30 2. BOTTOM
U(?)
P(?) ?!
1
P
x
(?) 2
?/2
P
z
(?) (?/2)!
1(?)
∞
?
2
Figure 2.3: Hierarchy of the Lie group U(n) and the finite groups P(n), P
x
(n), P
z
(n), and 1(n).
2.2.3 CONTROLLED CIRCUITS VS. CONTROLLED GATES
We summarize the present section by Figure 2.3, drawing the group hierarchy: both the group
graph and the graph of the corresponding group orders. We recall the duality.
• Members of P
z
(2
w
) have one controlling bit and w 1 controlled bits, whereas members
of P
x
(2
w
) have one controlled bit and w 1 controlling bits.
• Matrices from P
z
(n) contain 2 blocks from P(n=2), whereas matrices from P
x
(n) contain
n=2 blocks from P(2).
• P
z
(n) is isomorphic to S
n=2
, whereas P
x
(n) is isomorphic to S
n=2
2
.
• P
z
(n) has order .n=2/Š, whereas P
x
(n) has order .2Š/
n=2
.
We close the present section by introducing yet another subgroup of P(2
w
). It consists of the
matrices of form
U D
U
11
0
0 I
:
Here again the bits A
2
, A
3
, …, and A
w
are controlled by the bit A
1
. However, here they are
affected iff A
1
equals 0. e symbol of this circuit has a white bullet instead of a black one: see
Figure 2.1d. We stress that such a circuit can be decomposed into two NOT gates and a positively
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