Arbitrary state controlled-unitary gate by adiabatic passage

0.18 MB·English

by  X. Lacour

Arbitrary state controlled-unitary gate by adiabatic passage
X. Lacour1, N. Sangouard2, S. Gu´erin1, and H.R. Jauslin1
1Laboratoire de Physique, Universit´e de Bourgogne,
UMR CNRS 5027, BP 47870,
21078 Dijon Cedex, France
2 Fachbereich Physik, Universit¨at Kaiserslautern,
67653, Kaiserslautern, Germany
(Dated: February 1, 2008)
Weproposea robustscheme involvingatoms fixedin an optical cavitytodirectly implement the
universalcontrolled-unitarygate. Thepresenttechniquebasedonadiabaticpassageusesnoveldark
states well suited for the controlled-rotation operation. We show that these dark states allow the
robust implementation of a gate that is a generalisation of the controlled-unitary gate to the case
6 wherethecontrolqubitcanbeselectedtobeanarbitrarystate. Thisgatehaspotentialapplications
0 totherapidimplementationofquantumalgorithmssuchasoftheprojectivemeasurementalgorithm.
0 This process is decoherence-free since excited atomic states and cavity modes are not populated
2 duringthedynamics.

n
PACSnumbers: 03.67.Lx,32.80.Qk
a
J

7 I. INTRODUCTION sage (f-STIRAP) [5, 6] in a tripod-type system [7] has
2
been proposed in Ref. [8]. A multi-controlled-unitary

gate acting on qubits fixed in an optical cavity has been
1 The realization of a universal quantum circuit is a
v great challenge in quantum information science. It is proposed in Ref. [9], but with an undesirable phase gate
3 that has to be compensated. The latter proposition is
known that an arbitrary quantum computation can be
8 based on the two-qubit adiabatic transfert described in
performed by combining quantum gates,i.e. unitary op-
1 Ref. [10] and f-STIRAP.
erators acting on qubits, that form a universal set. We
1
0 distinguish two types of universal sets. The first one is Since in experimental implementations of quantum
6 composedofageneralone-qubitgate[correspondingtoa computations the errors grow with the number of quan-
0 generaloperatorU of SU(2)] and a two-qubitentangling tum gates involved, it is advantageous to implement di-
/ gate [1]; the secondtype is composedby a single kindof rectlycertaingatesinsteadofrelegatingthemtoacombi-
h
p gates, called universal gates, like the controlled-unitary naisonofelementarygates,asillustratedinRef.[13]with
- gate (C-U) [2]. the direct implementation of the SWAP gate. There is
t a double advantage: to reduce the errors with a smaller
n In order to make quantum computations, the imple-
a mentations of these quantum gates have to be robust, number of gates and to decrease decoherence effects by
u reducing the computational time.
i.e. they have to be insensitive to fluctuations or to par-
q
tialknowledgeofexperimentalparameters. Furthermore In this paper we propose a direct implementation by
:
v theyhavetobeinsensitivetodecoherenceeffects,suchas adiabatic passage along dark states of a arbitrary state
Xi spontaneous emission. Those conditions can be fulfilled controlled-unitarygate usingonlysevenpulses. Thisgate
if the qubit is encoded in atomic metastable states, and can be writen as
r
a if the gates are implemented by adiabatic passage along
dark states, i.e. instantaneous eigenstates with time in- 11 0
(1)
dependent eigenvalue (equal to the energy of the ground 0 U
(cid:20) (cid:21)
states) and with zero projection on the excited states.
However,adiabaticityisnotsufficienttoinsurethe ro- in the basis φ 0 , φ 1 , φ 0 , φ 1 , where U ( re-
nc nc c c
{| i | i | i | i}
bustness of certainquantum gates. The parametersthat spectively 11) are an unitary (the identity) operator of
determinetheactionofthegatesonqubits,liketheargu- SU(2),φ anarbitrarycontrolstateofthefirstqubit,φ
c nc
ment of the rotation gate or the phase of the controlled- itsorthogonalstate. Thisprocessisbasedondarkstates
phase gate, have to be controlled with high accuracy to generalisingthoseoff-STIRAPandofRef.[10],andpar-
performcomputation[3]. We thereforehavetoavoidthe ticularly adapted for a two-qubit controlled-rotationop-
use of the non-robust dynamical phases, depending on eration. Thisgatehaspotentialapplicationsfortherapid
the area under the adiabatic pulses, and of geometric realisationofquantumalgorithms. We showforinstance
phases [4] that require the control of a loop in the pa- that it allows a direct implementation of the projective
rameterspace. Analternativetechniqueconsistsinusing measurement algorithm. The paper is organised as fol-
elliptic polarisation and static phase difference of lasers, lows. The system is introduced in section II. The defini-
which can be easily controlledexperimentally. Following tionandthedynamicsofthegateareshowninsectionIII.
this idea, the implementation of a general single qubit Section IV is devoted to the numerical demonstration,
gatebasedonfractionalstimulatedRamanadiabaticpas- and section V presents some conclusions.
2
II. SYSTEM represented on Fig. 2, where U,V are unitary operators
and q , q the control and target qubit.
0 1
| i | i
e 0,0
| i≡| i q0 V V†
| i •
q1 U
| i
Ω (σ+) g(π) Ω (σ )
0 1 −
FIG. 2: Quantum circuit representing the decomposition of
the arbitrary state controlled-unitary gate from elementary
gates.
0 1, 1 a 1,0 1 1,1
| i≡| − i | i≡| i | i≡| i
FIG. 1: Schematic representation of a Zeeman sublevel sys-
temwiththepolarisation ofthefieldsdrivingeachtransition
A. Background
associated withthestates|J, MiforJ =0, 1. Forthegener-
ation ofsingle qubitgates, each transition is drivenbylaser;
forthetwo-qubitstatesrotationacavityfielddrivesthetran- We use the following notation: the states of the sys-
sition |ai−|ei, theothers are drivenby lasers. temarewritten s1s2 n ,wheretheindicess1, s2 denote
| i| i
respectively the states of the first and second qubit, and
n the photon number state of the cavity-mode.
As in Refs. [9, 10, 11] we consider a register of
qubits fixed in an optical cavity. Each qubit is en- In Ref. [10], a robust tool has been established to
codedinatripod-typeZeemansystemcomposedofthree drive a complete population transfer between two-qubit
metastable states and one excited state, and can be ad- states s1s2 0 . For instance, if the states a of each
| i| i | i
dressed individually by laser fields. The qubits interact atom are coupled by the cavity, then a counter-intuitive
witheachotherthroughthecavitymode[10]. Thesingle pulse sequence Ω(2),Ω(1) induces the populationtransfer
1 0
qubit gates can be implemented in this system by cou- 0a 0 a1 0 . Such a coherent manipulation of the
| i| i → | i| i
pling the three ground states by lasers [12]. We choose two-atomstate s1s2 0 offersvariouspossibilitiesforthe
| i| i
heretoidentifytheZeemansublevels J =1, M = 1 to implementation of two-qubit quantum gates. This tool
| ± i
the computationalstates 0 and 1 . The ancillary state is at the heart of the SWAP gate in Ref. [13] and of the
| i | i
J =1, M =0 andtheexcitedstate J =0, M =0 are CNOT gate in Ref. [14].
| i | i
respectively denoted a and e (see Fig. 1). As STIRAP [5] can be extended to f-STIRAP [6], one
| i | i
During the dynamics, the excited state e is coupled can extend this process to the creation of coherent su-
| i
to the computational states 0 and 1 by circularly po- perpositions of the two-atomstates. In this case, we use
larisedlaserfields of Rabifre|quiencie|s Ωi(k) andΩ(k) (the three laser fields ofthe form E(k)cos(ωt+φ(k)) [i=0,1,
0 1 i i
superscriptklabelstheatoms),andtotheancillarystate k = 1,2], coupling respectively the states 1 and e of
| i | i
a by the linearly polarised cavity mode of Rabi fre- the first atom, the states 0 e and 1 e of the
|quiencyg(k) whichistimeindependent. Eachfieldisone- second atom. In the intera|citi−on|piicture|aind−u|nider the
photon resonant, and their polarisation and frequencies rotating wave approximation the Hamiltonian is given
are such that they drive a unique transition. The choice by
ofthe polarizationsis guided by geometricalconstraints,
wtohtehnewceaviimtypoasxeist.hTathetheessleansteiraslpprooinptagisattehaotrtthhoegopnoalallry- H = Ω(11)e−iφ(11)|e(1)ih1(1)|+g(1)aˆ|e(1)iha(1)|
isation of the cavity mode is orthogonal to the plane of + Ω(02)e−iφ(02)|e(2)ih0(2)|+Ω(12)e−iφ(12)|e(2)ih1(2)|
the circular polarisation of the lasers. + g(2)aˆe(2) a(2) +h.c. (2)
| ih |
with aˆ the anihilation operator of the cavity mode, Ω(k)
III. DYNAMICS i
the Rabi frequencies associated to the laser amplitudes
E(k).
In this section we describe the sequence of pulses i
Theinteractionofthe secondqubitisparametrisedby
that will permit to generate a arbitrary state controlled-
the following laser Rabi frequencies:
unitary gate (Cas-U). We recall that a standard
controlled-unitarygate yields a unitary operationonthe
Ω(2)(t) = Ω(2)(t) sinθ, (3a)
target qubit if the control qubit is in state 1 . We de- 0
fine its generalisation as follows. The arb|itriary state Ω(2)(t) = Ω(2)(t) cosθ, (3b)
1
controlled-unitarygate yields a unitary operationonthe
target qubit if the control qubit is in an arbitrary pre- which can be generated in a robust way using a single
selected state that we can choose robustly by the laser laserofappropriateellipticpolarisation. Werefertosuch
pulseparameters. Itisequivalenttothe quantumcircuit a laser as Ω(2) in what follows. We define for the second
3
qubit one non-coupled and three coupled states as constant ratio. It guarantees that there is no geometric
phase, which would be detrimental for robustness.
|ΦΦnci == scionsθθeeiiφφ((22))|11i+−csoinsθθ|00i,, ((44ba)) noTthpeardtiacrikpastteaitnesth|Ψe1d,3yinaamreicsst.atFiuorntnhaerrymsotraet,essinacnedg(dko)
| ci | i | i is time independent, we remark that for an arbitrary
|Φc2i = sinθeiφ(2)|1i−cosθ|0i, (4c) pulse sequence involving Ω(2) and Ω(11), the initial pop-
Φc3 = cosθeiφ(2) 1 +sinθ 0 , (4d) ulation of the states |00i|0i,|01i|0i,|10i|0i,|11i|0i stays
| i | i | i always unchanged at the end of such a process.
where φ(2) =φ(2) φ(2). Thedarkstate|Ψ4iistheprincipaleigenstateinvolved
1 − 0 inthe gateoperation. It evolvesaccordingtothe linkage
The hamiltonian admits the following dark states, i.e.
pattern represented in Fig. 3. There are similarities be-
instantaneouseigenstatesofnulleigenvaluesandnotcon-
nected to excited atomic states, which belong to three
ae 0 ea 0
orthogonalsubspaces: | i| i | i| i
Ψ = 0Φ 0 , (5a)
1 nc
| i | i| i
Ψ2 = cosη 0Φc 0 sinηe−iφ(02) 0a 1 , (5b) Ω(02) Ω(12) g(2) g(1) Ω(11)
| i | i| i− | i| i
for the first one,
a0 0 a1 0 aa 1 1a 0
Ψ3 = aΦnc 0 , (6a) | i| i | i| i | i| i | i| i
| i | i| i
Ψ4 = sinϕaΦc 0 +cosψcosϕei(φ(11)−φ(02)) 1a 0 FIG. 3: Linkage pattern associated to thedark state |Ψ4i.
| i | i| i | i| i
sinψcosϕe−iφ(02) aa 1 , (6b)
− | i| i tweenthislinkagepatternandtheoneofthetripod-type
system used for single qubit rotations [8]. It shows that
for the second one, and
the states a0 0 , a1 0 , 1a 0 can respectively evolve
| i| i | i| i | i| i
|Ψ5i= cosψeiφ(11)|1Φc2i|0i−sinψ|aΦc2i|1i, (7a) sinintghleeqsaumbiet rwoatyataiosnt,hleeasdtaintegst|o0ia,|r1oib,|uasitirnovtoaltvioednionf tthhee
|Ψ6i= √2cosη cosψeiφ(11)|1Φc3i|0i−sinψ|aΦc3i|1i two-atom states {|a0i|0i, |a1i|0i}.
We first describe precisely the dynamics of this two-
hsinηe−iφ(cid:16)(02) √2cosψeiφ(11) 1a 1 +sinψ aa 2(cid:17) atom states rotation, before applying it to the construc-
− | i| i | i| i
tion of the arbitrary state controlled-unitary gate.
/ (1+cos2ψ(cid:16)+sin2ψcos2η), (7(cid:17)bi)
for the third one.
B. Robust rotation of two-atom states
ThemixinganglesaredeterminedbytheRabifrequen-
cies through the relations
We start with the initial state
tanη = Ω(2)/g(2) (8a) Φ = αa1 0 +β a0 0
i
| i | i| i | i| i
tanψ = Ω(1)/g(1), tanϕ=sinψ/tanη. (8b) = a (α1 +β 0 ) 0
1 | i⊗ | i | i ⊗| i
= aφ 0 (10)
i
The dark states Ψ , Ψ , Ψ drive respec- | i| i
1,2 3,4 5,6
tively the populatio|n ofi |the ist|atesi 00 0 , 01 0 ; whereα,βarecomplexnumberssuchthat α2+ β 2 =1.
a0 0 , a1 0 and 10 0 , 11 0 in the a|diaib|aitic| limi|iti. –Step1: weinducetheinitialconnecti|on|to|th|edark
| Sii|ncie|thei|ciouplin|g bie|twie|enit|hie dark states of a same states
subspace are respectively of the form
Φ =α Ψ +α Ψ (11)
i 3 3 4 4
| i | i | i
d
Ψ Ψ = θ˙cosη, (9a) with the constant coefficients
2 1
h |dt| i −
Ψ d Ψ = θ˙sinϕ, (9b) α3 =hΦnc|φii (12)
h 4|dt| 3i − (α4 = Φc φi
h | i
d
Ψ6 Ψ5 = θ˙√2cosη, (9c) using the partially overlapping pulse sequence Ω(1),Ω(2)
h |dt| i 1
such that ϕ decreases from π/2 to 0. In the adiabatic
thesixdarkstatescanevolvefreelyandindependentlyin limit the dark states evolve independently, such that at
the adiabatic limit under the condition θ θ0 = const. the end of the pulse sequence the statevector becomes
≡
This condition can be satisfied when the amplitudes of
the lasers interacting with the second atom vary with a Φ =α3 Ψ3 +α4ei(φ(11)−φ(02)) 1a 0 (13)
| i | i | i| i
4
since Ψ is a stationary state. state a . This can be done using two lasers of appropri-
3
| i | i
– Step 2: we use a pulse sequence in the reversed ate polarisations of Rabi frequencies: Ω(1) and
a(sti)
order, i.e. Ω(2),Ω(1) such that ϕ increases from 0 to
1
π/2. The phases φ(2) are unchanged, while we shift by Ω(1) (t) = Ω(1) (t)cosχ , (18a)
0,1 0(sti) (sti)
δ the phase φ(11) of the laser pulse addressing the first Ω(1) (t) = Ω(1) (t)sinχ . (18b)
qubit. This induces the connection 1a 0 e iδ Ψ , 1(sti) (sti)
− 4
| i| i → | i
and therefore the statevector becomes
ThelatterofellipticalpolarisationisreferredtoasΩ(1) .
(sti)
Φ =α Ψ +α e iδ Ψ , (14) They drive respectively in a non-resonant way the tran-
3 3 4 − 4
| i | i | i sition of states a , 0 , 1 of the first qubit to e with
| i | i | i | i
a one-photon detuning. Alternatively, we prefer to use
and at the end of the pulse sequence
moreefficientone-photonresonanttransitionstoasecond
Φ = a (α Φ +α4e iδ Φ )0 excited atomic state. The important point is to discard
f 3 nc − c
| i | i | i | i | i the transition a e by the cavity in this preliminary
=e−iδ/2 a U(δ,n)φi 0 , (15) step. | i−| i
| i | i| i
They define the control-state of the control-qubit and
where its orthogonalstate
U(δ,n)=exp( iδn σˆ) (16) φ = sinχeiφ(1) 1 +cosχ0 , (19a)
c
− 2 · | i | i | i
φ = cosχeiφ(1) 1 sinχ0 . (19b)
nc
| i | i− | i
isageneralrotationofSU(2)ofangleδaroundthevector
n. The components of this vector n are The fuul process is decomposed in three steps:
– Step1: westartfromageneralinitialstatewritten
n=(sin2θcosφ(2),sin2θsinφ(2),cos2θ), (17) in the basis φnc , φc 0 , 1
{| i | i}⊗{| i | i}
and σˆ = (σ ,σ ,σ ) are the Pauli operators defined for ψi = φnc (α1 0 +α2 1 )0 + φc (α3 0 +α4 1 )0 ,
x y z | i | i | i | i | i | i | i | i | i
(20)
the second qubit: σ = 0 1 + 1 0, σ = i(0 1
x y
| ih | | ih | | ih |− where α are complex numbers such that
1 0), σ = 0 0 1 1. We notice that the initial i=1,..,4
z
|p1oi1phu0|la)t,iownhiocfh|tahirehes|cto−anten|seic|ht0e1|di|t0oitahnedd|a0r0ki|s0tiat(e|1s0iΨ|0i aanndd W4i=1e|uαsie|2a=f-1S.TIRAP-process with the pulse sequence
|Ψi| (iΨ and Ψ ), stay unchanged at the en|d1oif the PΩ(1) ,Ω(1) with relative phase ξ in order to transfer for
| 2i | 5i | 6i a(sti)
process. the first qubit the population of state φ to state a .
c
Thefirstqubitcontrolstherotationappliedonthesec- This gives the state | i | i
ondqubit. Indeed,equations(15)and(16)showthatthe
process described induces, up to a global phase δ/2, a ψ = φ (α 0 +α 1 )0 eiξ a (α 0 +α 1 )0 .
1 nc 1 2 3 4
− | i | i | i | i | i− | i | i | i | i
rotation of the second qubit of angle δ around the vec- (21)
tor n on the Bloch sphere only if the first one is in the – Step 2: we apply the process previously described
ancillary state a . Then, by transferring the popula- toimplementtherotationU(δ,n)[seeEqs.(15,16)]. The
| i
tion of an arbitrary preselected state of the first qubit state (21) becomes
on state a before realising the two-atom rotation, we
| i ψ = φ (α 0 +α 1 )0 (22)
get a controlled-unitary gate generalised to the case of 2 nc 1 2
| i | i | i | i | i
an arbitrarystate for the controlqubit. Moreover,if the eiξe−iδ2 a U(δ,n)(α3 0 +α4 1 )0 ).
couplings between the cavity mode and the atoms are − | i | i | i | i
much stronger than the classical laser field interaction, –Step3: wemaketheinverseoperationofstep1,i.e.
then the cavity is negligibly populated and the coupling af-STIRAP-processwiththe pulsesequenceΩ(1),Ω(1)
a(sti)
between the atoms are given by a virtual photon. The
withrelativephaseξ inordertotransferinthefirstqubit
′
proposed process is then decoherence-free in the sense
thepopulationofstate a tostate φ . Thefinalsystem
that spontaneous radiation from the excited state and | i | ci
state reads
cavity damping are avoided.
ψ = φ (α 0 +α 1 )0 (23)
3 nc 1 2
| i | i | i | i | i
C. The arbitrary state controlled-unitary gate +ei(ξ−ξ′)e−iδ2|φciU(δ,n)(α3|0i+α4|1i)|0i.
Under the condition ξ ξ δ/2 = 0 the undesirable
′
− −
The extension of the previous process to the imple- phase factor of the state (23) vanishes and one obtains
mentation of the arbitrary state controlled-unitary gate directlythearbitrarystatecontrolled-unitarygatewhich
isnowsimple: itconsiststotransferasapreliminarystep makesthe unitary operationU(δ,n)on the secondqubit
the controlledstate φ of the first qubit to its ancillary only if the first one is in state φ .
c c
| i | i
5
IV. NUMERICAL SIMULATION 1
|−0〉|0〉
0.5 (a)
We show the numerical simulation of the arbitrary
0
state controlled-unitary gate on Figs. 4-5. We have
1
chosen Rabi frequencies of gaussian shape and of full |−1〉|0〉
s0.5 (b)
width at half maximum T = 100ns. The couplings are n
P o
parametrisedbyΩmax/2π=14MHzandg/2π=34MHz ati 0
wcehnitchteccahnnobleogciuersre[1n5tl,y1o6b].taiTnhede esximpeurliamtieonntailslymwaidtehfroer- Popul0.15 |+0〉|0〉 |a0〉|0〉 |1a〉|0〉 |+0〉|0〉,|+1〉|0〉 (c)
the state of the controlqubit |+i≡ √12(|0i+|1i) (its or- 0 |a1〉|0〉
thogonal state is denoted 1 (0 1 )). We have 1
represented in Fig. 4 the|−timi≡e e√v2ol|utiio−n|oif the phases 0.5 |+1〉|0〉 |a1〉|0〉 |1a〉|0〉 |+0〉|0〉,|+1〉|0〉 (d)
associated to the probability amplitudes for the initial 0 |a0〉|0〉
states +0 0 , +1 0 . Fig.5exibitsthetimeevolution
ofthe p|opuil|atiio|nsfoi|rtiheinitialstates 0 0 , 1 0 , p | 30 g
| + 0i|0i, | + 1i|0i. They show that |w−heni|tihe|−conit|roil Ω× T 15 Ω(a1()sti) Ω((1st)i) Ω(11) Ω(2) Ω(2) Ω(11) Ω((1st)i) Ω(a1()sti) (e)
qubit is in state , the state of the target qubit is un- | 0
|−i 0 4 8 12 16 20
changed[Figs.5(a)and5(b)];andwhenthecontrolqubit
time (t/Tp)
is in state + , a R(π/4) gate is applied on the target
| i
qubit,suchthatitsstates 0 and 1 becomerespectively
| i | i FIG. 5: (Colour online) Time evolution of the populations
(0 + 1 )/√2 [Figs 4(a) and 5(c)] and ( 0 + 1 )/√2
| i | i −| i | i representedrespectivelyfortheinitialstates|−0i|0i,|−1i|0i,
[Figs 4(b) and 5(d)].
|+0i|0i, |+1i|0i [frames (a)-(d)]. The Rabi frequencies are
represented in thelower frame.
π
0 |a1〉|0〉 |+0〉|0〉 (a)
ses −π |a0〉|0〉 |1a〉|0〉 |+1〉|0〉 todhnaeerk-spthsatotaettosens|−s|tΨa05ti,e|60si,i, tawhnhedicc|ho−hae1rreei|n0sciuepeoevfropltvohesesitipporanorstcieaoslfslysiseavlseoernnag-l
a
h sitive to the cavity decay rate. These losses are neg-
P π |1a〉|0〉 |+0〉|0〉 ligible under the condition g(i) Ω(i) where the
0 |a0〉|0〉 |+1〉|0〉 (b) cavity is negligibly populated, an≫d cajv,mityaxdamping is
−π |a1〉|0〉 thus avoided. We have calculated the gate fidelity
, in Tab. I for different values of the parame-
+
30 tFe−rsF(Ω(i) , g(i), κ). , stand respectively for
× Tp | 15 Ωg(1) Ω(1) Ω(1) Ω(2) Ω(2) Ω(1) Ω(1) Ω(1) (c) n|ho−tied|r−esjnp,umemcatxii|v2,ely|h+thied|i+dneauFlm−fii|n2Fal+wshtaetreew|±itihdoiu,|t±cnauvmitiy ddee--
Ω a(sti) (sti) 1 1 (sti) a(sti)
| cayandthefinalstateofthenumericalsimulationforthe
0 evolution of an initial state with a control-qubit in state
0 4 8 12 16 20
, and a target qubit in state 0 or 1 .
time (t/Tp)
|±i | i | i
FprIoGb.a4li:ty(Caomlopulirtuodnelisnefo)rTtimheeienvitoilaultisotnatoefst|he+p0hia|0sies(oufpptheer (Ω(ji,m) ax, g(i), κ)/2π (MHz) F− F+
(14, 34, 4.1) 0.281 0.854
frame), |+1i|0i (middle frame). The Rabi frequencies are
(14, 34, 2.05) 0.488 0.918
represented in thelower frame.
(14, 34, 1) 0.680 0.954
(14, 68, 4.1) 0.668 0.954
(14, 68, 2.05) 0.799 0.966
(14, 68, 1) 0.892 0.976
V. DISCUSSION AND CONCLUSION
TABLE I: Fidelity of the arbitrary state controlled-unitary
The implementaion of the arbitrary state controlled- gate.
unitary gate proposedis robustunder the adiabaticcon-
ditions Ωj(i,)maxTP,g(i)TP ≫ 1, Ω(ji,)max,g(i) ≫ κ,1/τ Wehavepresentedaschemeadaptedfortheimplemen-
where κ,1/τ are the cavity decay rate and the spec- tationofa universaltwoqubit quantumgatethatgener-
tral linewidth of the excited atomic states. It does alise the controlled-unitary gate to an arbitrary control
not involve spontaneous emission since the dynamics state of the first qubit. This arbitrary state controlled-
follows dark states. However, since the population of unitary gate opens up novel applications for the rapid
6
implementation of quantum algorithms. For instance, Fig. 6, where M is an unitary operator of eigenvalues
1. The output qubit of this circuit is an eigenvector of
±
M dependingoftheresultofthemeasurementofthefirst
0 H H 0
| i • | i qubit. This circuit offers many applications in quantum
≡ Uas
error corrections [2, 17].
q M q
| i | i
FIG. 6: Quantum circuit realising a projective measurement
Acknowledgments
anditsequivalentusingthearbitrarystatecontrolled-unitary
gate denoted Uas. H stands for the Hadamard gate.
N.S.acknowledgesfinancialsupportsfromtheEUnet-
the main part of the projective measurement circuit [2] work QUACS under contract No. HPRN-CT-2002-0039
can be built directly from this gate, as represented in and from La Fondation Carnot.
[1] M.J. Bremner, C. M.Dawson, J. L.Dodd,A.Gilchrist, [10] T. Pellizzari, S. A. Gardiner, J. I. Cirac and P. Zoller,
A. W. Harrow, D. Mortimer, M. A. Nielsen, T. J. Os- Phys. Rev.Lett. 75, 3788 (1995).
borne, Phys.Rev.Lett. 89, 247902 (2002). [11] K. T. Kapale, G. S. Agarwal, M. O. Scully, Phys. Rev.
[2] I. L. Chuang and M. A. Nielsen, Quantum Computation A 72, 052304 (2005).
and Quantum Information,CambridgeUniversityPress, [12] X. Lacour, S. Gu´erin, N. V. Vitanov, L. P. Yatsenko,
Cambridge (2000). H. R.Jauslin, preprint.
[3] J. Preskill, Proc. R. Soc. London A 454, 385 (1998). [13] N.Sangouard,X.Lacour,S.Gu´erin,H.R.Jauslin,Phys.
[4] L. M. Duan, I. J. Cirac, P. Zoller, Science 292, 1695 Rev. A 72, 062309 (2005).
(2001). [14] N.Sangouard, X.Lacour, S. Gu´erin,H.R.Jauslin, Eur.
[5] K. Bergmann, H. Theuer and B. W. Shore, Rev. Mod. Phys. J. D DOI:10.1140/epjd/e2005-00315-2 (2005).
Phys.70, 1003 (1998). [15] R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca,
[6] N.V.Vitanov,K.A.SuominenandB.W.Shore,J.Phys. A. D. Boozer, H. J. Kimble, J. Phys. B: At. Mol. Phys.
B 32, 4535 (1999). 38, S551 (2005).
[7] R. Unanyan, M. Fleischhauer, B. W. Shore and [16] N. V. Vitanov, M. Fleischhaeur, B. W. Shore,
K.Bergmann, Opt.Commun. 155, 144 (1998). K. Bergmann, Adv.Ad. Mol. Opt.Phys. 46, 55 (2001).
[8] Z.Kis and F. Renzoni, Phys.Rev.A 65, 032318 (2002). [17] J. Vala, K. B. Walley, D. S. Weiss, e-print
[9] H. Goto and K. Ichimura, Phys. Rev. A 70, 012305 arXiv:quant-ph/0510021.
(2004).