88 5. TOP-DOWN
from n=2 blocks of size 2 2 and of type
1
˙1
. erefore, if n D 2
w
, such matrices rep-
resent multiply controlled Z gates acting on the first qubit. us the corresponding products
G
I
C
G satisfy the equality
H Z H X
D
D
:
Analogously, the intersection of the groups Q(n), defined in Section 3.8, and bZU(n) is
a group which we will denote Q
z
(n). It consists of all n n diagonal matrices with exclusively
entries 1 in the upper half diagonal and ˙1 in the lower half diagonal. It has order 2
n=2
and is
isomorphic to the direct product S
n=2
2
; see Figure 5.3.
S(?)
U(?)
P
x
(?)
bXU(?)
Q
z
(?)
bZU(?)
1(?)
Figure 5.3: Hierarchy of the Lie groups U(n), bXU(n), and bZU(n) and the finite groups S(n),
P
x
(n), Q
z
(n), and 1(n).
Although Figures 3.3 and 5.3 resemble each other, there is an important quantitative dif-
ference. Whereas in Figure 3.3 the groups XU(n) and ZU(n) have quite different dimensions
(i.e., .n 1/
2
and n 1, respectively), in Figure 5.3 the groups bXU(n) and bZU(n) have the
same dimension, i.e., .n=2/
2
. Whereas in Figure 3.3 the groups P(n) and Q(n) have quite dif-
ferent orders (i.e., nŠ and 2
n1
, respectively), in Figure 5.3 the groups P
x
(n) and Q
z
(n) have the
same order, i.e., 2
n=2
. us, Figure 5.3 has a nice symmetry, absent in Figure 3.3. is symmetry
is also expressed by the fact that bXU(n) and bZU(n) are conjugate and so are P
x
(n) and Q
z
(n):
bXU.n/ D G bZU.n/ G and bZU.n/ D G bXU.n/ G I