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EvenifthepreviousIsingtransformationhasinterestingproperties,thisuni-
tary transformation has to be simulated using quantum measurements only, in
order to get a relevant comparison between the one-way quantum computer
andthemodelbasedonstatetransfers.Sinceanyunitarytransformationcanbe
simulatedusingquantummeasurementsonly,eachC whichcomposestheIsing
Z
transformation can be simulated with measurements, but this simulation needs
an additional auxiliary qubit [6]. Therefore one may wonder if the transforma-
tion which creates the initial cluster state can be simulated without additional
auxiliary qubits.
4.1 Simulation of CZ on |φi⊗|+i without auxiliary qubit
Figure 9 - Simulation of CZ on |φi⊗|+i without auxiliary qubit
6
Lemma 1. For a given qubit a in an unknown state |φi and a given qubit b
in the state |+i, the sequence of measurements {Z(b),Z(a) ⊗X(b)} (see fig. 9)
simulates the unitary transformation C on |φi⊗|+i up to a two-qubit Pauli
Z
operator.
Proof. If |φi = α|0i+β|1i and if the outcome of Z(b) is i ∈ {−1,1}, then the
state |ψ i of the register a,b after this measurement is:
1
1−i
|ψ i=(I ⊗σ 2 )[α|00i+β|10i].
1 x
IftheoutcomeofZ(a)⊗X(b) isj ∈{−1,1},thenthestate|ψ ioftheregister
2
a,b after this measurement is:
1−i 1−j
|ψ i= 1 (σ 2 ⊗σ 2 )[α|00i+α|01i+β|10i−β|11i].
2 √2 z z
Since C (|φi⊗|+i) = α|00i+α|01i+β|10i−β|11i, the 2-qubit unitary
Z
transformation C is simulated on |φi⊗|+i up to a 2-qubit Pauli operator. (cid:3)
Z
4.2 Creation of a one-dimensional Cluster State
In order to create the initial cluster state on a one-dimensional n-qubit lattice
composed of a unique input qubit, a cascade of C can be performed as follow:
Z
Figure 10 - Cascade of CZ for creating the initial cluster state |φCi
For each C of the previous cascade the state of the second input qubit is
Z
|+i, thus, according to Lemma 1, the previous cascade of C is simulated by
Z
the following cascade of measurements. Note that this simulation requires no
additional auxiliary qubit.
Figure 11 - Cascade of measurements for creating the initial cluster state |φCi up to a
Pauli operator
7
5 Executions on a One-Way Quantum Computer
An execution on a one-way quantum computer is a sequence of one-qubit mea-
surements on a cluster state. For instance, for a given five-qubit cluster state, if
the first qubit is considered as an input qubit |φi and if the sequence of mea-
surements{X,Y,Y,Y}isperformedonthefirstfourqubits(seefig12),thenthe
state of the last qubit is σH|φi. Thus the one-way quantum computer of figure
12 simulates the Hadamard transformation.
Figure 12 - Simulation of the Hadamard transformation
Ifthephaseofpreparationoftheclusterstateandthephaseofexecutionare
bothrepresented(seefig13a),thenthemeasurementsimpliedinthepreparation
of the cluster state and those implied in the execution can be decomposed into
a succession of generalized state transfers (see fig 13b).
(a) (b) (c)
Figure 13 - Execution on a one-way quantum computer
This decomposition offers a natural translation from any one-dimensional
one-way quantum computer to quantum computation via projective measure-
ments only. Moreover a straightforwardinterpretation of the action of any one-
dimensional one-way quantum computer is obtained. For instance, the one-way
quantum computer of figure 12 can be decomposed (see fig 13c) into a step of
simulation of H (fig 4), and three steps of simulation of HS (fig 5), thus the
†
actionU ofthisone-wayquantumcomputerisU =(HS )(HS )(HS )(H)=H.
† † †
Figure 14 - Simulation of S†
Similarly,theactionU oftheone-wayquantumcomputerpresentedinfigure
14 is U =(H)(HS )(H)(H)=S .
† †
8
More generally, the measurements allowed in a one-way quantum compu-
tation are in the basis B(ϕ) = {|0 i+eiϕ |1 i,|0 i−eiϕ |1 i} for all ϕ. The observable
√2 √2
associatedwithB(ϕ)isO(ϕ)=cos(ϕ)X+sin(ϕ)Y.EachO(ϕ)-measurementis
associatedwith a generalizedstate transfer with V =R (−ϕ) and V =H (see
1 zˆ 2
fig 8).
Thus the action U of the one-way quantum computer of figure 15 is U =
HR (−ζ)HR (−η)HR (−ξ)H.
zˆ zˆ zˆ
Figure 15 - Simulation of a general one-qubit unitary transformation
The connexions between the one-way quantum computer and generalized
state transfer may also be used for designing new one-way quantum computers.
For instance the one-way quantum computers introduced in figure 16 simulate
respectivelyH andS whilerequiringlessqubitsthanthoseintroducedbyBriegel
and Raussendorf [7,8].
Figure 16 - Left: simulation of H - Right: simulation of (HS†)(HS†)(H)=S
6 Conclusion
In this paper, we have introduced an exact quantum universal family of ob-
servables including only one two-qubit observable, while others are one-qubit