PauliAlgebra is a module to deal with exact calculations of Pauli matrices
The interface to the module is the class PauliVector that can be used to instantiate any 2x2 Hermitian matrix
from PauliAlgebra import PauliVector
M = PauliVector([
1, # Identity
2, # sigma x
0, # sigma y
1j # sigma z
])
PauliVectoris fully compatible withsympyexpressions!
Alternatively, the module exposes the standard matrices
-
Id:2x2Identity -
sigma_x -
sigma_y -
sigma_z -
sigma_plussigma_plus = (sigma_x + 1j*sigma_y)/2 # ((0,1) # (0,0))
-
sigma_minus=(sigma_x - 1j*sigma_y)/2sigma_minus = (sigma_x - 1j*sigma_y)/2 # ((0,0) # (1,0))
-
P_upP_up = (Id + sigma_z)/2 # ((1,0) # (0,0))
-
P_downP_down = (Id - sigma_z)/2 # ((0,0) # (0,1))
PauliVector supports the following arithmetic operations:
- Addition and subtraction with another
PauliVector - Multiplication with a scalar or another
PauliVector(performs matrix multiplication) - Division by a scalar
So the above example could have been written as
from PauliAlgebra import (
Id,
sigma_x,
sigma_z
)
M = Id + 2*sigma_x + 1j*sigma_zThis module allows for fast adn exact computation of commutators and anticommutators of two PauliVector using the relationship
and the (anti)commutativity of dot and cross product.
A = sigma_x
B = sigma_y
commAB = PauliVector.commutator(A,B) # = 2j*sigma_z
anticommAB = PauliVector.anticommutator(A,B) # = 0PauliVector supports exponentiation with the standard formula
M = -1j*np.pi*sigma_x
expM = M.exponentiate() # == IdPauliVector is fully compatible with sympy expressions!
theta = sp.symbols(r'\theta')
M = (Id* sp.sin(theta) + sigma_z*sp.cos(theta))/sp.sqrt(2)
M.to_sp().applyfunc(sp.trigsimp)
# [sqrt(2)*sin(\theta + pi/4), 0],
# [ 0, -sqrt(2)*cos(\theta + pi/4)]]For complex expressions
M.simplify()will simplify theId,x,y, andzcomponents
Once you are done performing algebra on a PauliVector you can turn it back into more common types
M.toMatrix()->np.ndarray(shape==(2,2))M.to_sp()->sp.Matrix(shape==(2,2))
PauliAlgebra is available on pypi!
It can be yours by simply
$ pip install PauliAlgebrain the environment of your choice!