diff --git a/clifford2.py b/clifford2.py
new file mode 100644
index 0000000..46ec079
--- /dev/null
+++ b/clifford2.py
@@ -0,0 +1,44 @@
+import numpy as np
+from tqdm import tqdm
+import os
+from qi import *
+import cPickle
+
+def find_up_to_phase(u):
+    """ Find the index of a given u within a list of unitaries, up to a global phase  """
+    global unitaries
+    for i, t in enumerate(unitaries):
+        for phase in range(8):
+            if np.allclose(t, np.exp(1j*phase*np.pi/4.)*u):
+                return i, phase
+    raise IndexError
+
+
+def compose_u(decomposition):
+    """ Get the unitary representation of a particular decomposition """
+    us = (elements[c] for c in decomposition)
+    return np.matrix(reduce(np.dot, us), dtype=complex)
+
+
+def construct_tables():
+    """ Constructs multiplication and conjugation tables """
+    conjugation_table = [find_up_to_phase(u.H)[0] for i, u in enumerate(unitaries)]
+    times_table = [[find_up_to_phase(u*v)[0] for v in unitaries] 
+            for u in tqdm(unitaries, "Building times-table")]
+    with open("tables.pkl", "w") as f:
+        cPickle.dump((conjugation_table, times_table), f)
+
+
+decompositions = \
+("xxxx", "xx", "zzxx", "zz", "zxx", "z", "zzz", "xxz",
+"xzx", "xzxxx", "xzzzx", "xxxzx", "xzz", "zzx", "xxx", "x",
+"zzzx", "xxzx", "zx", "zxxx", "xxxz", "xzzz", "xz", "xzxx")
+elements = {"x": sqx, "z": msqz}
+unitaries = [compose_u(d) for d in decompositions]
+
+# Build / reload lookup tables
+if not os.path.exists("tables.pkl"):
+    construct_tables()
+
+if __name__ == '__main__':
+    pass
diff --git a/cz.py b/cz.py
index d2d69f4..e93bca0 100644
--- a/cz.py
+++ b/cz.py
@@ -2,11 +2,13 @@ from matplotlib import pyplot as plt
 from graph import *
 from viz import draw
 
-def local_complementation(g, vops, a):
-    for i in g[a]:
-        for j in g[a]:
+def local_complementation(g, vops, v):
+    for i in g[v]:
+        for j in g[v]:
             if i<j:
                 toggle_edge(g, i, j)
+        # vops[i] = times_table[vop[i]][sqrtmiz]
+        # vops[v] = times_table[vop[v]][sqrtix]
 
 if __name__ == '__main__':
     g, vops = graph(10)
diff --git a/tests/test_clifford.py b/tests/test_clifford.py
index e0fd872..3117cf4 100644
--- a/tests/test_clifford.py
+++ b/tests/test_clifford.py
@@ -3,7 +3,7 @@ from numpy import *
 from scipy.linalg import sqrtm
 from qi import *
 from tqdm import tqdm
-
+import itertools as it
 
 
 def identify_pauli(m):
@@ -45,10 +45,9 @@ def test_we_have_all_useful_gates():
 def test_group():
     """ Test we are really in a group """
     matches = set()
-    for a in tqdm(lc.unitaries, "Testing this is a group"):
-        for b in lc.unitaries:
-            i, phase = lc.find_up_to_phase(a*b)
-            matches.add(i)
+    for a, b in tqdm(it.combinations(lc.unitaries, 2), "Testing this is a group"):
+        i, phase = lc.find_up_to_phase(a*b)
+        matches.add(i)
     assert len(matches)==24