Euler¶

`graphqomb.euler` module¶

Euler angles and related functions.

This module provides:

euler_decomposition: Decompose a 2x2 unitary matrix into Euler angles.
bloch_sphere_coordinates: Get the Bloch sphere coordinates corresponding to a vector.
LocalUnitary: Class to represent a local unitary.
LocalClifford: Class to represent a local Clifford.
meas_basis_info: Return the measurement plane and angle corresponding to a vector.
update_lc_lc: Update a LocalClifford object with another LocalClifford object.
update_lc_basis: Update a LocalClifford object with a MeasBasis object.

Local Operators¶

class graphqomb.euler.LocalUnitary[source]¶

Bases: object

Class to represent signle-qubit unitaries.

\(U(\alpha, \beta, \gamma) = R_z(\gamma)R_x(\beta)R_z(\alpha)\)

alpha¶

angle for the first \(R_z\), by default 0

Type:: float

beta¶

angle for the \(R_x\), by default 0

Type:: float

gamma¶

angle for the last \(R_z\), by default 0

Type:: float

__init__(alpha=0, beta=0, gamma=0)[source]¶

print_angles()[source]¶

Print the Euler angles.

conjugate()[source]¶

Return the conjugate of the LocalUnitary object.

Returns:: conjugate LocalUnitary
Return type:: LocalUnitary

matrix()[source]¶

Return the 2x2 unitary matrix corresponding to the Euler angles.

Returns:: 2x2 unitary matrix
Return type:: numpy.typing.NDArray[numpy.complex128]

class graphqomb.euler.LocalClifford[source]¶

Bases: LocalUnitary

Class to represent a local Clifford.

\(U(\alpha, \beta, \gamma) = R_z(\gamma)R_x(\beta)R_z(\alpha)\) Each angle must be integer multiples of \(\pi/2\).

alpha¶

angle for the first \(R_z\). The angle must be a multiple of \(\pi/2\), by default 0

Type:: float

beta¶

angle for the \(R_x\). The angle must be a multiple of \(\pi/2\), by default 0

Type:: float

gamma¶

angle for the last \(R_z\). The angle must be a multiple of \(\pi/2\), by default 0

Type:: float

__init__(alpha=0, beta=0, gamma=0)[source]¶

conjugate()[source]¶

Return the conjugate of the LocalClifford object.

Returns:: conjugate LocalClifford
Return type:: LocalClifford

Functions¶

graphqomb.euler.euler_decomposition(u)[source]¶

Decompose a 2x2 unitary matrix into Euler angles.

\(U \rightarrow R_z(\gamma)R_x(\beta)R_z(\alpha)\)

Parameters:: u (numpy.typing.NDArray[numpy.complex128]) – unitary 2x2 matrix
Returns:: euler angles (\(\alpha\), \(\beta\), \(\gamma\))
Return type:: tuple[float, float, float]

graphqomb.euler.bloch_sphere_coordinates(vector)[source]¶

Get the Bloch sphere coordinates corresponding to a vector.

\(|\psi\rangle = \cos(\theta/2)|0\rangle + \exp(i\phi)\sin(\theta/2)|1\rangle\)

Parameters:: vector (numpy.typing.NDArray[numpy.complex128]) – 1 qubit state vector
Returns:: Bloch sphere coordinates (\(\theta\), \(\phi\))
Return type:: tuple[float, float]

graphqomb.euler.meas_basis_info(vector)[source]¶

Return the measurement plane and angle corresponding to a vector.

Parameters:: vector (numpy.typing.NDArray[numpy.complex128]) – 1 qubit state vector
Returns:: measurement plane and angle
Return type:: tuple[Plane, float]
Raises:: ValueError – if the vector does not lie on any of 3 planes