Department of Physics, SUNY at Binghamton N.D. Mermin...
70
1 Operator method in Quantum Computing Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton Binghamton, NY (Date: November 27, 2014) N.D. Mermin, Quantum Computer Science An Introduction (Cambridge, 2007). 1. Definition of the operator n ˆ We define the operator n ˆ from the eigenvalue problem x x x n ˆ , with 0 0 0 0 ˆ n , 1 1 1 1 ˆ n where x = 0 and 1. n ˆ is the projection operator and is defined by 1 1 ˆ n . The matrix of n ˆ under the basis of { 0 , 1 } is given by 1 0 0 0 ˆ n The operator n ˆ is the projection operator on the state . 1 2. Definition of the operator n m ˆ 1 ˆ ˆ We define the operator m ˆ as n m ˆ 1 ˆ ˆ where 1 ˆ is the identity matrix of 2x2, 1 0 0 1 1 ˆ For simplicity, here, we use m ˆ instead of n ~ . We note that
Transcript of Department of Physics, SUNY at Binghamton N.D. Mermin...
1
Operator method in Quantum Computing Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton Binghamton, NY
(Date: November 27, 2014) N.D. Mermin, Quantum Computer Science An Introduction (Cambridge, 2007). 1. Definition of the operator n
We define the operator n from the eigenvalue problem
xxxn ˆ ,
with
0000ˆ n , 1111ˆ n
where x = 0 and 1. n is the projection operator and is defined by
11ˆ n .
The matrix of n under the basis of { 0 , 1 } is given by
10
00n
The operator n is the projection operator on the state . 1
2. Definition of the operator nm ˆ1ˆ We define the operator m as
nm ˆ1ˆ
where 1 is the identity matrix of 2x2,
10
011
For simplicity, here, we use m instead of n~ . We note that
2
xxxxxxnxm )1()ˆ1(ˆ
Thus we have
00ˆ m , 0101ˆ m
The matrix of m under the basis of { 0 , 1 } is given by
00
01m
The operator n is the projection operator on the state . 0
00ˆ m .
3. Properties of n and m
nn ˆˆ2
mm ˆˆ 2
0ˆˆˆˆ nmmn
1ˆˆ nm 4. Pauli matrix
01
10ˆˆ
xX ,
0
0ˆˆ
i
iY y ,
10
01ˆˆ
zZ
mXXn ˆˆˆˆ , nXXm ˆˆˆˆ
1ˆˆˆ 222 ZYX
XZZX ˆˆˆˆ
3
)ˆ1(2
1ˆ Zn , )ˆ1(
2
1ˆ Zm , mnZ ˆˆˆ
XZiZXiY ˆˆˆˆˆ , YXiXYiZ ˆˆˆˆˆ , ZYiYZiX ˆˆˆˆˆ 5. Hadamard operator
)ˆˆ(2
1
11
11
2
1ˆ ZXH
1)ˆˆˆˆˆˆ(2
1)ˆˆ)(ˆˆ(
2
1ˆ 222 ZXZZXXZXZXH
ZHXH ˆˆˆˆ , XHZH ˆˆˆˆ . Note that
Z
XZZXZX
ZXZZXZX
ZXZX
ZXXZXHXH
ˆ
)ˆˆˆˆˆˆ(2
1
)ˆˆˆˆˆˆ(2
1
)ˆˆ1)(ˆˆ(2
1
)ˆˆ(ˆ)ˆˆ(2
1ˆˆˆ
22
2
X
XZZX
XZZXZX
XZZX
ZXZZXHZH
ˆ
)ˆˆˆˆ(2
1
)ˆˆˆˆˆˆ(2
1
)ˆˆ1)(ˆˆ(2
1
)ˆˆ(ˆ)ˆˆ(2
1ˆˆˆ
22
)ˆˆˆˆˆˆˆˆ(2
1)ˆˆ()ˆˆ(
2
1ˆˆ ZZXZZXXXZXZXHH
6. Calculation of matrices by using Mathematica
4
Clear "Global` " ; I2 IdentityMatrix 2 ;
n0 00 1
; m I2 n; X PauliMatrix 1 ;
Y PauliMatrix 2 ; Z PauliMatrix 3 ; 010
;
101
;
m MatrixForm
1 00 0
n MatrixForm
0 00 1
n.n n MatrixForm
0 00 0
m.m m MatrixForm
0 00 0
5
n.X X.m MatrixForm
0 00 0
m.X X.n MatrixForm
0 00 0
m n MatrixForm
1 00 1
Z.X X.Z MatrixForm
0 00 0
12
I2 Z MatrixForm
0 00 1
12
X Z MatrixForm
12
12
12
12
6
H11
2X Z ; H1 MatrixForm
12
12
12
12
X.X MatrixForm
1 00 1
Z.Z MatrixForm
1 00 1
X.Z Z.X MatrixForm
0 00 0
H1.H1 MatrixForm
1 00 1
H1.X.H1 Z MatrixForm
0 00 0
7
H1.Z.H1 X MatrixForm
0 00 0
H1. 0 MatrixForm
12
12
H1. 1 MatrixForm
1212
f11 KroneckerProduct m, I2
KroneckerProduct n, X ;
f11 MatrixForm
1 0 0 00 1 0 00 0 0 10 0 1 0
8
f1212
KroneckerProduct I2, I2 X
KroneckerProduct Z, I2 X ;
f12 MatrixForm
1 0 0 00 1 0 00 0 0 10 0 1 0
f1312
KroneckerProduct I2 Z, I2
KroneckerProduct I2 Z, X ;
f13 MatrixForm
1 0 0 00 1 0 00 0 0 10 0 1 0
S11 KroneckerProduct n, n
KroneckerProduct m, m
KroneckerProduct X.n, X.m
KroneckerProduct X.m, X.n ;
S11 MatrixForm
9
1 0 0 00 0 1 00 1 0 00 0 0 1
C11 KroneckerProduct m, I2
KroneckerProduct n, X ;
C11 MatrixForm
1 0 0 00 1 0 00 0 0 10 0 1 0
Z X .Y MatrixForm
0 00 0
Y Z.X MatrixForm
0 00 0
X Y.Z MatrixForm
0 00 0
10
7. The CNOT gate with CNOTU
The CNOT gate is defined by
KroneckerProduct H1, H1 MatrixForm
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
U1u11 u12u21 u22
;
KroneckerProduct I2, U1 MatrixForm
u11 u12 0 0u21 u22 0 0
0 0 u11 u120 0 u21 u22
h1
KroneckerProduct H1, H1 .KroneckerProduct X, X
MatrixForm
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
11
)ˆˆˆ11ˆ11(2
1
ˆ)ˆ1(2
11)ˆ1(
2
1
ˆˆ1ˆˆ
XZXZ
XZZ
XnmUCNOT
with
1ˆ 2 CNOTU
)1ˆˆ1ˆ111ˆ111(2
1
1ˆ)21(ˆ12
XZXZ
CUCNOT
)ˆ1ˆˆ1111ˆ111(2
1)31(ˆ XZXZUCNOT
((Mathematica))
Clear "Global` " ;
I2 IdentityMatrix 2 ;
X PauliMatrix 1 ; Y PauliMatrix 2 ;
Z PauliMatrix 3 ;
UCNOT12
KroneckerProduct I2, I2
KroneckerProduct Z, I2
KroneckerProduct I2, X
KroneckerProduct Z, X ;
UCNOT MatrixForm
1 0 0 00 1 0 00 0 0 10 0 1 0
12
8. Swap (exchange) operator
UCNOT1212
KroneckerProduct I2, I2, I2
KroneckerProduct Z, I2, I2
KroneckerProduct I2, X, I2
KroneckerProduct Z, X, I2 ;
UCNOT12 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
UCNOT1312
KroneckerProduct I2, I2, I2
KroneckerProduct Z, I2, I2
KroneckerProduct I2, I2, X
KroneckerProduct Z, I2, X
MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 1 0 00 0 0 0 1 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
swapG
and 2.
G
Then the ________9. T
Only if a
a
p is called a
1(2
1
1(2
1
1(2
1
ˆ[4
1
1(4
1
ˆˆˆ
X
nnGSWAP
swap opera
__________Toffoli gate
1ba
aa ' ,
swap (excha
ˆ1
)ˆˆ1
)ˆˆ1
[)]ˆ1(ˆ
ˆ1()ˆ
ˆˆˆ
XX
ZZ
ZZ
XZX
ZZ
mmn
ator becomes
___________
bb '
ange) operato
ˆˆˆ
ˆˆ(4
1)
ˆˆ(4
1)
)]ˆ1(
)ˆ1(4
1)ˆ
ˆˆ()ˆˆ(
ZYYX
YiX
ZXX
ZX
ZZ
mXnX
s equivalent t
__________
13
or, which sim
)ˆˆ
)ˆ()ˆ
ˆ()ˆ
)ˆ1()
()ˆˆ()
ZZ
YiXY
XXZ
Z
mX
to the Dirac
___________
mply interch
ˆˆ(4
1)
ˆ(4
1)ˆˆ
)]ˆ1(ˆ[4
1
)ˆˆ(
YiX
XZX
ZX
nX
exchange sp
__________
hanges the st
ˆ()ˆ
()ˆˆ
ˆ1([]
YiXY
XZX
ZX
pin operator
___________
tates of qubi
)ˆ
)ˆˆ
)]
Y
ZX
Z
_________
ts 1
14
cUc ˆ'
Other wise
aa ' , bb '
cc '
Truth table
a b c a’ b’ c’ 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1
1 1 0 1 1 100ˆ2111 uuU
1 1 1 1 1 101ˆ2212 uuU
U
I
I
I
000
000
000
000
000000
000000
00100000
00010000
00001000
00000100
00000010
00000001
)](ˆ
2221
2211
uu
uu
UGToffli
10. Example: equivalent quantum circuits
We show that a two-qubit controlled gate canted using a combination of one qubit controlled gate as
1323ˆ)1ˆ])([ˆ1)(1ˆ(ˆ][ˆ
VCNOTCNOTVToffoli GGVGGGUG .
This can
G
which me
The left h
G
where
be also desc
ˆ][ˆToffoli GUG
eans that GT
hand-side is
][ˆ UGToffoli
cribed by
13ˆ(ˆ
CNOTV GG
][UToffoli is a
the Toffoli g
0000
0000
0000
0000
1000
0100
0010
0001
(ˆ1)[1 VG
universal ga
gate with the
2
1
00
00
010
001
000
000
000
000
U
U
15
ˆ)]( CNOTG
ate (reversibl
e matrix U ,
221
121
0
0
0
0
0
0
U
U
23ˆ)1 VG
le).
,
16
2221
1211ˆUU
UUU
Suppose that the matrix U is expressed by 2ˆˆ VU , where V is the unitary operator. The CNOT operator is given by
X
I
0
0
0100
1000
0010
0001
ˆCNOTG ,
1ˆ CNOTG is obtained as
0ˆ00
ˆ000
00ˆ0
000ˆ
10
01ˆ
2
2
2
2
I
I
I
I
GCNOT X
I
The control V gate is given by
V
I
0
0][ˆ VG ,.
V
I
0
0][ˆ VG
Then we have
V
I
V
I
000
000
000
000
000000
000000
00100000
00010000
000000
000000
00000010
00000001
][ˆ1ˆ
2221
1211
2221
1211
23
VV
VV
VV
VV
VUGV
13ˆ
VC is the matrix obtained from the matrix ][ˆ1 VG by the appropriate interchange of row and
column,
17
V
V
I
I
000
000
000
000
000000
000000
000000
000000
00001000
00000100
00000010
00000001
][ˆ
2221
1211
2221
121113
VV
VV
VV
VVVGV
((Mathematica))
18
Clear "Global` " ; I2 IdentityMatrix 2 ; CV
1 0 0 00 1 0 00 0 v11 v120 0 v21 v22
;
CVP
1 0 0 00 1 0 00 0 v11c v21c0 0 v12c v22c
;
V1v11 v12v21 v22
;
CUV13 A : Module A1, U, U1, U11, U12 , A1 A;
U KroneckerProduct I2, A1 ;
U1 U All, 1 , U All, 2 , U All, 5 , U All, 6 ,
U All, 3 , U All, 4 , U All, 7 , U All, 8 ;
U11 Transpose U1 ;
U12 U11 1 , U11 2 , U11 5 , U11 6 , U11 3 ,
U11 4 , U11 7 , U11 8 ;
CV13 CUV13 CV ; CV13 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 v11 v12 0 00 0 0 0 v21 v22 0 00 0 0 0 0 0 v11 v120 0 0 0 0 0 v21 v22
19
UCNOT
1 0 0 00 1 0 00 0 0 10 0 1 0
; UCNOTI2 KroneckerProduct UCNOT, I2 ;
UCNOTI2 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0
CV23 KroneckerProduct I2, CV ;
CV23 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 v11 v12 0 0 0 00 0 v21 v22 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 v11 v120 0 0 0 0 0 v21 v22
20
CVP23 KroneckerProduct I2, CVP ; CVP23 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 v11c v21c 0 0 0 00 0 v12c v22c 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 v11c v21c0 0 0 0 0 0 v12c v22c
K1 CV23.UCNOTI2.CVP23.UCNOTI2.CV13 FullSimplify;
vvcv11 v12v21 v22
.v11c v21cv12c v22c
;
vcvv11c v21cv12c v22c
.v11 v12v21 v22
;
U1 V1.V1 Simplify;
rule1 vvc 1, 1 1, vvc 1, 2 0, vvc 2, 1 0,
vvc 2, 2 1, vcv 1, 1 1, vcv 1, 2 0, vcv 2, 1 0,
vcv 2, 2 1 ;
rule2 U1 1, 1 U11, U1 1, 2 U12, U1 2, 1 U21,
U1 2, 2 U22 ;
K11 K1 . rule1; K12 K11 . rule2; K12 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 U11 U120 0 0 0 0 0 U21 U22
________
________10. E
We s
ˆ1( GCNO
where
L1 C
L11
1 00 10 00 00 00 00 00 0
__________
__________Equivalent c
how the equ
ˆ)(GCNOTOT
CV13.UC
L1 .
0 0 00 0 01 0 00 1 00 0 10 0 00 0 00 0 0
___________
___________circuits uivalence bet
ˆ1)(1 GCNOT
NOTI2.C
rule1;
0 00 00 00 00 01 00 U110 U21
_______
__________
tween two qu
00
00
00
00
00
00
10
01
ˆ() GCNOTT
CVP23.U
L12 L1
000000
U12U22
21
_________
uantum circu
0100
1000
0000
0000
0010
0001
0000
0000
(ˆ)1 13 CNOC
UCNOTI2.
11 . r
uits as show
000
00
010
100
000
000
000
000
)OT
.CV23
rule2; L
wn below,
FullS
L12 M
Simplify
MatrixFo
y;
orm
22
00100000
00010000
10000000
01000000
00001000
00000100
00000010
00000001
1ˆCNOTG
01000000
10000000
00100000
00010000
00000100
00001000
00000010
00000001
ˆ1 CNOTG
01000000
10000000
00010000
00100000
00001000
00000100
00000010
00000001
ˆ13CNOTG
)ˆˆˆ11ˆ11(2
1
ˆ)ˆ1(2
11)ˆ1(
2
1
ˆˆ1ˆ][ˆ
VZVZ
VZZ
VnmVG
23
2221
2221
1211
121112
000000
000000
000000
000000
00001000
00000100
00000010
00000001
1][ˆ
VV
VV
VV
VVVG
V
I
V
I
000
000
000
000
000000
000000
00100000
00010000
000000
000000
00000010
00000001
][ˆ1][ˆ
2
2
2221
1211
2221
1211
23
VV
VV
VV
VV
VGVG
V
V
I
I
000
000
000
000
000000
000000
000000
000000
00001000
00000100
00000010
00000001
)ˆ1ˆˆ1111ˆ111(2
1][ˆ
2
2
2221
1211
2221
1211
13
VV
VV
VV
VV
VZVZVG
((Mathematica))
24
Clear "Global` " ; I2 IdentityMatrix 2 ;
X PauliMatrix 1 ; Y PauliMatrix 2 ;
Z PauliMatrix 3 ;
VV11 V12V21 V22
;
UV12
KroneckerProduct I2, I2
KroneckerProduct Z, I2
KroneckerProduct I2, V
KroneckerProduct Z, V ;
UV MatrixForm
1 0 0 00 1 0 00 0 V11 V120 0 V21 V22
C12V KroneckerProduct UV, I2 ;
C12V MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 V11 0 V12 00 0 0 0 0 V11 0 V120 0 0 0 V21 0 V22 00 0 0 0 0 V21 0 V22
25
_____________________________________________________________________________ 11 Toffoli gate:
The Toffoli gate is given by
C23V KroneckerProduct I2, UV ;
C23V MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 V11 V12 0 0 0 00 0 V21 V22 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 V11 V120 0 0 0 0 0 V21 V22
C13V12
KroneckerProduct I2, I2, I2
KroneckerProduct Z, I2, I2
KroneckerProduct I2, I2, V
KroneckerProduct Z, I2, V ;
C13V MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 V11 V12 0 00 0 0 0 V21 V22 0 00 0 0 0 0 0 V11 V120 0 0 0 0 0 V21 V22
26
X
I
I
I
000
000
000
000
01000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
ˆˆˆ1ˆˆ11ˆ
)ˆˆ1ˆ(ˆ11ˆ
ˆˆ11ˆˆ
Xnnmnm
Xnmnm
GnmG CNOTtoffoli
where
XnmGCNOTˆˆ1ˆˆ
((Mathematica))
27
12. Fredkin gate
The Fredkin gate is given by
Clear "Global` " ; I2 IdentityMatrix 2 ; X PauliMatrix 1 ;
Y PauliMatrix 2 ; Z PauliMatrix 3 ; n0 00 1
;
m1 00 0
;
UCNOT
1 0 0 00 1 0 00 0 0 10 0 1 0
;
TOF1 KroneckerProduct m, I2, I2
KroneckerProduct n, UCNOT ;
TOF1 MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0
28
10000000
01000000
00001000
00010000
00100000
00000100
00000010
00000001
ˆ)ˆˆˆˆˆ11(2
1ˆ11
ˆˆˆ11ˆ
nZZYYXXm
nUmG SWAPFredkin
where SWAPG is the SWAP operator,
1000
0010
0100
0001
)ˆˆˆˆˆ11(2
1ˆ ZZYYXXGSWAP
((Mathematica))
29
13. Controlled V gate
Clear "Global` " ; X PauliMatrix 1 ; Y PauliMatrix 2 ;
Z PauliMatrix 3 ; n0 00 1
; m1 00 0
;
I2 IdentityMatrix 2 ;
SWAP
1 0 0 00 0 1 00 1 0 00 0 0 1
;
Fredkin
KroneckerProduct I2, I2, m
KroneckerProduct SWAP, n Simplify;
Fredkin MatrixForm
1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 1 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1
SWAP12
KroneckerProduct I2, I2 KroneckerProduct X, X
KroneckerProduct Y, Y KroneckerProduct Z, Z ;
SWAP MatrixForm
1 0 0 00 0 1 00 1 0 00 0 0 1
30
2221
1211
00
00
0010
0001
)ˆˆˆ11ˆ11(2
1][ˆ
VV
VVVZVZVC
2221
1211
00
0100
00
0001
)ˆˆ1ˆˆ111(2
1][ˆ
VV
VVZVVZVR
31
____________________________________________________________________________ 14. Controlled-U gate
U
I
0
0
00
00
0010
0001
)ˆˆˆ11ˆ11(2
1][ˆ
2221
1211
UU
UU
UZUZUC
Clear "Global` " ; I2 IdentityMatrix 2 ;
X PauliMatrix 1 ; Y PauliMatrix 2 ; Z PauliMatrix 3 ;
VV11 V12V21 V22
;
UV12
KroneckerProduct I2, I2 KroneckerProduct Z, I2
KroneckerProduct I2, V KroneckerProduct Z, V ;
RUV12
KroneckerProduct I2, I2 KroneckerProduct I2, Z
KroneckerProduct V, I2 KroneckerProduct V, Z ;
UV MatrixForm
1 0 0 00 1 0 00 0 V11 V120 0 V21 V22
RUV MatrixForm
1 0 0 00 V11 0 V120 0 1 00 V21 0 V22
32
2221
1211
00
0100
00
0001
)ˆˆ1ˆˆ111(2
1][ˆ
UU
UUZUUZUR
)(ˆ HR
0010
0100
1000
0001
ˆCNOTR
15. Controlled-CNOT
X
I
0
0
0100
1000
0010
0001
ˆˆ1ˆˆ XnmGCNOT
16. Fredkin gate
10000000
01000000
00001000
00010000
00100000
00000100
00000010
00000001
ˆ)ˆˆˆˆˆ11(2
1ˆ11
ˆˆˆ11ˆ
nZZYYXXm
nGmG SWAPFredkin
17. Toffoli gate
33
X
I
I
I
000
000
000
000
01000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
ˆˆˆ1ˆˆ11ˆ
)ˆˆ1ˆ(ˆ11ˆ
ˆˆ11ˆˆ
Xnnmnm
Xnmnm
GnmG CNOTtoffoli
18. R-CNOT
0010
0100
1000
0001
ˆˆˆ1ˆ nXmRCNOT .
19. Swap gate
1000
0010
0100
0001
)ˆˆˆˆˆ11(2
1
ˆˆˆˆˆˆˆˆˆˆˆˆ
ZZYYXX
mXnXnXmXnnmmGSWAP
_________________________________________________________________________
10
011 ,
01
10X ,
0
0ˆi
iY ,
10
01Z
34
20. Hadamard gate:
11
11
2
1H
i
i
e
eP
0
0ˆ
ie
T0
01ˆ
ie
S0
01ˆ
_____________________________________________________________________________
2221
2221
1211
121112
000000
000000
000000
000000
00001000
00000100
00000010
00000001
1][ˆ][ˆ
VV
VV
VV
VVVGVGV
V
I
V
I
000
000
000
000
000000
000000
000000
000000
00001000
00000100
00000010
00000001
][ˆ1ˆ
2
2
2221
2221
1211
1211
23
VV
VV
VV
VV
VGGV
35
V
V
I
I
000
000
000
000
000000
000000
000000
000000
00001000
00000100
00000010
00000001
)ˆ1ˆˆ1111ˆ111(2
1ˆ
2
2
2221
1211
2221
1211
13
VV
VV
VV
VV
VZVZGV
______________________________________________________________________________
10. Expression for CNOTG in terms of Pauli operators X and Z
We know that the controlled-CNOT gate can be expressed by
XnmGCNOTˆˆ1ˆˆ .
Here we consider another expressions for the controlled-CNOT gate. We start with
)10(2
1 , )10(
2
1
The projection operators are defined by
11
11
2
111
1
1
2
1P ,
11
11
2
111
1
1
2
1P
We note that
1ˆˆ PP .
The operator X can be described as
36
PPX ˆˆ
01
10ˆ .
The operator Z can be expressed by
nmZ ˆˆ10
01ˆ
Note that
)ˆ1(2
1ˆ Zm , )ˆ1(
2
1ˆ Zn
since 1ˆˆ nm .
We now introduce the operator (controlled-CNOT gate), which can be generated from
0100
1000
0010
0001
ˆˆ1ˆˆ XnmGCNOT
CNOTG is equivalent to the controlled-X gate
X
I
0
0][ˆˆ XCGCNOT .
This can be derived using the Kronecker product. Since
1ˆˆ PP , XPP ˆˆˆ
we get
)ˆ1(2
1ˆ XP , )ˆ1(2
1ˆ XP
Then the controlled-CNOT operator can be rewritten as
37
PZP
PnmPnm
PnPnPmPm
PPnPPmGCNOT
ˆˆˆ1
ˆ)ˆˆ(ˆ)ˆˆ(
ˆˆˆˆˆˆˆˆ
)ˆˆ(ˆ)ˆˆ(ˆˆ
or
)]ˆ1(ˆ)ˆ1(1[2
1ˆ XZXGCNOT
11. The expression of CNOTG in terms of Hadamard gate H
The Hadamard gate is defined by
)ˆˆ(2
1
11
11
2
1ˆ XZH
.
Then we get
00
11
2
1
11
11
00
01
2
1ˆˆHm .
)ˆ1(2
1ˆ11
11
2
1
00
11
11
11
2
1ˆˆˆ XPHmH
Noting that 1112 and using the property of the Kronecker product, we get
)ˆ1)(ˆ1)(ˆ1()ˆˆ1)(ˆ1()ˆˆˆ(1ˆ1 HmHHmHHmHP (1)
Similarly, we have
11
00
2
1
11
11
10
00
2
1ˆˆHn .
)ˆ1(2
1ˆ11
11
2
1
11
00
11
11
2
1ˆˆˆ XPHnH
.
38
Noting that ZZ ˆ1ˆ and using the property of the Kronecker product, we get
)ˆ1)(ˆˆ)(ˆ1()ˆˆ1)(ˆˆ()ˆˆˆ(ˆˆˆ HnZHHnHZHnHZPZ (2)
From Eqs.(1) and (2), thus we have
)ˆ1)(ˆˆˆ1)(ˆ1(
)ˆ1)(ˆˆ)(ˆ1()ˆ1)(ˆ1)(ˆ1(
ˆˆˆ1ˆ
HnZmH
HnZHHmH
PZPGCNOT
which can be rewritten as
)ˆ1(ˆ)ˆ1()ˆ1(ˆ)ˆ1(ˆˆ HGHHRHGG ZZXCNOT .
since
ZZ GR ˆˆ .
12. Matrix representation of Kronecker product for two qubits
00
01m ,
10
00n , 1ˆˆ nm .
11
11
2
111
1
1
2
1P ,
11
11
2
111
1
1
2
1P
0000
0000
0010
0001
1m ,
1000
0100
0000
0000
1n ,
0000
0100
0000
0001
ˆ1 m ,
1000
0000
0010
0000
ˆ1 n ,
39
0000
0000
0000
0001
ˆˆ mm ,
0000
0000
0010
0000
ˆˆ nm
0000
0100
0000
0000
ˆˆ mn ,
1000
0000
0000
0000
ˆˆ nn
0000
0000
0001
0010
ˆˆ Xm ,
0100
1000
0000
0000
ˆˆ Xn ,
0000
0001
0000
0100
ˆˆ mX ,
0010
0000
1000
0000
ˆˆ nX
0000
0000
00
00
ˆˆ 2221
1211
uu
uu
Um ,
2221
1211
00
00
0000
0000
ˆˆ
uu
uuUn
0000
00
0000
00
ˆˆ2221
1211
uu
uu
mU ,
2221
1211
00
0000
00
0000
ˆˆ
uu
uunU
0000
0010
0000
0000
ˆˆˆˆ nXmX ,
0000
0000
0100
0000
ˆˆˆˆ mXnX ,
40
0000
0010
0100
0000
ˆˆˆˆˆˆˆˆ mXnXnXmX ,
0000
0000
0011
0011
2
1ˆˆ Pm ,
1100
1100
0000
0000
2
1ˆˆ Pn
0000
0000
0011
0011
2
1ˆˆ Pm ,
1100
1100
0000
0000
2
1ˆˆ Pn
13. Equivalence of quantum circuits between ZG and ZR
We show that the controlled-Z gate ZG is equivalent to the quantum circuit with ZR .
((Method-1)) The use of matrices
1000
0100
0010
0001
1000
0100
0000
0000
0000
0000
0010
0001
ˆˆ1ˆˆ ZnmGZ
R
Fig. Q
((Method
G
R
00
00
10
01
00
00
00
01
ˆ1ˆ mRZ
Quantum gate
d-II) Opera
1
)ˆˆ(
1ˆ
1ˆ
1ˆ(ˆ
2
2
nm
m
m
mGZ
1
ˆ(1
ˆ1
ˆ1
ˆ1(ˆ
2
2
m
m
m
mRZ
10
01
00
00
0
0
0
0
00
01
00
00
ˆˆ nZ
es with ZG a
ation method
1)
1ˆ
ˆˆˆ
ˆ)(ˆˆ
n
Znm
mZn
)ˆ
ˆ1
ˆˆˆ
1)(ˆˆ
n
n
nmZ
nZ
100
000
001
000
and ZR .
d
ˆˆˆˆ
)ˆˆ12
nZmn
Zn
ˆˆˆˆ
)ˆˆˆ
ZmnZ
nZm
41
ˆ
)22 Z
ˆ
)22 n
42
1
)ˆˆ()ˆˆ(
ˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆ
)ˆˆˆ1)(ˆˆ1ˆ(ˆˆ
nmnm
nnmnnmmm
nZZnmZnnZmmm
nZmZnmRG ZZ
1
)ˆˆ()ˆˆ(
ˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆ
)ˆˆ1ˆ)(ˆˆˆ1(ˆˆ
nmnm
nnmnnmmm
ZnnZnmZZmnmm
ZnmnZmGR ZZ
where
0ˆˆ nm , 1ˆˆ nm
mmZZm ˆˆˆˆˆ , nnZZn ˆˆˆˆ . Since
ZZZZ GRGG ˆ)ˆˆ(ˆ , RGRR ZZˆ)ˆˆ(ˆ ,
we get the relation
ZZ RG ˆˆ .
The two quantum circuits are equivalent. ______________________________________________________________________________ 14. Quantum circuits related to the controlled U-gate
(a) Quantum circuit with UnmGUˆˆ1ˆˆ .
Fig. C
(b) Q
Controlled-U
00
00
10
01
00
00
10
01
1ˆˆ mGU
Quantum cir
U gate with G
2221
1211
0
0
001
000
000
000
001
000
ˆˆ
uu
uu
Un
rcuit with R
mGU 1ˆˆ
21
11
00
00
000
000
u
u
mRU ˆ1ˆ
43
Un ˆˆ . U i
22
12
0
0
u
u
nU ˆˆ
is the 2x2 mmatrix.
Fig. Q
R
________15. Q
Fig. C
G
________
Quantum circ
21
11
0
00
0
01
ˆ1ˆ
u
u
mRU
__________Qauntum cir
Controlled-CN
0
0
0
1
0
0
0
1
ˆˆ mGCNOT
__________
cuit with RUˆ
22
12
0
01
0
00
ˆˆ
u
u
nU
___________rcuits relate
NOT with G
010
100
001
000
000
000
001
000
ˆˆ1 Xn
___________
Um ˆˆ1
__________ed to the con
mGCNOT ˆˆ
100
000
000
000
__________
44
nU ˆˆ . U is
___________ntrolled-CN
Xn ˆˆ1
0
1
0
0
___________
the 2x2 mat
__________NOT
__________
trix.
___________
___________
__________
__________
____
__
Fig. Q
G
G
________
Quantum ˆˆ
12 GG CNCNOT
0
0
0
0
0
0
0
1
ˆ
ˆˆ12
m
GG CNCNOT
__________
gate wit
ˆ1 mNOT
000
000
000
000
100
010
001
000
ˆ11
1
n
NOT
___________
th Contro
ˆ11 n
0010
0001
1000
0100
0000
0000
0000
0000
1X
__________
45
olled-CNOT
1ˆ X .
0
0
1
0
0
0
0
0
___________
T between
__________
n 1 an
___________
nd 2.
__________
with
__
Fig. Q
G
G
________
Quantum
GCNOT 1ˆ23
0
0
0
0
0
0
0
1
1
1ˆ23GCNOT
__________
gate wit
GCNOT 1ˆ
000
000
000
000
010
100
001
000
11ˆ
ˆ
m
GCNOT
___________
th Contro
m 11ˆ
0100
1000
0010
0001
0000
0000
0000
0000
ˆˆ Xn
__________
46
olled-CNOT
Xn ˆˆ .
0
1
0
0
0
0
0
0
___________
T between
__________
n 2 an
___________
nd 3.
__________
with
__
Fig. Q
G
G
________
16 R
Quantum
mGCNOT ˆˆ13
0
0
0
0
0
0
0
1
ˆˆ13 mGCNOT
__________
R-CNOT gat
gate wit
n11
000
000
000
000
100
010
001
000
ˆ11 n
___________
te with RCNOˆ
th Contro
X1 .
0100
1000
0001
0010
0000
0000
0000
0000
ˆ1 X
__________
mOT ˆ1
47
olled-CNOT
0
1
0
0
0
0
0
0
___________
nX ˆˆ
T between
__________
n 1 an
___
nd 3. with
Fig. Q
R
17. Q
(a) S
Fig. S
Quantum gate
0
0
0
1
0
0
0
1
ˆ1ˆ mRCNOT
Quantrum ci
wap gate w
wap gate wi
e with RCNOTˆ
001
010
100
000
000
010
000
000
ˆˆˆ nXm
ircuits relat
with GSWAPˆ
ith mGSWAPˆ
XmTˆˆ1
010
000
000
000
ted to the SW
nmm ˆˆˆ
nmm ˆˆˆ
48
nX ˆˆ
0
0
1
0
WAP gate
XmXn ˆˆˆˆ
nXmXn ˆˆˆˆ
mXnXnX ˆˆˆˆˆ
mXnXnX ˆˆˆˆˆ
m
m
G
(b) Q
Fig. Q
0
0
0
1
0
0
0
1
ˆˆ mGSWAP
Quantum cir
Quantum circ
100
001
010
000
000
000
000
000
ˆˆˆ nnm
rcuit ˆ12SWAPG
cuit includin
000
000
000
000
ˆˆˆˆ XnXmX
1ˆ2 SWAPG
g Swap gate
49
00
10
00
00
1
0
0
0
ˆˆˆˆ mXnX
1
e between 1 a
0
0
0
0
00
00
00
00
and 2. ˆSWAPG
000
000
010
000
ˆ12 SWAPP G
1
G
(c) Q
Fig. Q
0
0
0
0
0
0
0
1
ˆ
ˆˆ12
m
GG SWSWAP
Quantum cir
Quantum circ
000
000
100
010
000
000
001
000
ˆ1ˆ
1
nm
WAP
rcuit with G
cuit includin
1000
0100
0000
0000
0010
0001
0000
0000
ˆˆ1ˆ mXn
SWAPG 1ˆ23
g Swap gate
50
0
0
0
0
0
0
0
1ˆˆnXm
SWAPG
e between 2 a
ˆˆˆˆ mXnX
and 3. SWAPG
1
SWP G123 WAP
G
(d) Q
Fig. Q
0
0
0
0
0
0
0
1
1
1ˆ23GSWAP
Quantum ga
Quantum circ
000
000
000
000
100
001
010
000
1ˆˆ
ˆ
mm
GSWAP
ate with ˆSWG
cuit includin
1000
0010
0100
0001
0000
0000
0000
0000
1ˆˆ nn
13WAP
g Swap gate
51
1
0
0
0
0
0
0
0
ˆˆˆˆ nXmX
e between 1 a
ˆˆˆ1 mXnX
and 3. ˆSWAPG
mX
13P
52
10000000
00001000
00100000
00000010
01000000
00000100
00010000
00000001
ˆˆ1ˆˆˆˆ1ˆˆˆ1ˆˆ1ˆ
ˆ13
mXnXnXmXnnmm
GSWAP
18. Matrix representation of typical tri-qubits
10000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
111 ,
00000000
00000000
00000000
00000000
00001000
00000100
00000010
00000001
11m ,
00000000
00000000
00000000
00000000
00000000
00000000
00000010
00000001
1ˆˆ mm ,
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000001
ˆˆˆ mmm
53
00000000
01000000
00000000
00010000
00000000
00000100
00000000
00000001
ˆ11 m ,
00000000
00000000
00000000
00010000
00000000
00000000
00000000
00000001
ˆˆ1 mm
00000000
00000000
00100000
00000000
00000000
00000000
00000010
00000000
ˆˆ1 nm
10000000
01000000
00100000
00010000
00000000
00000000
00000000
00000000
11n ,
10000000
01000000
00000000
00000000
00000000
00000000
00000000
00000000
1ˆˆ nn ,
10000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
ˆˆˆ nnn
m
19 Q
(a) Q
Fig. T
1ˆˆ nm
Quantum ga
Quantum ga
Toffoli gate w
0000
0000
0000
0000
1000
0100
0000
0000
ates related t
ate toffoliG
with Gtoffoliˆ
0000
0000
0000
0000
0001
0000
0000
0000
to the Toffo
m 11ˆ
54
0
0
0
0
0
0
0
0
, n
oli gate
mn 1ˆˆ
0
0
0
0
0
0
0
0
1m
Xnn ˆˆˆ
000
000
000
000
000
000
000
000
X
0000
0000
0010
0001
0000
0000
0000
0000
55
X
I
I
I
000
000
000
000
01000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
01000000
10000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00100000
00010000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00001000
00000100
00000010
00000001
ˆˆˆ1ˆˆ11ˆˆ XnnmnmGtoffoli
(b) Quantum gate with toffoliR
In the expression of
321321321ˆˆˆ1ˆˆ11ˆˆ XnnmnmGtoffoli
we change the number of subscript as 1→3, 2→2, 3→1,
123123123ˆˆˆ1ˆˆ11ˆ Xnnmnm
This can be rewrirren as
nnXnmm
nnXnmmRtoffoli
ˆˆˆˆˆ1ˆ11
ˆˆˆˆˆ1ˆ11ˆ321321321
Fig. Q
We note
R
20. Q
(a) Q
Quantum gate
that
11Rtoffoli
Quantum ga
Quantum ga
e Rtoffoli 1ˆ
ˆ1ˆ mm
ate related to
ate with FreG
m 1ˆ1
ˆˆˆ nXn
o the Fredk
edkin m 1ˆ
56
Xnm ˆˆˆ
0
0
0
0
0
0
0
1
ˆˆ nn
kin gate
SWGn ˆˆ1
nn ˆˆ
0100
0000
0000
1000
0000
0010
0001
0000
WAP
0000
0100
0010
0001
1000
0000
0000
0000
Fig. F
G
where
G
(b) Q
redkin gate w
0
0
0
0
0
0
0
1
ˆ
ˆ
ˆˆ
m
m
mGFredkin
0
0
0
1
ˆˆ mGSWAP
Quantum ga
with FredkinG
0000
0000
0000
1000
0100
0010
0001
0000
ˆ11
(ˆ11
ˆ11
mn
n
n
100
001
010
000
ˆˆˆ nnm
ate with FreR
m 11ˆ
1000
0010
0100
0001
0000
0000
0000
0000
ˆˆˆ
ˆˆˆ(
ˆ
nmm
nmm
GSWAP
ˆˆˆˆ XnXmX
edkin
57
SWAPGn ˆˆ ’
ˆˆˆ
ˆˆˆ
nnn
XmXn
ˆˆˆˆ mXnX
’
ˆˆˆˆ
ˆˆˆˆ
nXmX
XnXnX
ˆˆˆˆ
)ˆˆ
XnXn
mX
ˆˆmX
Fig. M
R
21. Q
Modified Fre
0
0
0
0
0
0
0
1
11
11RFredkin
Quantum cir
dkin gate wi
0000
0000
0100
1000
0000
0010
0001
0000
ˆˆ1
ˆˆ1
mm
Gm SWA
rcuit equiva
ith RFredkinˆ
1000
0100
0000
0001
0010
0000
0000
0000
ˆˆˆ
ˆ
nnm
nAP
alence to the
58
Gm11
ˆˆˆˆ mXnn
e Fredkin ga
nGSWAP ˆˆ
ˆˆˆ nnXm
ate
ˆˆˆˆ mXnX n
59
nXmXnmXnXnnnnmmnm
XmnXnXnXnnXn
nmXmXnn
Xnmnnmmnmm
nXnXnXnmXnnXn
mnXmnnXmmnXmnXmXnn
mXmnnmmnnmXmmm
nXXnnmXnnnXmnmmn
nXmmmnXm
nXm
XnnmnmnXmRGR CNOTToffoliCNOT
ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ11ˆ
ˆˆˆˆˆˆˆˆˆˆˆˆ
ˆ1ˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆ1ˆ
)ˆˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆˆ1ˆ(
)ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆ1ˆ()ˆˆ1ˆ11(
)ˆˆ1ˆ11(
)ˆˆˆ1ˆˆ11ˆ()ˆˆ1ˆ11(ˆˆˆ
2
2
22
22
2323
where
0ˆˆ nm , 0ˆˆ mn , mm ˆˆ 2 , nn ˆˆ2 , 1ˆ 2 X ,
1ˆˆ nm ,
mXXn ˆˆˆˆ , nXXm ˆˆˆˆ , and
60
00100000
01000000
10000000
00010000
00000010
00000100
00001000
00000001
ˆˆ1ˆ11
ˆ1ˆ23
nXm
RR CNOTCNOT
,
01000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
ˆˆˆ1ˆˆ11ˆˆ XnnmnmGtoffoli
Thus we have
FredkinCNOTToffoliCNOT GRGR ˆˆˆˆ2323
where
G
22. Q
Fig.a Q
Fig.b S
0
0
0
0
0
0
0
1
ˆˆ mGFredkin
Quantum ci
Quantum circ
wap gate wi
0000
0000
0000
1000
0100
0010
0001
0000
ˆ11 mn
ircuit CNOTG
cuit with CNG
ith mGSWAPˆ
1000
0010
0100
0001
0000
0000
0000
0000
ˆˆˆ nmm
CNOTCNOTT GR ˆˆ
CNCNOTNOT GR ˆˆ
nmm ˆˆˆ
61
ˆˆˆ nnn
equivalent
NOT .
XmXn ˆˆˆˆ
ˆˆˆˆ nXmX
t to SWAPG
mXnXnX ˆˆˆˆˆ
ˆˆˆˆ XnXn
mX ˆ .
ˆˆmX
We show
R
G
Then we
G
which is
G
23. E
We s
Fig.a Q
which is
w that this qu
mRCNOT ˆ1ˆ
mGCNOT ˆˆ
get
GRG CNOTCNOTˆˆˆ
the same as
mmGSWAP ˆˆ
Equivalent q
how that the
Quantum circ
equivalent t
uantum circu
nXm ˆˆˆ ,
Xn ˆˆ1 .
mm
mGCNOT
ˆ
ˆ(
Xnnm ˆˆˆ
quantum cir
ese two quan
cuit with ˆ(X
to a new type
uit is equival
Xnnm
Xn
ˆˆˆ
1)(ˆˆ1
XnXmX ˆˆˆˆˆ
rcuits: X(
ntum circuits
(ˆ)1 XGX CNOT
e of quantum
62
ent to the SW
XmXnX
Xm
ˆˆˆˆ
ˆˆ1
mXnX ˆˆˆˆ .
XGCNOTˆ(ˆ)1
s are equival
)1ˆ X
m gate
WAP gate. W
nXmX
nmn
ˆˆˆˆ
1ˆ)(ˆ
n 1ˆ)1
lent to each o
We note that
Xn )ˆˆ
Xm ˆˆ
other.
t
63
Fig.b Quantum circuit with Xmn ˆˆ1ˆ . Controlled operation with a NOT gate being performed on the second qubit, conditional on the first qubit being set to zero.
We note that
XnmGCNOTˆˆ1ˆˆ ,
Then we have
Xmn
XXmXn
XXnXXmX
XXnXmX
XXnmXXGX CNOT
ˆˆ1ˆ
ˆˆˆ1ˆˆ
ˆˆˆˆ1ˆˆˆ
)ˆˆˆ1ˆˆ)(1ˆ(
)1ˆ)(ˆˆ1ˆ)(1ˆ()1ˆ(ˆ)1ˆ(
22
In fact we obtain
1000
0100
0001
0010
)1ˆ(ˆ)1ˆ( XGX CNOT .
24. Equivalence of two quantum circuits: XZ GHGH ˆ)ˆ1(ˆ)ˆ1(
We show that these two quantum circuits are equivalent to each other.
Fig.a Q
Fig.b E
G
The quan
1(
Note that
H
Quantum circ
Equivalent qu
mGZ 1ˆˆ
ntum circuit
ZGH 1(ˆ)ˆ1
t
ˆ(2
1ˆ 2 XH
cuit with 1(
uantum circu
Zn ˆˆ ,
can be repre
XG
m
m
H
ˆ
ˆ
ˆ
1(
1()ˆ
)ˆˆ)(ˆ ZXZ
1(ˆ)ˆ GH Z
uit with XG
mGX ˆˆ
esented by
Xn
HnH
HmH
mH
ˆˆ1
ˆˆˆ
ˆˆ)(ˆ
1ˆ)(ˆ
2
ˆˆ(2
1 22 ZX
64
)H .
Xnm ˆˆ1ˆ
HZH
HZn
HZn
ˆˆˆ
)ˆˆˆ
1)(ˆˆ
ˆˆˆˆ2 XZZX
X .
H )ˆ
1) X
H
where
X
25. E
We s
Fig.a Q
Fig.b E
X
Xi
X
X
XHZH
ˆ
ˆ(2
1
ˆ(2
1
ˆ(2
1
ˆ(2
1ˆˆˆ
XYYX ˆˆˆˆ
Equivalence
how that the
Quantum circ
Equivalent qu
YZiXYX
YiZ
ZXZZ
XZZ
ˆˆˆˆ
)1ˆ)(ˆ
ˆˆ)(ˆ
ˆ(ˆ)ˆ
Zi ˆ , ZY ˆˆ
of two quan
ese two quan
cuit with ˆ(H
uantum circu
ZY
Z
Z
)ˆˆ
)
)ˆ
)ˆ
2
XiYZZ ˆˆˆˆ
ntum circui
ntum circuits
ˆ)ˆ GHH CNOT
uit with RCNOˆ
65
X , XZ ˆˆ
its; HH ˆˆ(
s are equival
)ˆˆ( HH .
mOT ˆ1
YiZX ˆˆˆ
CNOT HGH ˆ(ˆ)
lent to each o
nX ˆˆ .
Y
CNOTRH ˆ)ˆ
other. T
66
We note that
XnmGCNOTˆˆ1ˆˆ , nXmRCNOT ˆˆˆ1ˆ .
The quantum circuit (Fig.a) is expressed by
ZHnHHmH
HXHHnHHmH
HXHHnHHHmH
HXHnHHmHH
HHXnmHHHHGHH CNOT
ˆˆˆˆ1ˆˆˆ
ˆˆˆˆˆˆ1ˆˆˆ
ˆˆˆˆˆˆˆˆˆˆ
)ˆˆˆˆˆˆˆ)(ˆˆ(
)ˆˆ)(ˆˆ1ˆ)(ˆˆ()ˆˆ(ˆ)ˆˆ(
2
or
nXm
ZXZ
ZXZX
ZXXHHGHH CNOT
ˆˆˆ1
)ˆ1(2
1ˆ)ˆ1(2
11
)ˆˆˆ11ˆ11(2
1
ˆ)ˆ1(2
11)ˆ1(
2
1)ˆˆ(ˆ)ˆˆ(
This agrees with
nXmRCNOT ˆˆˆ1ˆ .
((Note))
ZHXH ˆˆˆˆ , XHZH ˆˆˆˆ ,
)ˆ1(2
1ˆ Zm , )ˆ1(
2
1ˆ Zn .
)ˆ1(2
1)ˆˆˆˆ(
2
1ˆ)ˆ1(ˆ2
1ˆˆˆ 2 XHZHHHZHHmH ,
)ˆ1(2
1)ˆˆˆˆ(
2
1ˆ)ˆ1(ˆ2
1ˆˆˆ 2 XHZHHHZHHnH .
26. C
Entangle
G
where
G
H
m
The matr
G
Construction
d qubits
)1ˆ(ˆ HGCNOT
mGCNOT ˆˆ
ˆ(2
1ˆ XH
Hm 1(2
1ˆˆ
rix of GCNOT
)1ˆ(ˆ HGCNOT
n of the Bell
1[(2
1
1ˆˆ
1ˆ()
Z
Hm
m
Xn ˆˆ1 ,
)Z ,
HZ ˆ)ˆ ,
)1ˆ( H is g
01
0
0
01
2
1)
l's states
1(1ˆ)ˆ
ˆˆˆ
ˆ)(ˆˆ
HZ
XHn
HXn
iXZ ˆˆ
ˆˆnH
iven by
010
101
101
010
67
ˆˆ)ˆ1
)1
XHZ
Y ,
)ˆ1(ˆ2
1ZH
]X
1
0
0
1
2
100
0
1
1
0
2
101
2
110
0(2
1
1
0
0
1
0(2
1
0
1
1
0
0(2
1
1
0
0
1
)110
)011
)1100
68
________REFERE R.P. FeynG.J. MilbG.P. Ber
CJ. Stolze
VG. Benne
BD.C. Ma
InM. HayaM. Pavič
(SP. Kaye, M.L. BelN.D. MeL. Diosi,G. Benne
BG. JaegerD. McMaZ. MegliN.S. Yan
2S.M. Bar
2
111
__________ENCES
nman, Feynmburn, The Ferman, G.D.
Computers (Wand D. Sute
VCH, 2004). et, G. Casati
Basic Conceparinescu andnc., 2005). shi, Quantumčić, QuantumSpringer 200R. Laflamm
llac, A Shortrmin, Quant A Short Coet, G. Casati
Basic Tools ar, Quantum Iahon, Quantcki, Quantumnofsky and M008). rnett, Quantu
0(2
1
0
1
1
0
___________
man Lectureeynman Proc Doolen, R
World Scienter, Quantum
i, and G. Strpts (World Sd G.M. Mar
m Informatiom Computa06).
me, and M. Mt Introductiotum Computurse in Quan
i, and G. Striand Special TInformation tum Computm ComputinM.A. Mannu
um Informat
)0110
__________
es on Compucessor (PerseR. Mainieri,tific, 1998).Computing
rini, PrinciplScientific, 20rinescu, App
on An Introdation and Q
Mosca, An Inon to Quantuter Science Antum Informini, PrincipleTopics (WorAn Overvie
ting Explaineng without Mucci, Quantu
tion (Oxford
69
___________
utation (Addeus Books, 1 and V.I.
A Short Co
les of Quant005). proaching Q
duction (SpriQuantum Com
ntroduction tum InformatiAn Introduct
mation Theores of Quanturld Scientificew (Springered (Wiley-In
Magic Deviceum Computi
d University P
__________
dison-Wesley1998). Tsifrinovich
urse from T
tum Comput
Quantum Com
inger, 2006)mmunicatio
to Quantum ion (Oxford tion (Cambriry (Springer,um Computac, 2007). r, 2007). nterscience, es (MIT Presing for Com
Press, 2009)
___________
y, 1996).
h, Introducti
Theory to Exp
tation and In
mputing (Pe
). n: Theory a
Computing University Pidge, 2007) , 2007). ation and Inf
2008) ss, 2008).
mputer Scient
).
__________
ion to Qua
periment (W
nformation v
eason Educa
and Experim
(Oxford, 20Press, 2006)
formation vo
tists (Cambr
__
antum
Wiley-
vol. I:
ation,
ments
07). .
ol. II:
ridge,
70
A.D. Vos Reversible Computing Fundamentals, Quantum Computing, and Applications (Wiley-VCH, 2010).
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2010).
D.C. Marinescu and G.M. Marinescu, Classical and Quantum Information (Elsevier, 2010). E. Rieffel and W. Polak, Quantum Computing A General Introduction (The MIT Press, 2011). M. Fayngold and V. Fayngold, Quantum Mechanics and Quantum Information (Wiley-VCH,
2013). S. Aaronson, Quantum Computing since Democritus (Cambridge University Press, 2013). _____________________________________________________________________________ Jacques Salomon Hadamard;(8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations. _____________________________________________________________________________ Tommaso Toffoli is a professor of electrical and computer engineering at Boston University where he joined the faculty in 1995. He has worked on cellular automata and the theory of artificial life (with Edward Fredkin and others), and is known for the invention of the Toffoli gate. ______________________________________________________________________________ Edward Fredkin (born 1934) is a distinguished career professor at Carnegie Mellon University (CMU) and an early pioneer of digital physics. His primary contributions include his work on reversible computing and cellular automata. While Konrad Zuse's book, Calculating Space (1969), mentioned the importance of reversible computation, the Fredkin gate represented the essential breakthrough. In recent work, he uses the term digital philosophy (DP). During his career Fredkin also served on the faculties of MIT in Computer Science, was a Fairchild Distinguished Scholar at Caltech, and Research Professor of Physics at Boston University. ______________________________________________________________________________ David Elieser Deutsch, FRS (born 18 May 1953) is a British physicist at the University of Oxford. He is a non-stipendiary Visiting Professor in the Department of Atomic and Laser Physics at the Centre for Quantum Computation (CQC) in the Clarendon Laboratory of the University of Oxford. He pioneered the field of quantum computation by formulating a description for a quantum Turing machine, as well as specifying an algorithm designed to run on a quantum computer. He is a proponent of the many-worlds interpretation of quantum mechanics. ______________________________________________________________________________
