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Figure 6.7: The Toffoli gate.
6Inevitably,languagetendstogetsloppy,and”bit”willbeusedwhenitisreallythequbit
basisstates thatarerefereedto.
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6.2.8 Some practical ”machinery”
Therelationbetweenaquantumcircuitandthecorrespondingunitarymatrixis
not entirely straightforwardto figure out. For the benefit of the reader we here
record some ”practical tools of the trade” often left implicit in the literature.
Gates acting independently
Let G denote a general single qubit gate. Then the action of G on the j-th
qubit on a multi-qubit quantum state is denoted by G(j) and defined by
G(j)x x x x = x x x G(j)x x x . (6.31)
1 2 j n 1 2 j 1 j j+1 n
| ··· ··· i | ··· − i⊗ | i ⊗| ··· i
(cid:0) (cid:1)
The generalization to several single qubit gates acting on different qubits
in the register is straightforward. In particular, specializing to the case of two
single qubit gates acting independently on a two-qubit state, we have
A(1) B(2)x x =A(1)B(2)x x = A(1)x B(2)x . (6.32)
1 2 1 2 1 2
⊗ | i | i | i ⊗ | i
(cid:0) (cid:1) (cid:0) (cid:1)
x A
1
x B
2
Figure 6.8: Two gates acting independently.
In order to work out this equation explicitly, let the states be given by
α α
x =α 0 +β 1 = 1 , x =α 0 +β 1 = 2 , (6.33)
| 1 i 1 | i 1 | i (cid:18)β 1(cid:19) | 2 i 2 | i 2 | i (cid:18)β 2(cid:19)
so that the composite state is
α α
1 2
α β
x x = 1 2. (6.34)
| 1 2 i β 1α 2
β β 
 1 2
Next, let the operators be
a a b b
A= 11 12 , B = 11 12 . (6.35)
(cid:18)a 21 a 22(cid:19) (cid:18)b 21 b 22(cid:19)
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The right hand side of equation (6.32) becomes
a a α b b α
11 12 1 11 12 2 =
(cid:18)a 21 a 22(cid:19)(cid:18)β 1(cid:19)⊗(cid:18)b 21 b 22(cid:19)(cid:18)β 2(cid:19)
(a α +a β )(b α +b β )
11 1 12 1 11 2 12 2
(a α +a β )(b α +b β )
 11 1 12 1 21 2 22 2 . (6.36)
(a α +a β )(b α +b β )
21 1 22 1 11 2 12 2
(a α +a β )(b α +b β )
 21 1 22 1 21 2 22 2 
The operatorproductin the left handside of(6.32) mustbe defined so that
it reproduce this expression. In analogywith the definition of the product of
⊗
vectors in (6.34), it is natural to define
a B a B
A B = 11 12 =
⊗ (cid:18)a 21B a 22B(cid:19)
a b a b a b a b
11 11 11 12 12 11 12 12
a b a b a b a b
 11 21 11 22 12 21 12 22. (6.37)
a b a b a b a b
21 11 21 12 22 11 22 12
a b a b a b a b 
 21 21 21 22 22 21 22 22
Then the left and side of (6.32) becomes
a b a b a b a b α α
11 11 11 12 12 11 12 12 1 2
a b a b a b a b α β
 11 21 11 22 12 21 12 22 1 2 (6.38)
a b a b a b a b β α
21 11 21 12 22 11 22 12 1 2
a b a b a b a b β β 
 21 21 21 22 22 21 22 22 1 2
Multiplyingout(6.36)and(6.38)anddoingsomecarefulbookkeeping,shows
the equality of these two expressions.
Two special cases are useful to note in order to gain some intuition. When
A=I, the unity matrix, we get in block diagonal form
B 0
I B = . (6.39)
⊗ (cid:18) 0 B(cid:19)
On the other hand, when B =I
a 0 a 0
11 12
0 a 0 a
A I = 11 12. (6.40)
⊗ a 21 0 a 22 0
 0 a 0 a 
 21 22
Note that general controlled operations cannot be achieved by direct prod-
ucts of operators, i.e. by operators acting independently. This can be seen by
inspecting the explicit product A B in 6.37.
⊗
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