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CNot=
0001
 
0010
 
Figure 7 - Step of simulation of CNot
3 Universal families of observables
There exist two types of quantum computation universalities:
4
– AfamilyF ofoperators(unitarytransformationsormeasurements)isquan-
tumuniversal iffforalloperatorO,thereexistsacombination(compositions
and tensor products) of some elements of F which simulates O.
– A family F of operators is approximatively quantum universal iff for all
operatorO andforallǫ>0,thereexistsanoperatorO andacombination
1 2
of some elements of F which simulates O , with ||O −O ||≤ǫ.
2 1 2
Since the family of unitary transformations F = {H,HT,CNot} is ap-
0
proximatively quantum universal [5,1], the family of observables F = {Z ⊗
1
X,X,Z,X Y} is also approximatively quantum universal[6]. The approximate
√−
2
quantum universality of F is based on the ability to simulate each element of
1
F using elements of F only (see fig. 4, 6 and 7).
0 1
Theorem 1. The family of observables F ={Z⊗X,Z,cos(θ)X+sin(θ)Y,θ ∈
2
[0 2π]} is quantum universal.
Proof. Since the family of unitary transformations F = {CNot}∪U (where
3 1
U is the set of all one-qubit unitary transformations)is quantum universal,the
1
quantum universality of F is reduced to the simulation of each element of F .
2 3
The proof consists in exhibiting a step of simulation (i.e. a simulation up to a
Paulioperator)of eachoperatorof F . The stages of correction,omitted in this
3
proof, are presented in [6] for F .
1
A step of simulation of CNot is presented in figure 7. For a given U ∈ U ,
1
U can be decomposed into three successive elementary rotations about the zˆ, xˆ
and zˆ axes in the Bloch sphere. The elementary rotation R can be expressed
xˆ
using an elementary rotation R and the Hadamard transformation H:
zˆ
∀ϕ,R (ϕ)=HR (ϕ)H.
xˆ zˆ
Thus for all one-qubit unitary transformation U, there exist ϕ ,ϕ ,ϕ such
1 2 3
that U =R (ϕ )HR (ϕ )HR (ϕ ), where
zˆ 3 zˆ 2 zˆ 1
1 0
R (ϕ)=
zˆ (cid:18)0eiϕ(cid:19)
So U can be decomposed into 4 operators:
U = (H)(HR (ϕ ))(HR (ϕ ))(HR (ϕ ))
zˆ 3 zˆ 2 zˆ 1
U4 U3 U2 U1
|{z}| {z }| {z }| {z }
A step of simulation of U = H is presented in figure 4, and the simulation of
4
U =HR (ϕ ),i∈[1,3]is obtainedusing the generalizedstate transferof figure
i zˆ i
3.SinceR (ϕ) ZR (ϕ)=Z andR (ϕ) XR (ϕ)=cos(ϕ)X−sin(ϕ)Y,itcomes
zˆ † zˆ zˆ † zˆ
the following step of simulation of HR (ϕ):
zˆ
Figure 8 - Step of simulation of HRzˆ(ϕ): V1=Rzˆ(ϕ) and V2=H.
(note that for all σ, there exists σ′ such that Hσ=σ′H)
5
Thus the simulation of any one-qubit unitary transformationcan be decom-
posedintofourunitarytransformationssuchthateachoftheseunitarytransfor-
mations can be simulated using only observables of F . Therefore the family of
2
observablesF ={Z⊗X,Z,cos(θ)X+sin(θ)Y,θ ∈[0,2π]}isquantumuniversal.
2
(cid:3)
4 The secret of the One-Way Quantum Computer is
hidden in the initial cluster state
One-wayquantumcomputationconsistsinmeasuringqubitafter qubita lattice
ofqubits,initiallypreparedinanentangledstate:theclusterstate.Thisisaone-
way computationbecausetheentanglementisconsumedstepbystep.Therefore
the creation of the intial cluster state is a crutial point.
In order to create the initial cluster state on a given lattice of qubits, the
following preparationis performed:
– Some qubits of the lattice are input qubits, i.e. qubits which are in an un-
knownstate|φi,othersareauxiliaryqubits.Eachauxiliaryqubitisinitialized
in the state |+i= 1 (|0i+|1i).
√2
– An Ising transformationis applied on the whole lattice. This Ising transfor-
mationisequivalenttotheapplicationofthe2-qubitunitarytransformation
Controlled-Z (C ) on each pair of neighboring qubits, where:
Z
100 0
010 0 
C =