linear operators
This commit is contained in:
Daniel Tsvetkov
parent
11ed1b44ac
commit
ba3be9b4ff
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@ -113,6 +113,18 @@ class Matrix(object):
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return str(self.m)
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class HorizontalVector(Matrix):
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"""Horizontal vector is basically a list"""
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def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
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super().__init__(m, *args, **kwargs)
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if not self._is_vector():
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raise TypeError("Not a vector")
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def _is_vector(self):
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return len(self.m.shape) == 1
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class Vector(Matrix):
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def __init__(self, m: ListOrNdarray = None, *args, **kwargs):
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super().__init__(m, *args, **kwargs)
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@ -337,38 +349,48 @@ def humanize(m):
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class Operator(object):
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def __init__(self, func=None, *args, **kwargs):
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def __init__(self, func: sp.Lambda, *args, **kwargs):
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"""An Operator turns one function into another"""
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self.func = func
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def on(self, *args, **kwargs):
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return self(*args, **kwargs)
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def on(self, *args):
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return self(*args)
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def __call__(self, *args, **kwargs):
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return self.func(*args, **kwargs)
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def __call__(self, *args):
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return self.func(*args)
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class LinearOperator(Operator):
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def __init__(self, func=None, *args, **kwargs):
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def __init__(self, func: sp.Lambda, *args, **kwargs):
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"""Linear operators satisfy f(x+y) = f(x) + f(y) and a*f(x) = f(a*x)"""
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super().__init__(func, *args, **kwargs)
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if not self._is_linear():
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raise TypeError("Not a linear operator")
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def _is_linear(self):
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# TODO: How to verify if the func is linear?
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# in case of Unitary Operator, self.func is a lambda that takes a Matrix (assumes has .m component)
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# A function is (jointly) linear in a given set of variables
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# if all second-order derivatives are identically zero (including mixed ones).
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# https://stackoverflow.com/questions/36283548/check-if-an-equation-is-linear-for-a-specific-set-of-variables
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expr, vars_ = self.func.expr, self.func.variables
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for x in vars_:
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for y in vars_:
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try:
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if not sp.Eq(sp.diff(expr, x, y), 0):
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return False
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except TypeError:
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return False
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return True
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# a, b = sympy.symbols('a, b')
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# expr, vars_ = a+b, [a, b]
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# for x in vars_:
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# for y in vars_:
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# try:
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# if not sympy.Eq(sympy.diff(expr, x, y), 0):
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# return False
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# except TypeError:
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# return False
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# return True
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def test_linear_operator():
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# Operators turn one vector into another
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# the times 2 operator should return the times two multiplication
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_x = sp.Symbol('x')
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_times_2 = LinearOperator(sp.Lambda(_x, 2 * _x))
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assert _times_2.on(5) == 10
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assert _times_2(5) == 10
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assert_raises(TypeError, "Not a linear operator", LinearOperator, sp.Lambda(_x, _x ** 2))
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class SquareMatrix(Matrix):
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@ -381,6 +403,39 @@ class SquareMatrix(Matrix):
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return self.m.shape[0] == self.m.shape[1]
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class LinearTransformation(LinearOperator, Matrix):
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def __init__(self, m: ListOrNdarray, func=None, name: str = '', *args, **kwargs):
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"""LinearTransformation (or linear map) inherits from both LinearOperator and a Matrix
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It is used to act on a State vector by defining the operator to be the dot product"""
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self.name = name
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Matrix.__init__(self, m=m, *args, **kwargs)
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self.func = func or self.operator_func
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LinearOperator.__init__(self, func=func, *args, **kwargs)
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def _is_linear(self):
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# Every matrix transformation is a linear transformation
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# https://www.mathbootcamps.com/proof-every-matrix-transformation-is-a-linear-transformation/
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return True
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def operator_func(self, other):
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return Vector(np.dot(self.m, other.m))
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def get_eigens(self):
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""" Returns (eigenvalue, eigenvector)
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M|v> = λ|v> ->
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M :is the operator (self)
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|v> :is eigenstate of M
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λ :is the corresponding eigenvalue
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"""
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eigenvalues, eigenvectors = np.linalg.eig(self.m)
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rv = []
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for i in range(0, len(eigenvectors)):
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eigenvalue = eigenvalues[i]
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eigenvector = HorizontalVector(eigenvectors[:, i])
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rv.append((eigenvalue, eigenvector))
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return rv
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class UnitaryMatrix(SquareMatrix):
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def __init__(self, m: ListOrNdarray, *args, **kwargs):
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"""Represents a Unitary matrix that satisfies UU+ = I"""
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@ -407,13 +462,13 @@ class HermitianMatrix(SquareMatrix):
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return np.isclose(self.m, self._conjugate_transpose()).all()
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class UnitaryOperator(LinearOperator, UnitaryMatrix):
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class UnitaryOperator(LinearTransformation, UnitaryMatrix):
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def __init__(self, m: ListOrNdarray, name: str = '', *args, **kwargs):
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"""UnitaryOperator inherits from both LinearOperator and a Unitary matrix
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"""UnitaryOperator inherits from both LinearTransformation and a Unitary matrix
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It is used to act on a State vector by defining the operator to be the dot product"""
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self.name = name
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UnitaryMatrix.__init__(self, m=m, *args, **kwargs)
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LinearOperator.__init__(self, func=self.operator_func, *args, **kwargs)
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LinearTransformation.__init__(self, m=m, func=self.operator_func, *args, **kwargs)
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def operator_func(self, other):
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return State(np.dot(self.m, other.m))
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@ -424,9 +479,9 @@ class UnitaryOperator(LinearOperator, UnitaryMatrix):
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return str(self.m)
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class HermitianOperator(LinearOperator, HermitianMatrix):
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class HermitianOperator(LinearTransformation, HermitianMatrix):
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def __init__(self, m: ListOrNdarray, name: str = '', *args, **kwargs):
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"""HermitianMatrix inherits from both LinearOperator and a Hermitian matrix
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"""HermitianMatrix inherits from both LinearTransformation and a Hermitian matrix
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It is used to act on a State vector by defining the operator to be the dot product"""
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self.name = name
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HermitianMatrix.__init__(self, m=m, *args, **kwargs)
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@ -596,6 +651,7 @@ def Rz(theta):
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[0, np.power(np.e, 1j * theta / 2)]],
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name="Rz")
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H = UnitaryOperator([[1 / np.sqrt(2), 1 / np.sqrt(2)],
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[1 / np.sqrt(2), -1 / np.sqrt(2)], ],
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name="H")
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@ -604,7 +660,7 @@ CNOT = TwoQubitOperator([[1, 0, 0, 0],
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[0, 1, 0, 0],
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[0, 0, 0, 1],
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[0, 0, 1, 0], ],
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C_partial, x_partial, I, X)
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C_partial, x_partial, I, X)
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def assert_raises(exception, msg, callable, *args, **kwargs):
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@ -667,7 +723,11 @@ class MeasurementOpeartor(HermitianOperator):
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return cls(matrix.conjugate_transpose().x(matrix).m)
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def get_prob(self, state: State):
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"""Returns result of <ψ|M†_m M_m|ψ>
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"""
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# <ψ|
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state_ct = state.conjugate_transpose()
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# This is: <ψ| . M†_m M_m . |ψ>
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return state_ct.m.dot(self.m.dot(state.m)).item()
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@ -707,6 +767,11 @@ def testRotPauli():
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assert np.allclose(abs_squared(Rz(np.pi).m), abs_squared(Z.m))
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def test_eigenstuff():
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assert LinearTransformation(m=[[1, 0], [0, 0]]).get_eigens() == \
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[(1.0, HorizontalVector([1., 0.])), (0., HorizontalVector([0., 1.]))]
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def test():
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# Test properties of Hilbert vector space
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# The four postulates of Quantum Mechanics
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@ -734,11 +799,9 @@ def test():
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# II: Dynamics | The evolution of a closed system is described by a unitary transformation
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#
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# Operators turn one vector into another
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# the times 2 operator should return the times two multiplication
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_times_2 = Operator(lambda x: 2 * x)
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assert _times_2.on(5) == 10
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assert _times_2(5) == 10
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test_linear_operator()
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test_eigenstuff()
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# Understanding the difference between unitary and hermitians
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test_unitary_hermitian()
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