Hadamard transform efficient (#103)

Added tests for Hadamard and improved its construction efficiency.

doc/source/qip-simulator.rst

@@ -40,7 +40,7 @@ method.
 
 .. testcode::
 
-  from qutip import tensor
+  from qutip import tensor, basis
   zero_state = tensor(basis(2, 0), basis(2, 0), basis(2, 0))
   result = qc.run(state=zero_state)
   wstate = result
@@ -94,42 +94,42 @@ outputs, we can use the :meth:`.QubitCircuit.run_statistics` function:
 .. testoutput::
   :options: +NORMALIZE_WHITESPACE
 
-    State:
-    Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
-    Qobj data =
-    [[0.]
-    [1.]
-    [0.]
-    [0.]
-    [0.]
-    [0.]
-    [0.]
-    [0.]]
-    with probability 0.33333257054168813
-    State:
-    Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
-    Qobj data =
-    [[0.]
-    [0.]
-    [1.]
-    [0.]
-    [0.]
-    [0.]
-    [0.]
-    [0.]]
-    with probability 0.33333257054168813
-    State:
-    Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
-    Qobj data =
-    [[0.]
-    [0.]
-    [0.]
-    [0.]
-    [1.]
-    [0.]
-    [0.]
-    [0.]]
-    with probability 0.33333485891662384
+  State:
+  Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
+  Qobj data =
+  [[0.]
+   [1.]
+   [0.]
+   [0.]
+   [0.]
+   [0.]
+   [0.]
+   [0.]]
+  with probability 0.3333325705416881
+  State:
+  Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
+  Qobj data =
+  [[0.]
+   [0.]
+   [1.]
+   [0.]
+   [0.]
+   [0.]
+   [0.]
+   [0.]]
+  with probability 0.3333325705416881
+  State:
+  Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
+  Qobj data =
+  [[0.]
+   [0.]
+   [0.]
+   [0.]
+   [1.]
+   [0.]
+   [0.]
+   [0.]]
+  with probability 0.33333485891662384
 
 The function returns a :class:`~.Result` object which contains
 the output states.
@@ -187,7 +187,7 @@ The :class:`.CircuitSimulator` class also enables stepping through the circuit:
    [0.        ]
    [0.        ]]
 
-This only excutes one gate in the circuit and
+This only executes one gate in the circuit and
 allows for a better understanding of how the state evolution takes place.
 The method steps through both the gates and the measurements.
 

src/qutip_qip/operations/gates.py

@@ -1387,14 +1387,10 @@ def hadamard_transform(N=1):
         Quantum object representation of the N-qubit Hadamard gate.
 
     """
-    data = 2 ** (-N / 2) * np.array(
-        [
-            [(-1) ** _hamming_distance(i & j) for i in range(2 ** N)]
-            for j in range(2 ** N)
-        ]
-    )
+    data = [[1, 1], [1, -1]]
+    H = Qobj(data) / np.sqrt(2)
 
-    return Qobj(data, dims=[[2] * N, [2] * N])
+    return tensor([H] * N)
 
 
 def _flatten(lst):

tests/test_gates.py

@@ -299,6 +299,23 @@ def test_cnot_explicit(self):
                              [0, 0, 0, 1, 0, 0, 0, 0]])
         np.testing.assert_allclose(test, expected, atol=1e-15)
 
+    def test_hadamard_explicit(self):
+        test = gates.hadamard_transform(3).full()
+        expected = np.array(
+            [
+                [1, 1, 1, 1, 1, 1, 1, 1],
+                [1, -1, 1, -1, 1, -1, 1, -1],
+                [1, 1, -1, -1, 1, 1, -1, -1],
+                [1, -1, -1, 1, 1, -1, -1, 1],
+                [1, 1, 1, 1, -1, -1, -1, -1],
+                [1, -1, 1, -1, -1, 1, -1, 1],
+                [1, 1, -1, -1, -1, -1, 1, 1],
+                [1, -1, -1, 1, -1, 1, 1, -1],
+            ]
+        )
+        expected = expected / np.sqrt(8)
+        np.testing.assert_allclose(test, expected)
+
     def test_cyclic_permutation(self):
         operators = [qutip.sigmax(), qutip.sigmaz()]
         test = gates.expand_operator(qutip.tensor(*operators), N=3,