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R y(θ)=e −iθY/2 =cos 2I −isin 2Y = (cid:18)sinθ2 − cosθ2 (cid:19), (6.19)
2 2
θ θ e iθ/2 0
R z(θ)=e−iθZ/2 =cos 2I −isin 2Z = − eiθ/2(cid:19). (6.20)
(cid:18) 0
The matricesinthe righthandsides ofthese equationscanbe derivedusing
the following general formula for functions of Pauli matrices
f(θ)+f( θ) f(θ) f( θ)
f(θn σ)= − I + − − n σ, (6.21)
· 2 2 ·
wherenˆ =(n ,n ,n )isathreedimensionalnormalvectorandσ =(σ ,σ ,σ )=
x y z x y z
(X,Y,Z).
From the definition of the rotation operators, it is clear that the following
equations hold
R (α)+R (β)=R (α+β) where i x,y,z . (6.22)
i i i
∈{ }
Some more useful identities are
XR (θ)X =R (θ), (6.23)
x x
XR (θ)X =R ( θ), (6.24)
y y
−
XR (θ)X =R ( θ). (6.25)
z z
−
Obviously, there are lots of such simple identities for single qubit gates.
6.2.5 The rotation operators and the Bloch sphere
The rotation operators can indeed be interpreted as rotations around the x, y
andz axesrespectively,inathreedimensionalspace. Insuchaninterpretation,
a general qubit ψ can be represented as
| i
θ θ
ψ =cos 0 +eiφsin 1 , (6.26)
| i 2| i 2| i
with angles θ and φ defined as in figure 6.5.
The basis vectors 0 and 1 are represented by (0,0,1) and (0,0, 1) (cor-
| i | i −
responding to θ =0,φ=0 and θ =π,φ=0), respectively.
Note that this representation (6.26), of the qubit follows from the general
form
ψ =α0 +β 1
| i | i | i
by writing α=eiγcosθ and β =eiγeiφsinθ so that
2 2
129
z
y
q
j y
x
Figure 6.5: The Bloch sphere.
θ θ
ψ =eiγ(cos 0 +eiφsin 1 )
| i 2| i 2| i
and then dropping the overallunphysical phase factor eiγ. Thus the number of
real physical degrees of freedom of a qubit is 2.
A general rotation by an angle α about an axis nˆ =(n ,n ,n ) is given by
x y z
the operator R (α)
nˆ
α α
R (α)=exp( iαnˆ σ/2)=cos I isin (n X +n Y +n Z). (6.27)
nˆ x y z
− · (cid:16)2 (cid:17) − (cid:16)2 (cid:17)
6.2.6 Single qubit phase-shift operators
The following phase-shift operators are sometimes useful
eiα 0
P(α)= =eiαI, (6.28)
(cid:18) 0 eiα(cid:19)
1 0
E(α)= =P(α/2)R (α). (6.29)
(cid:18)0 eiα(cid:19) z
6.2.7 Some special controlled operations
Two of the most important and useful controlled operations are the two-qubit
CNOT-gate and the three-qubit Toffoli-gate.
130
Controlled NOT
The CNOT-gateis anexample ofa two-qubitcontrolledoperation. It alsogoes
underthename(quantum)XOR.Itsmatrixrepresentationinthecomputational
basis is
1 0 0 0
0 1 0 0
CNOT=Λ (X)=  (6.30)
1 0 0 0 1
0 0 1 0
 
c c
t t c
Figure 6.6: Controlled NOT gate.
The CNOT-gate performs a NOT (i.e. an X) operation on the target bit t
conditioned on the control bit c being 1.6
Toffoli gate
The Toffoli gate is Λ (X). As noted in chapter 2, it is universal for classical
2
reversible computation.
a a
b b
t t (ab)