67
C H A P T E R 4
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4.1 PRELIMINARY CIRCUIT DECOMPOSITION
According to (3.14), in other words, according to the decomposition
U D e
i˛
Z
1
XZ
2
;
a quantum schematic (here for w D 3 and thus n D 8) looks like
Z
2
X
Z
1
e
i˛
:
As here n is even, we can apply the identity (3.16). We thus can transform (3.14) into a decom-
position containing exclusively XU and ZU matrices:
U D P
0
Z
0
P
1
0
Z
0
1
XZ
2
;
where Z
0
D diag.1; e
i˛
; 1; e
i˛
; 1; : : : ; 1; e
i˛
/ is a ZU matrix which can be implemented by a
single (uncontrolled) PHASOR gate (acting on the w th qubit) and where Z
0
1
is the product Z
0
Z
1
:
Z
2
X
Z
0
1
P
1
0
P
0
Z
0
:
For convenience, we rewrite Equation (3.14) as
U D e
i˛
n
L
n
X
n
R
n
;
where the left matrix L
n
and the right matrix R
n
are members of ZU(n) and X
n
belongs to
XU(n). As a member of the .n 1/
2
-dimensional group XU(n), X
n
has the form (3.13):
X
n
D T
n
1
U
n1
T
1
n
;
where U
n1
is a member of U(n 1) and T
n
is the n n Fourier matrix F .
68 4. TOP
Again according to the ZXZ-theorem (3.14), U
n1
can be decomposed as
e
i˛
n1
l
n1
x
n1
r
n1
;
a product of a scalar, a ZU(n 1) matrix, an XU(n 1) matrix, and a second ZU(n 1) matrix.
We thus obtain for X
n
the product T
n
L
n1
X
n1
R
n1
T
1
n
, where
L
n1
D
1
e
i˛
n1
l
n1
; X
n1
D
1
x
n1
; and R
n1
D
1
r
n1
:
Hence, we have U D e
i˛
n
L
n
T
n
L
n1
X
n1
R
n1
T
1
n
R
n
. By applying such decomposition again
and again, we find a decomposition
e
i˛
n
L
n
T
n
L
n1
T
n1
L
n2
: : : T
2
L
1
X
1
R
1
T
1
2
R
2
: : : R
n2
T
1
n1
R
n1
T
1
n
R
n
of an arbitrary member of U(n). As automatically X
1
and R
1
equal the n n unit matrix, we
thus obtain
U D e
i˛
n
L
n
T
n
L
n1
T
n1
L
n2
: : : T
2
L
1
T
1
2
R
2
: : : R
n2
T
1
n1
R
n1
T
1
n
R
n
; (4.1)
where all n matrices L
j
and all n 1 matrices R
j
belong to the .n 1/-dimensional group
ZU(n). e n 1 matrices T
j
are block-diagonal matrices:
T
j
D
1
.nj /.nj /
F
j
; (4.2)
where 1
.nj /.nj /
is the .n j / .n j / unit matrix and F
j
is the j j Fourier matrix. For
w D 2 (and thus n D 4), Equation (4.1) thus looks like the following cascade of six constant
matrices, seven ZU circuits, and one overall phase:
R
4
T
1
4
R
3
T
1
3
R
2
T
1
2
L
1
T
2
L
2
T
3
L
3
T
4
L
4
e
i˛
3 0 2 0 1 0 1 0 2 0 3 0 3
1 ;
where the T
j
blocks represent the n 1 constant matrices
T
2
D
0
B
B
@
1
1
1=
p
2 1=
p
2
1=
p
2 1=
p
2
1
C
C
A
; T
3
D
0
B
B
@
1
1=
p
3 1=
p
3 1=
p
3
1=
p
3 !=
p
3 !
2
=
p
3
1=
p
3 !
2
=
p
3 !=
p
3
1
C
C
A
;
and T
4
D
0
B
B
@
1=2 1=2 1=2 1=2
1=2 i=2 1=2 i=2
1=2 1=2 1=2 1=2
1=2 i=2 1=2 i=2
1
C
C
A
;
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