Coins

Cx

\[ C_x = \frac{1}{\sqrt{2}}\left(\begin{matrix} 1 & i \\ i & 1 \\ \end{matrix}\right) \]

Cy

\[ C_y = \frac{1}{\sqrt{2}}\left(\begin{matrix} 1 & -i \\ -i & 1 \\ \end{matrix}\right) \]

H

\[ H = \frac{1}{\sqrt{2}}\left(\begin{matrix} 1 & 1 \\ 1 & -1 \\ \end{matrix}\right) \]

I

\[ I = \left(\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix}\right) \]

S

\[ S = \left(\begin{matrix} 1 & 0 \\ 0 & i \\ \end{matrix}\right) \]

T

\[ T = \left(\begin{matrix} 1 & 0 \\ 0 & e^{i\frac{\pi}{4}} \\ \end{matrix}\right) \]

X

\[ X = \left(\begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix}\right) \]

Z

\[ Z = \left(\begin{matrix} 1 & 0 \\ 0 & -1 \\ \end{matrix}\right) \]

generalized_coin(theta, phi, lbd)

General coin

\[ U(\theta,\phi,\lambda) = \left(\begin{matrix} \cos \theta & -e^{i\lambda} \sin \theta \\ e^{i\lambda} \sin \theta & e^{i(\lambda+\phi)} \cos \theta \\ \end{matrix}\right) \]

Parameters:

Name	Type	Description	Default
theta	float	The angle \(\theta\).	required
phi	float	The angle \(\phi\).	required
lbd	float	The angle \(\lambda\).	required

Returns:

Type	Description
(complex numpy array)	\(U(\theta,\phi,\lambda)\)

Source code in qwgraph/coins.py

def generalized_coin(theta, phi, lbd):
	""" General coin

	$$
	U(\\theta,\\phi,\\lambda) = \\left(\\begin{matrix}
	\\cos \\theta & -e^{i\\lambda} \\sin \\theta \\\\
	e^{i\\lambda} \\sin \\theta & e^{i(\\lambda+\\phi)} \\cos \\theta \\\\
	\\end{matrix}\\right)
	$$

	Args:
		theta (float): The angle $\\theta$.
		phi (float): The angle $\\phi$.
		lbd (float): The angle $\\lambda$.

	Returns:
		(complex numpy array): $U(\\theta,\\phi,\\lambda)$
	"""
	return np.array([[np.cos(theta) , -np.exp(1j*lbd)*np.sin(theta)],
		[np.exp(1j*lbd)*np.sin(theta) , np.exp(1j*(lbd+phi))*np.cos(theta)]],dtype=complex)

phase_shift(phi)

Phase shift coin

\[ P(\phi) = \left(\begin{matrix} 1 & 0 \\ 0 & e^{i(\phi)}\\ \end{matrix}\right) \]

Parameters:

Name	Type	Description	Default
phi	float	The angle \(\phi\).	required

Returns:

Type	Description
(complex numpy array)	\(P(\phi)\)

Source code in qwgraph/coins.py

def phase_shift(phi):
	""" Phase shift coin

	$$
	P(\\phi) = \\left(\\begin{matrix}
	1 & 0 \\\\
	0 & e^{i(\\phi)}\\\\
	\\end{matrix}\\right)
	$$

	Args:
		phi (float): The angle $\\phi$.

	Returns:
		(complex numpy array): $P(\\phi)$
	"""
	return np.array([
		[1,0],
		[0,np.exp(1j*phi)]],dtype=complex)