} 1 + i & 1 + i \\ 1 - i & -1 + i \end{pmatrix} $$

Gate $W$:

$$ W = V V $$

Combinations of the gates:

$$ { V, \\ V X, \\ V Y, \\ V Z, \\ W X, \\ W Y, \\ W Z, \\ H X, \\ H Y, \\ H Z, \\ H, \\ H V, \\ H V X, \\ H V Y, \\ H V Z, \\ H W, \\ H W X, \\ H W Y, \\ H W Z, \\ W } $$