Projective measurements with hermitians

lib_q_computer_math.py

@@ -236,6 +236,14 @@ class State(Vector):
         choices = [format_str.format(i) for i in range(len(weights))]
         return random.choices(choices, weights)[0]
 
+    def measure_with_op(self, mo):
+        """
+        Measures with a measurement operator mo
+        TODO: Can't define mo: MeasurementOperator because in python you can't declare classes before defining them
+        """
+        m = mo.on(self) / np.sqrt(mo.get_prob(self))
+        return State(m)
+
     def measure_n(self, n=1):
         """measures n times"""
         measurements = defaultdict(int)
@@ -402,6 +410,24 @@ class UnitaryOperator(LinearOperator, UnitaryMatrix):
         return str(self.m)
 
 
+class HermitianOperator(LinearOperator, HermitianMatrix):
+    def __init__(self, m: ListOrNdarray, name: str = '', *args, **kwargs):
+        """HermitianMatrix inherits from both LinearOperator and a Hermitian matrix
+        It is used to act on a State vector by defining the operator to be the dot product"""
+        self.name = name
+        HermitianMatrix.__init__(self, m=m, *args, **kwargs)
+        LinearOperator.__init__(self, func=self.operator_func, *args, **kwargs)
+
+    def operator_func(self, other):
+        """This might not return a normalized state vector so don't wrap it in State"""
+        return np.dot(self.m, other.m)
+
+    def __repr__(self):
+        if self.name:
+            return '-{}-'.format(self.name)
+        return str(self.m)
+
+
 # TODO - How to add a CNOT gate to the Quantum Processor?
 #        Imagine if I have to act on a 3-qubit computer and CNOT(q1, q3)
 #
@@ -597,8 +623,8 @@ def assert_not_raises(exception, msg, callable, *args, **kwargs):
 def test_unitary_hermitian():
     # Unitary is UU+ = I; Hermitian is U = U+
     # Matrixes could be either, neither or both
-    # Quantum operators are described *only* by unitary transformations
-    # TODO: What are Hermitians?
+    # Quantum operators (gates) are described *only* by unitary transformations
+    # Hermitian operators are used for measurement operators - https://towardsdatascience.com/understanding-basics-of-measurements-in-quantum-computation-4c885879eba0
     h_not_u = [
         [1, 0],
         [0, 2],
@@ -628,6 +654,40 @@ def test_unitary_hermitian():
     assert_raises(TypeError, "Not a Unitary matrix", UnitaryMatrix, not_u_not_h)
 
 
+class MeasurementOpeartor(HermitianOperator):
+    """Measurement operators are Hermitians: <ψ|M†_m M_m|ψ>"""
+
+    @classmethod
+    def create_from_prob(cls, matrix: Matrix):
+        """returns |M†_m><M_m|"""
+        return cls(matrix.conjugate_transpose().x(matrix).m)
+
+    def get_prob(self, state: State):
+        state_ct = state.conjugate_transpose()
+        return np.asscalar(state_ct.m.dot(self.m.dot(state.m)))
+
+
+def test_measurement_ops():
+    m0 = MeasurementOpeartor.create_from_prob(Matrix([1, 0]))
+    m1 = MeasurementOpeartor.create_from_prob(Matrix([0, 1]))
+    assert m0 == Matrix([[1, 0],
+                         [0, 0]])
+    assert m1 == Matrix([[0, 0],
+                         [0, 1]])
+
+    # p(0) -> probability of measurement to yield a 0
+    assert m0.get_prob(_0) == 1.0
+    assert m1.get_prob(_0) == 0.0
+    assert m0.get_prob(_1) == 0.0
+    assert m1.get_prob(_1) == 1.0
+
+    # Post-state measurement of qubit with operator
+    assert _p.measure_with_op(m0) == _0
+    assert _p.measure_with_op(m1) == _1
+    assert _m.measure_with_op(m0) == _0
+    assert _m.measure_with_op(m1) == s([0, -1])
+
+
 def abs_squared(x):
     return np.abs(x) ** 2
 
@@ -703,6 +763,9 @@ def test():
     assert np.isclose(_p.get_prob(0), 0.5)  # Probability for |+> in 0 is 0.5
     assert np.isclose(_p.get_prob(1), 0.5)  # Probability for |+> in 1 is 0.5
 
+    # Test measurement operators
+    test_measurement_ops()
+
     # IV: Compositing | The state space of a composite physical system
     #                   is the tensor product of the state spaces
     #                   of the component physical systems.

requirements.txt

@@ -9,7 +9,7 @@ cryptography==2.8
 cvxopt==1.2.4
 cycler==0.10.0
 Cython==0.29.14
-dataclasses==0.7
+dataclasses==0.6
 decorator==4.4.0
 dill==0.3.1.1
 dlx==1.0.4