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quantum

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print;
print "SIMULASI PENCARIAN KUANTUM MENGGUNAKAN ALGORITMA GROVER";
print;
input "Masukkan bilangan bulat yang ingin dicari:",bil;
algoritma(bil);
print;
print "-----------------------------------------";
}
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Unifying Quantum Computation with Projective
Measurements only and One-Way Quantum Computation
Philippe Jorrand, Simon Perdrix
LeibnizLaboratory
4002 46 avenueF´elix Viallet 38000 Grenoble, France
philippe.jorrand@imag.fr, simon.perdrix@imag.fr
rpA
Abstract. Quantum measurement is universal for quantum computa-
tion(Nielsen[4],Raussendorf[7,8]).Twomodelsforperformingmeasure-
ment-basedquantumcomputationexist:theone-wayquantumcomputer
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was introduced by Briegel and Raussendorf [7], and quantum computa-
tion via projective measurements only by Nielsen [4]. The more recent
1v5214040/hp-tnauq:viXra development of thissecond model is based on state transfers [6] instead
ofteleportation. From thisdevelopment,afinitebutapproximatequan-
tumuniversalfamilyofobservablesisexhibited,whichincludesonlyone
two-qubit observable, while others are one-qubit observables [6]. In this
article, an infinite but exact quantum universal family of observables is
proposed, includingalso only one two-qubitobservable.
The rest of the paper is dedicated to compare these two models of
measurement-based quantum computation, i.e. one-way quantum com-
putation and quantum computation via projective measurements only.
From this comparison, which was initiated by Cirac and Verstraete [9],
closer and more natural connections appear between these two models.
These close connections lead to a unified view of measurement-based
quantum computation.
1 Introduction
Quantummeasurementisuniversalforquantumcomputation(Nielsen[4],Raus-
sendorf[7,8]).Thereexisttwomodelsforperformingquantumcomputationwith
measurements only: one-way quantum computation, introduced by Briegel and
Raussendorf [7], and quantum computation via projective measurements only,
introduced by Nielsen [4] and improved successively by Leung [2,3] and Perdrix
[6]. One-way quantum computation consists in performing one-qubit measure-
ments on a lattice of qubits initialized in a specific entangled state: the cluster
state, whereas quantum computation via projective measurements only consists
in simulating any unitary transformation using a teleportation-like scheme.
These two families of measurement-based quantum computations have been
recently linked by Cirac and Verstraete [9], who introduced a concept of vir-
tual qubits. We give another approach to closer connections between these two
families by considering on the one hand, one-way quantum computation, and
ontheotherhand,quantumcomputationviameasurementsonlybasedonstate
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transfer[6].Theseconnectionsareestablishedbyanalyzinghowthepreparation
of the cluster state can be obtained starting from a non-entangled state, while
usingmeasurementsonly.Theseconnectionsleadtoanaturaltranslationofany
(one dimensional) one-way quantum computer into a sequence of generalized
state transfers and vice versa.
2 Survey of quantum computation via measurements
only based on state transfer
The computation introduced by Nielsen [4], developed by Leung [2], is based
on teleportation. State transfer is an alternative to teleportation for purpose of
computation. State transfer needs less measurements and less auxiliary qubits
than teleportation, but in return, state transfer cannot replace teleportation in
non-local applications.
Figure 1 - State Transfer
Measurements are defined by the Pauli observables X, Y and Z.
For a given qubit a and an auxiliary qubit b, the sequence of measurements
{X(b),Z(a)⊗Z(b),X(a)} (see fig. 1), transfers the state |φi=α|0i+β|1i from
a to b up to a Pauli operator which depends on the classical outcomes of the
measurements.
Figure 2 - State Transfer with additional unitary transformations V1 and V2.
Figure 3 - Generalized State Transfer.
By modifying the measurements performed during the state transfer, all 1-
qubitunitarytransformationsU maybe simulateduptoaPaulioperator,using
generalized state transfers, see fig. 2 and fig. 3 with V = I and V = U. This
1 2
step of simulation of U (i.e. the simulation of U up to a Pauli operator σ) is
followedby astageofcorrectionwhichconsistsinsimulatingσ.The readermay
reffer to [6] for details on the stage of correction.
Generalizedstate transfers which simulate H, HS and HT are given in fig.
†
4, 5 and 6, where:
3
1 1 1 0 10
H = 1 , T = , S =
√2(cid:18)1−1(cid:19) (cid:18)0ei 4π (cid:19) (cid:18)0 i(cid:19)
Figure 4 - Step of simulation of H: V1=Id and V2 =H
(note that for all σ, there exists σ′ such that Hσ=σ′H)
Figure 5 - Step of simulation of HS†: V1=S† and V2 =H.
Figure 6 - Step of simulation of HT: V1=T and V2 =H.
For a given 2-qubit register a,b and one auxiliary qubit c, the sequence of
measurements {Z(c),Z(a)⊗X(c),Z(c)⊗X(b),X(c)} (see fig. 7), simulates the 2-
qubit unitarytransformationCNot onthe state |φi ofa,b up toa 2-qubitPauli
operator which depends on the classical outcomes of the measurements, where:
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