Robust implementations of Quantum Repeaters

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by  Alexander Klein

Robust implementations of quantum repeaters
Alexander Klein, Uwe Dorner, Carolina Moura Alves, and Dieter Jaksch
Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
(Dated: February 1, 2008)
We show how to efficiently exploit decoherence free subspaces (DFSs), which are immune to
collective noise, for realizing quantum repeaters with long lived quantum memories. Our setup
consists of an assembly of simple modules and weshow howto implement themin systems of cold,
neutralatomsinarraysofdipoletraps. Wedevelopmethodsforrealizingrobustgateoperationson
qubits encoded in a DFS using collisional interactions between the atoms. We also give a detailed
analysis of the performance and stability of all required gate operations and emphasize that all
modules can berealized with current or near futureexperimental technology.
6 PACSnumbers: 03.67.Mn,03.67.Hk,03.67.Pp,32.80.Pj
0
0
2 I. INTRODUCTION transmission line into segments with a length of the or-
der of the attenuation length of the channel. On each
n
segment entangled particle pairs are created and by ap-
a One of the major obstacles in the realization of quan-
J tuminformationprocessors(QIP)isdecoherence,caused plying entanglement swapping [24] and purification pro-
tocols [17, 25, 26] entangled pairs of larger distances
4 by the unwanted interaction of the system with its en-
2 vironment. Even the feasibility of simple special pur- are produced. By repeating these steps a distant pair
posequantumdevices,suchasquantumrepeaters,issig- of qubits with high entanglement fidelity is obtained.
2 Entanglement purification, distillation, and concentra-
nificantly affected by the presence of decoherence and
v tion have been realized in a number of seminal experi-
a special precaution to counteract the resulting loss of
6
ments [27, 28, 29, 30] using photonic states for encod-
5 quantuminformationhas to be taken. A number ofcon-
ing the qubits. Techniques for converting flying pho-
0 cepts have been developed to fight decoherence and in-
tonic qubits into atomic ones via cavity-enhanced inter-
1 creasethereliabilityofQIPincludingactiveerrorcorrec-
1 tion [1, 2, 3] and fault tolerant quantum computing [4]. actions [31, 32] or the usage of atomic ensembles [33, 34]
5 Asidefromthisitwasshownthat,dependentonthetype have been proposed. Very recently a demonstration of
0 longdistanceentanglementofmassiveparticleshasbeen
of interactionprocesses betweenthe qubits and their en-
h/ vironment, subspaces of the system’s Hilbert space can achieved [35].
p existwhichareimmunetodecoherenceprocessesinduced Here we develop a proposal for a quantum repeater
- by this interaction [5, 6, 7, 8]. These decoherence free with DFS quantum memory using the entanglement pu-
t
n subspaces are a method of passive errorcorrectionor er- rification scheme introduced in [17, 36] (also known as
a ror prevention and can significantly increase the lifetime the “Innsbruck protocol”). During the purification pro-
u
ofquantuminformationandreliabilityofQIPasalready cedure the entanglement of a primary pair of qubits is
q
: demonstrated with ion traps, NMR and optical meth- increased by sacrificing auxiliary entangled qubit pairs.
v ods [9, 10, 11, 12, 13, 14, 15, 16]. However, this robust- The entanglement fidelity of the final entangled qubit
i
X ness makes states within the decoherence free subspace pair does not converge to 1, but to a value which de-
(DFS) difficult to manipulate via controlled operations pends on the fidelity of the initial primary pair and on
r
a and,ingeneral,complicatedinteractionswitha DFSare the constant fidelity of the auxiliary pairs. However,the
necessary for realizing universal QIP [8]. scheme [17, 36] is favorable from a pragmatic point of
AparticularlypromisingapplicationofDFSsarequan- view since it requires fewer physical resources than the
tum repeaters [17, 18] which require very long quantum purification schemes used in [25, 26, 37]. On the other
informationstoragetimesbutonlyalimitedsetofquan- side, the protocol [17, 36] takes longer and thus quan-
tummanipulations. Inthispaperweshowthatthequan- tum information has to be stored reliably for the whole
tum operations necessary for a quantum repeater can duration of the protocol. We overcome this apparent
be realized on DFS qubits with nearest neighbor hop- disadvantage by the usage of DFS qubits of four neutral
ping [19, 20] and interactions with ancilla qubits [21] atomswhichformthe keyelements ofourDFSquantum
only, thus circumventing the difficulties usually associ- repeater.
ated with the usage of DFS [8]. Quantum repeaters are Our scheme is based on current and near future tech-
used to distribute maximally entangled pairs of qubits nologyandismotivatedbytheextraordinaryprogressof
over long distances which is a necessary prerequisite the field of cold, neutral atoms in dipole traps over the
for important applications such as quantum cryptogra- pastfewyearsbothonthetheoretical[38,39,40,41]and
phy [22] and quantum teleportation [23]. They are thus on the experimental [42, 43] side. Atomic quantum reg-
consideredoneofthe mostimportantnearfuture special isters have been realized in arraysof microlenses [44, 45]
purpose QIP. andbyexploitingthesuperfluidtoMott-Insulatortransi-
The basic idea of a quantum repeater is to divide the tioninopticallattices[38,46,47],singleandmanyatoms
2
have been manipulated coherently [48, 49], atoms stored intermediate
A node B
in standing waves have been transported over macro-
(i)
scopic distances using conveyor belt techniques [50, 51,
52, 53], and multipartite entangled states of atoms were Entanglement swapping
created [49, 54]. Furthermore, extremely versatile tech-
(ii)
nologies such as optical tweezers and holographic traps
thatarecontrolledbyliquidcrystalspatiallightmodula- State transfer
tors [55, 56, 57] are currently developed. They allow for
(iii)
essentially arbitrary designs of optical potentials.
We willshowthata DFSquantumrepeatercanbe as-
sembledfrommoduleswhichonlyrequirerotationsofthe (iv)
logical DFS qubits around the x and z axes by angles π
andπ/2andacontrolledphase(CPHASE)gatewherean Entanglement swapping
ancilla atom acts as the control qubit. The single qubit (v)
rotations will be realized by selectively lowering poten-
tialbarriersbetweenatomsandthesystemwillnotleave Purification
the DFS during these gates. The ancilla qubit, which is (vi)
required to interact with a DFS qubit for realizing the
CPHASE gate, will not be decoherence free. However,
FIG. 1: Illustration of the entanglement purification scheme.
we will show that its influence on the DFS memory is
Pairs of entangled flyingqubits(open circles) are created via
sufficiently small to allow for gate fidelities better than
entanglement swapping. These pairs are then used to purify
98.7%. Furthermore,wewillshowthatourschemeallows
an entangled pairof DFS qubits(solid circles; seetext). The
for deviations of the system parameters of a few percent
framed steps (iv)-(vi) are repeated until a desired entangle-
without significantly affecting the gate fidelities. ment fidelity is achieved.
This paper is structured as follows: In Sec. II we will
briefly summarize the repeater protocol and show how
DFSs can be embedded into this scheme. The goal of
this section is to show what resourcesare needed for the intermediatenodesandtransmittedtothenextnode. In
implementation of the quantum repeater. In Sec. III we Fig. 1 this is illustrated for the example of one interme-
propose methods of how the operations of Sec. II can be diatenode. After this,entanglementswapping[24]atall
physically realized with cold, neutral atoms in arrays of intermediate nodes is performed which yields a partially
dipole traps. The results are summarized in Sec. IV. entangled pair of flying qubits between A and B and its
state is transferred to a DFS qubit pair (step (ii)). Fi-
nally, as in (i) and (ii), auxiliary entangled qubit pairs
II. QUANTUM REPEATERS WITH are generated which are sacrificed in order to increase
DECOHERENCE FREE SUBSPACES the entanglement of the DFS qubit pair (step (iv)-(vi)).
Steps (iv)-(vi) are repeated until a desired degree of en-
In the following we will describe the purification and tanglement is obtained for the DFS qubit pair.
repeater protocol of Refs. [17, 36] and extend it by us- However,astraightforwardextensionofthisschemein
ing DFS. The basic steps of the purification scheme are order to connect arbitrary separated points A and B is
illustrated in Fig. 1. Only one quantum channel (repre- not possible since the distance between the intermediate
sented by the lines with an arrow) and two DFS qubits nodes has to be chosen such that the fidelity of the fi-
(representedby the solid circles)are needed. The goalis nal pair is not lower than F . This is, in general, not
min
toshareahighfidelityBellstatebetweenAandB which possible if the distance between A and B is too large.
is done by transmissionof “one-half”of a maximally en- Therefore, a quantum repeater concept by means of a
tangled flying qubit pair (open circles in Fig. 1(i)). In nested purification scheme was proposed [17, 36] which
general, the flying qubit (e.g., a photon) is not directly uses the above purification protocol to create a number
sentfromAtoB sincethepresentavailabletransmission ofentangled DFS qubit pairs which are then used to pu-
quantum channels (particularly optical fibers) degrade rify alowernumberofadditionalDFSpairswith greater
the entanglement of the flying qubit pair exponentially distance. These pairs are then again used to purify an
with the distance between the nodes. Thus, if the chan- even lower number of pairs, etc. until merely one entan-
nel is too long, the minimum fidelity required for the gled pair is left. This procedure is illustrated in Fig. 2:
successful application of the purification protocol (F ) The boxes stretching over lines (i) and (ii) represent the
min
is not achieved. To overcome this problem a sufficiently purification protocol of Fig. 1. The entangled pairs of
high number of intermediate nodes between A and B is line (ii) are then used to purify the pairs in line (iii) in
introduced such that the distance between two nodes is the same way as in Fig. 1 except that the flying qubits
ofthe orderofthe opticalfiber’s attenuationlength[36]. have to be replaced by DFS qubits, i.e., entanglement
Entangled flying qubit pairs are prepared at each of the swapping between DFS qubits, a state transfer between
3
...
(i) is given by
(ii)
H = σxB +σyB +σzB . (1)
(iii) I i x i y i z
i
X
(iv)
.. Here, i labels the physical qubits, B = (Bx,By,Bz) de-
. scribes the fields and fluctuations, and the σ are Pauli
i
matrices acting on the ith physical qubit.
FIG.2: Thequantumrepeaterscheme. Thedashedboxesin-
dicatethesubroutineshowninFig.1. Thecreatedentangled The corresponding decoherence free subspace is
DFS qubit pairs are used to purify further DFS qubit pairs spanned by the states with total zero angular momen-
(in line (iii)). The resulting pairs are then used to purify the tum, and the lowest number of physical qubits giving a
pairs of line (iv) and so on. twofold degenerate DFS is 4. The DFS qubit is then
represented by [8, 64]
1
0 = (01 10 ) (01 10 ),
DFS qubits (to initialize the pairs in line (iii) and (iv)) | iL 2 | i−| i ⊗ | i−| i
(2)
1
and purification by using entangled DFS pairs have to 1 = (2 1100 +2 0011 (01 + 10 )⊗2),
be performed. The diagram shown in Fig. 2 can be ex- | iL 2√3 | i | i− | i | i
tended in horizontaland verticaldirections such that an
where ijkl = i j k l with i,j,k,l = 0,1 de-
arbitrarydistancebetweenthefinalpaircanbeachieved. | i | i1| i2| i3| i4
scribesthecomputationalbasisstatesofthefourphysical
Itwasshownthatthe number of(DFS) qubits necessary
qubits. Eachofthefourphysicalqubitsisrepresentedby
for the scheme increases merely logarithmically with the
two internal states of an atom which are trapped in an
distance of the final entangled pair [17, 36].
array of dipole traps, for example, an optical lattice. By
Insummary,thepurificationprocedureshowninFig.1
themethoddescribedinSec.IIIAthesystemisprepared
canbe realizedby assemblingthe followingmodules: (0)
inthe logicalstate 0 andin Sec.IIIB itis shownthat
The creationof pairs of flying qubits and performing en- | iL
by tuning the system parameters appropriately we can
tanglementswappingbetweenthem,(1)therealizationof
implement the Hamiltonians H and H with
twoDFSqubits,(2)thetransferofthestateoftheflying X Z
qubits to the DFSqubits, and(3)the actualpurification 1
H = X = (V +2V ), (3)
of the entangled DFS qubit pair. Additionally, for the X L 12 23
√3
nested purification protocol, i.e., for the full implemen-
H = Z = V , (4)
tation of the quantum repeater, we have to (4) perform Z L − 12
entanglementswappingbetweenneighboringDFSqubits,
where X (Z ) is the X-gate (Z-gate) for the logical
L L
(5) we need a state transfer between DFS qubits (analo-
qubit and V are permutations of the physical qubits i
ij
gously to module (2)), and (6) we must perform entan-
and j. Note that the logical Z-gate can also be repre-
glement purification using two DFS qubit pairs. Module
sented as Z = V . Since the DFS defined by Eq. (2)
L 34
(0) can be implemented using standardquantum optical −
isinvariantunderpermutationsofthephysicalqubits[8],
methods as demonstrated in [58, 59, 60, 61, 62]. It does
the time evolution of the system generates rotations
not involve DFS qubits and will thus not be discussed
further in this paper. Rx(θ) = e−iθ2XL, (5)
L
Inthefollowingwewillshowhoweachoftheremaining
Rz(θ) = e−iθ2ZL (6)
modulescanberealized. Fortheoperationsusingancilla L
qubits and thus leaving the DFS we then calculate the
without ever leaving the DFS, where θ = 2t and t is the
achievable fidelities.
timenecessaryforperformingtherotations. TheindexL
indicatesthattheseareoperationsonthelogicalqubit(in
thefollowingwewillalsousethisnotationforoperations
on physical qubits, where we omit the index L).
A. Realization of the repeater modules
Althoughitisnotrequiredintherepeaterprotocol,we
can thus perform arbitrary rotations of the logical qubit
1. Module 1: Decoherence free memory and single qubit (and hence we can prepare arbitrary states in the DFS)
rotations
sinceeverytwolevelunitaryoperatorcanbedecomposed
in Rx and Rz [65].
L L
The main sources of decoherence in systems of cold
atomsarefluctuationsofthe opticalpotentialandexter-
nal electromagnetic (stray-)fields [63]. These sources are 2. Module 2: State transfer between flying and DFS qubit
homogeneousacrosstheregionofthephysicalqubitsand
the interactionwith them thus only depends on their in- Thequantumcircuitforthestatetransferoftheflying
ternal states. The Hamiltonian describing the situation qubit to the logical qubit is shown in Fig. 3. The state
4
ation of the physicalqubits 3 and 4 leads to a controlled
(cid:13)
|ψiq • Rx(−π2) FE(cid:13) • (−ZL) (CPHASE) gate on the logical qubit. As can be
seen from Fig. 3, apart from rotations given in Eqs. (5)
|0iL RLx(π2) −ZL RLx(−π2) RLz(π) |ψiL and (6), we need to perform a rotation Rx( π/2) and
−
a measurement of the control atom which can be done
by standard techniques such as laser induced Rabi oscil-
FIG.3: Quantumcircuitforthestatetransferoftheauxiliary
lations and state selective laser excitations. Eventually,
qubit to theDFS qubit.
the state of the atom is transferred to the logical qubit,
i.e., ψ =α 0 +β 1 .
| iL | iL | iL
of the flying qubit is denoted as ψ = α 0 +β 1 .
| iq | iq | iq
As described in Sec. III the state of the flying qubit
(a photon) is first transferred to an appropriate inter-
3. Module 3: Entanglement purification
nal state of an auxiliary atom. The state of this atom
then controls the interaction between physical qubits 3
and 4 leading to a controlled SWAP operation between The local operations required for entanglement purifi-
theseatoms. SinceZ = V ,acontrolledSWAPoper- cation are shown in Fig. 4.
L 34
−
(cid:13)
auxiliary qubit UA,B Rz(−π2) Rx(−π2) • Rx(π2) Rz(π2) FE(cid:13)
DFS qubit ULA,B RLx(−π) −ZL RLx(π)
FIG. 4: Quantumcircuit for entanglement purification. The procedureis performed on both sides, A and B.
This circuit corresponds to the scheme proposed in [26], 4. Module 4: Entanglement swapping between DFS qubits
buthereitisexpressedintermsofoperationsthatwecan
implementinoursetup. Thecircuithastobeappliedon
both ends of the transmission line A and B, where
UA = UB † =Rx(π/2). (7) Entanglement swapping between neighboring DFS
L L L
qubit pairs canbe done by a Bell measurementandsub-
As in subsection IIA(cid:0)2, th(cid:1)e auxiliary qubit is again an sequent local operations Rx(π/2) and Rz(π/2) that are
L L
atomthatcontrolsaphaseoperationonthelogicalqubit. controlledby the outcome of the measurement[36]. The
Ifthe outcomeofthe measurementsatAandB coincide Bellmeasurementcanbedecomposedinacontrollednot
we end up with an entangled DFS qubit pair of higher (CNOT) gate, a Hadamard gate, and a measurement in
fidelity. Iftheydonotcoincidethepurificationprocedure the computational basis (the Hadamard gate can be re-
has to be repeated. placed by, e.g., the sequence Rz(π/2)Rx(π/2)Rz(π/2)).
L L L
|ψiL1 RLx(−π) −ZL RLx(π) RLz(π)
(cid:13)
0 Rx( π) Rx(π) Rx( π) (cid:13)
| iq −2 • 2 • −2 FE •
|ψiL2 RLz(−π2) RLx(π2) −ZL RLx(−π2) RLz(π2)
FIG. 5: Quantum circuit for a CNOT operation between two logical qubits.
The implementation of a decoherence free CNOT opera- oftwo-particleinteractions withpairwise individualtun-
tionisrelativelycomplicatedincludinglengthysequences
5
ing[8]. Thereforeweuseanancillaatom(asinSec.IIA2 A quantitative measure of the operation fidelity is de-
and IIA3) to mediate the interaction between different rived from Ref. [65]. A distance measure of two density
DFSqubits. ApossiblequantumcircuitforaCNOTgate matrices ̺, σ is given by
betweenDFSqubitsisshowninFig.5. Thecenterlinein
2
the circuit represents the ancilla qubit. The shown cir- f(̺,σ)= Tr √σ̺√σ , (9)
cuitusesonlygatesthatwecanimplementinourscheme
(cid:18) q (cid:19)
and leads to a CNOT gate with the upper DFS qubit as
where,in contrastto [65], we use the squaredtracesince
the control qubit. We will discuss in Sec. IIB how the
for pure states this corresponds to a probability, i.e. the
decoherence of the auxiliary atom influences the fidelity
absolute square of the overlap of the two states. The
of this operation.
operation- (or gate-) fidelity can be derived from this
ForthemeasurementoftheDFSqubitweusethesame
quantity: Let be a quantum operation describing a
ancilla as for the CNOT gate. The correspondingcircuit E
quantumcircuitwheredecoherenceistakenintoaccount,
is shown in Fig. 6. The final measurement of the ancilla
and the same operation without decoherence. We de-
0
E
fine the operation fidelity by
(cid:13)
|1iq Rx(π2) • Rx(−π2) FE(cid:13) F(E0,E)=m̺inf(E0(̺),E(̺)), (10)
|ψiL −ZL where it is sufficient to minimize over pure input
states ̺ [65].
Let us first consider the state transfer from the atom
FIG. 6: Measurement of thelogical qubit.
to the DFS qubit (Fig. 3). To minimize decoherence ef-
fects, the Rx(π/2) and Rx( π/2) gates are applied be-
L L −
is equivalent to a measurement of the DFS qubit. foretheauxiliaryatominteractswiththeDFSqubitand
after the measurement, respectively. In addition, we ne-
glectdecoherenceduringtheRx( π/2)rotationsincethe
−
5. Modules 5 and 6: State transfer between DFS qubits and time to perform such a gate on one atom is, in general,
entanglement purification using pairs of DFS qubits much smaller than the expected decoherence time 1/γ
(seeSec.IIID). Thus,decoherenceoccursmainlyduring
The state transfer ofone DFS qubit to anothercanbe the CPHASE gate. We examined the dynamics of the
doneanalogouslytoFig.3,where ψ hastobereplaced system during the application of the CPHASE gate by
| iq solving Eq. (8) analytically. The resulting gate fidelity
by ψ . The necessary CPHASE gate between DFS
| iL for the CPHASE operation is given by
qubits can be constructed similarly as the CNOT gate:
We merely have to omit the single qubit rotations in the 1+e−γt γt
last line of Fig. 5. Also the entanglement purification FCPHASE( 0, )= 1 . (11)
E E 2 ≥ − 2
scheme of Fig. 4 can be adapted in this way. Again,
AswillbediscussedinSec.IIIDthisbehaviorissufficient
the auxiliary qubit is replaced by a DFS qubit and the
to ensure a working repeater scheme.
CPHASE(orCNOT)gateisrealizedasdescribedabove.
Note that, since unitary operations do not change the
fidelity of the circuits [65], all operations which only in-
volve one CPHASE gate have the same fidelity given in
B. Operation time and decoherence
Eq.(11). Thisincludesparticularlytheoperationsshown
in Figs. 4 and 6.
Someoftheaboveoperationsrequiretheuseofasingle InasimilarwaywecalculatedthefidelityoftheCNOT
auxiliaryqubitwhichisnotdecoherencefree. Entangling gate between the two DFS qubits (Fig. 5). Again, all
this qubit with the DFS qubits will therefore lower the R(x,z) gates are applied before the auxiliary atom inter-
overall fidelity of the corresponding gate operations on L
acts with the DFS qubit and after the measurement, re-
the DFS qubits. The auxiliary qubit is represented in
spectively. Duringtheseperiodstheauxiliaryatomisnot
our scheme by two long-lived internal states of an atom
entangledwitheitherDFSqubitandthusitsdecoherence
andthusthemajorexperimentallyexpecteddecoherence
hasnoinfluenceonthem. Furthermore,weassumeagain
mechanism is dephasing [66]. This process is described
that decoherence during the R(x,z) rotations can be ne-
by the master equation
glected. The gatefidelity ofthe CNOToperationis then
given by
i γ
∂ ̺ = [Hˆ,̺ ]+ (σ ̺ σ ̺ ) , (8)
t aux −~ aux 2 z aux z − aux 1+e−γt 2
F ( , )= 1 γt, (12)
CNOT 0
E E 2 ≥ −
where̺ isthedensitymatrixoftheauxiliaryatom,Hˆ (cid:18) (cid:19)
aux
is the Hamiltonian of the system, 1/γ characterizes the where t is the time which is needed to perform one
decoherencetime,andσ isaPaulimatrix. Decoherence CPHASE gate. Since the two DFS qubits are not en-
z
of the DFS qubits is not taken into account here. tangled before the CNOT gate is applied it is sufficient
6
to use only pure, unentangled states for the minimiza-
tion [65]. The fidelity for a CPHASE gate between two
DFS qubits is the same as for the CNOT gate because
they differ only by unitary operations. As in the case
of the CPHASE gate, the decay of the CNOT fidelity is
slow enough to ensure a working quantum repeater (see
Sec. IIID).
1 2 3 aux 4
The fidelity of the state transfer between two DFS
qubits described in Sec. IIA5 can also be calculated by y
explicitly solving the time evolution. The state transfer
fidelity FST is given by 1 2 3 4 x
1 3 z
F ( , )= 1+e−2γt 1+e−γt 1 γt. (13)
ST 0
E E 4 ≥ − 2
(cid:0) (cid:1)(cid:0) (cid:1)
The fidelity of the entanglement purification using
pairs of DFS qubits (see Sec. IIA5) is calculated using
the chaining property for fidelity measures [65]. Assum-
FIG. 7: Schematic picture of a repeater node. The four
ing that the two DFS qubits on one side A or B are not register atoms per qubit (black circles) are stored in a one-
entangled with each other we find as a lower bound dimensionalarrayofdipoletraps,e.g.,anopticallattice. Be-
tween thethird and fourth atom thereis a free lattice site in
3 which the auxiliary atom (gray) can be moved, e.g., via an
F ( , ) 1 +√2 γt. (14)
EP E0 E ≥ − 2 optical conveyor belt.
(cid:18) (cid:19)
In both formulas, Eqs. (13) and (14), the time t is again
the time needed for performing one CPHASE gate. As is transferred to the auxiliary atom, e.g., by a cavity-
wewillshowinSec IIIDalloftheseinfidelities aresmall enhanced interaction [31, 32]. The atom is then moved
for typical experimental decoherence times. to the DFS qubit and the operation shown in Fig. 3 is
In summary, we have shown in this section that the performed. In order to implement the rotations on the
DFS quantum repeater can be divided into simple mod- DFS qubits we also have to be able to selectively lower
uleswhichrequire(apartfromsinglequbitrotationsand lattice barriersbetweencertain lattice sites. This can be
measurements of an ancilla atom which can be realized done by using suitable superlattices or by applying sin-
by standard methods) merely the operations Rx( π), gle laser pulses as described in [19]. The addressability
Rx( π), Rz( π), Rz(π), and a CPHASE gatLe ±on2 a of single atoms is only required for the initialization of
DLFS±qubitwLit±h2the anLcillaatomascontrolqubit. Inthe the register, cf. Sec. IIIA.
followingsectionweproceedby describinghowthese ba- The general Hamiltonian for a one-dimensional atom
sic operationscanbe implementedinsystems oftrapped chainforM latticesites(withoutauxiliaryatom)isgiven
neutral atoms. by [38]
M−1
Hˆ = J(a)aˆ†aˆ J(b)ˆb†ˆb +h.c.
III. PHYSICAL IMPLEMENTATION IN − j j j+1− j j j+1
ARRAYS OF DIPOLE TRAPS Xj=1 (cid:16) (cid:17)
M
U U
+ aaˆ†aˆ†aˆ aˆ + bˆb†ˆb†ˆb ˆb +U aˆ†aˆ ˆb†ˆb ,
The physical system we consider for the implementa- 2 j j j j 2 j j j j ab j j j j
tion of a DFS qubit is a one-dimensional chain of four Xj=1(cid:18) (cid:19)
(15)
atoms in an arrayofoptical dipole traps,e.g., an optical
lattice [38] with one atom per site. Each of the atoms
whereJ(α) arestatedependenthoppingmatrixelements
hastwolong-livedinternalstatesa andbwhichencodea j
physicalqubit aˆ† vac 0 ,ˆb† vac 1 . InFig.7 two fromsitej tositej+1andUa,b,Uab arestatedependent
| i≡| i | i≡| i on-siteinteractionstrengths. Weshouldmentionthatfor
DFS qubits are schematically shown. The thick, diago-
the rotations Rx and Rz and for the CPHASE gate the
nal line represents an optical conveyor belt [53] storing L L
hopping constants for both kinds of atoms are required
the auxiliary atom. This atom can be transported to
the DFSstringswhereitoccupiesanempty sitebetween to be the same, Jj(a) = Jj(b), as well as Ua = Ub = Uab,
the third and the fourth atom, which is necessary for cf. alsoAppendix A. Apartfromtheinitializationofthe
theCPHASEoperationsdescribedinSec.II. Asidefrom register we need M =5 lattice sites.
this, the auxiliary atom is needed for the state transfer In the following subsections we will show how the reg-
oftheflyingqubittotheDFSqubit: Theflyingqubitsin istercanbe preparedinstate |0iL andhowthe rotations
(x,z)
ourrepeaterschemearerepresentedbyentangledphoton R andtheCPHASEgatecanbeimplemented. These
L
pairs. After a photonarrivesata repeater node its state threeingredientsaresufficienttorealizetherepeaterpro-
7
tocol. 1. Analytical treatment
The Hamiltonian H is, apart from a sign, equivalent
Z
to a swap of the first two atoms V [see Eq. (4)]. If we
12
A. Initialization of state 0
| iL assume high potential barriers between the second, the
third, and the fourth atom, i.e., J(a,b) =J(a,b) =0, such
2 3
The preparation of state 0 requires the creation of that these atoms form a separate, decoupled system, we
| iL
apairofsingletstates(01 10 )/√2. Thesestatesare merely have to describe the dynamics of the first and
| i−| i
decoupled from global operations but can be realized by the second atom. If we further assume that Ua,b,Uab are
localoperationsandbyselectivelyloweringthepotential much larger than the hopping terms J(a,b) we can adia-
j
barriers between neighboring atoms. Starting out from baticallyeliminateallstateswithmorethanoneatomin
theproductstate 0110 thepotentialbarrierbetweenthe a lattice site. By choosing the parameters according to
| i
first and second atom and between the third and fourth
atomarelowered(e.g.viaasuperlatticepotential). This J(a) =J(b) =J, (18)
1 1
generates two double well type potentials. The parame-
U =U =U =U, (19)
a b ab
ters of the Hamiltonian Eq. (15) are then given by
we arrive at a the effective Hamiltonian
J2 =0, J1(,a3) =J1(,b3) =J, Ua =Ub =Uab =U. (16) HZ =−JU2 (σ1xσ2x+σ1yσ2y +σ1zσ2z +1112)
(20)
Assuming that J/U is sufficiently small (see also 2J2 2J2
Sec. IIIB), states with two atoms in one site can be adi- = V12 = ZL,
− U U
abaticallyeliminatedleadingtoaneffective Hamiltonian
[67] of the form where we omitted a constant term. The time evolution
of the system then generates the rotation
J2
Hini =−U (σ1xσ2x+σ1yσ2y+σ1zσ2z RLz(θ)=exp −i2θZL , (21)
+σxσx+σyσy +σzσz). (17) (cid:18) (cid:19)
3 4 3 4 3 4
where θ = 4J2t/U~. Hence, the time for a Rz(π)-
L
Turning this Hamiltonian on for a time t=π~U/8J2 we rotation is given by t = πU~/4J2. As an example we
consider Na atoms in an optical lattice with a wave-
obtainthe state (01 +i 10 ) (10 +i 01 )/2. Finally,
| i | i ⊗ | i | i length of λ = 514nm. The height of the transverse po-
alocaloperationonthefirstandfourthphysicalqubitof
the form exp( iπσz/4) and exp( iπσz/4), respectively, tential barriers are assumed to be Vy = Vz = 50ER,
− 1 − 4 which is in the range of today’s experiments [43]. Here,
yieldsthedesiredproductofsingletstates. Alternatively,
E = ~2(2π)2/2mλ2 is the recoil energy and m is the
methods proposedin[68]canbeusedtorealizethe state R
mass of the atoms. If we increase the s-wave scatter-
(01 + 10 ) (10 + 01 )/2. Localsinglequbitrotations
| i | i ⊗ | i | i ing length by a factor of 5 using a Feshbach resonance
of the first and fourth qubit can be used to bring this
andchooseV =7.7E betweenthe firstandsecondlat-
state into the desired from. If laser addressing of single x R
tice site we get a ratio U/J = 75 which is well in the
physical qubits in adjacent lattice sites is not possible
regime where the adiabatic elimination is valid, as will
we can move the first and the fourth atom into their
be discussed in more detail below. Note that for these
neighboring, free sites (and back again) by lowering the
parameters the assumptions for the validity of the Bose-
corresponding potential barriers.
HubbardHamiltonianEq.(15)arefulfilled. Thehopping
In order to write information into the register we re-
term is then calculated to be J =0.033E which results
quire a free lattice site between the third and fourth R
inanoperationtimeoft=8.7msfortheRz(π)-rotation.
atom. Hence, in a last step the fourth atom is moved L
We emphasize that we never leave the DFS during the
intothefifthlatticesitebyloweringthe potentialbarrier
rotationanditisexpectedthatthistimeismuchshorter
between the fourth and fifth site.
than the decoherence time of the DFS qubit.
Nevertheless, the time for a Rz(π)-rotation can be
L
made considerably shorter by tuning the interaction be-
tween the register atoms to zero (e.g., via a Feshbach
B. Rotations of the logical qubit
resonance). As we show in Appendix B, the qubits still
remain in the DFS for all times. In this interaction-
Inthissectionweanalyticallyinvestigatehowtoimple- free case, the exact dynamics of the two atom system
ment the rotations RLx and RLz in the two limiting cases (i.e., without adiabatic elimination) results after a time
ofaveryhighinteractionbetweentheregisteratomsand t=π~/2J in the mapping
a vanishing interaction. The system is simulated numer-
ically to check our results. α,β β,α , α,β =0,1. (22)
| i12 →−| i12
8
−1 (a) (b)
10
100 100
−3 10−3 10−3
ππ) / 10 1−fZ10−6 1−fZ10−6
−
φ( 10−5 101−90−4 10−2 100 10−09.9 1 1.1
J / J U /U
2 1 ab
−7
10
1 2 3
10 10 10
U/J FIG. 9: Fidelity Eq. (27) for the rotation Rz(π). In (a) the
L
hoppingtermbetweenthesecondandthirdatom isdetuned,
in (b) the interaction between register atoms of kind a and
FIG. 8: Numerical studies of the phase difference φ between b. The other parameters are (a) Ua,b = Uab = 100J1, (b)
states|0iL and|1iL afterapplyingtherotation RLz(π)versus Ua,b=U =100J.
the ratio U/J calculated with the full Hamiltonian Eq. (15)
and the parameters Eqs. (18) and (19) given by adiabatic
0
10
elimination.
−4
10
Up to a global phase this is equivalent to a Rz(π)-
L
rotation on the logical qubits. The time needed for a
−8
lattice as above is t = 0.23ms. The disadvantage of this 10
scheme is that only rotations with a phase of π can be 1−f
performed. 10−12 Z
(φ − π) / π
According to Eq. (3) the implementation of H in-
X
volvesthree atoms. Inthe samespiritas for the ZL-gate 10−5 10−4 10−3 10−2 10−1 100
we require that the potential barrier between the third U/J
and fourth lattice site is high, i.e., J(a,b) =0. By choos-
3
ing the remaining parameters as
FIG.10: Numericalstudiesforafastrotationwithnooronly
weak interaction between the atoms. The fidelity Eq. (27)
J2(a) =√2J1(a) =√2J1(b) =J2(b) ≡√2J, (23) of the rotation RLz(π) and the phase difference between the
U =U =U =U, (24) two logical qubits after this rotation versus the ratio U/J is
a b ab
calculated with the full Hamiltonian Eq. (15). The accuracy
andrequiringthatJ/U 1,weagainadiabaticallyelim- of the gate gets better with a smaller interaction U between
inate all states with mo≪re than one atom in one well. A theatoms. Note that here no adiabatic elimination has been
used.
detailed calculation is given in Appendix A. The result-
ing Hamiltonian on the logical qubits is then given by
chosen according to Eqs. (18)-(19).We numerically cal-
culated the fidelity of the DFS qubit remaining in its
2√3J2
H = X , (25) state during an Rz-rotation,
X L L
− U
f l exp( iH t/~) l 2 (27)
where we omitted a constant term. This Hamiltonian Z ≡|h |L − Z | iL|
generates a rotation with l = 0,1. For U/J = 75 it turns out that the in-
fidelity 1 f is smaller than 3 10−3 for l = 1 and
Z
Rx(θ)=exp iθX , (26) smaller th−an 10−9 for l=0 during×the whole durationof
L − 2 L a Rz(2π)-rotation.
(cid:18) (cid:19) L
Since this fidelity does not take the phase difference φ
where θ = 4√3J2t/U~. Taking the numbers of the between states 0 and 1 into account, φ was calcu-
above examp−le, the time required for a RLx(−π) rotation lated as well. In| FiLig. 8 th|eiLphase difference is shown for
is t=5.0ms. the rotation Rz(π). As expected, for increasing ratios
L
U/J the phase difference gets closer to π. For U/J 75
≈
the inaccuracy of the achieved phase is already smaller
2. Numerical simulation than10−3,whichshowsthatthe adiabaticeliminationis
well justified.
Totestthevalidityoftheadiabaticeliminationwesim- From an experimental point of view it is also impor-
ulated the system numerically. In case of Rz-rotations tanttoknowwhathappenswhensomeoftheparameters
L
the parameters of the exact Hamiltonian Eq. (15) are aredetuned. SomeresultsareshowninFig.9. Theerror
9
0 (a) (b)
10
10−1 10−1
−2
10 10−2 10−2
X X
−fX10−4 1−f 10−3 1−f 10−3
1
−6 10−4 10−4
10 0.9 1 1.1 0.9 1 1.1
J2/√2J1 Uab / U
−8
10
1 2 3 4
10 10 10 10
U/J
FIG. 12: Fidelity Eq. (28) for therotation Rx(π). In (a) the
L
hoppingtermbetweenthesecondandthirdatom isdetuned,
FIG. 11: Numerical studies of the fidelity Eq. (28) of the in (b) the interaction between register atoms of kind a and
rotation RLx(π) versus the ratio U/J calculated with the full b. The other parameters are (a) Ua,b = Uab = 100J1, (b)
Hamiltonian Eq. (15) and the parameters Eqs. (23) and (24) Ua,b=U =100J.
given byadiabatic elimination.
duetothenonvanishinghoppingbetweenthesecondand
third lattice site J2 is negligible as long as J2 <10−2J1. as mentioned in Sec. IIA1, ZL = −V34. We assume
thatthe potentialbarriersbetweenthe first,second,and
Because of this we have completely neglected the influ-
third atom of the DFS qubit are high enough to ignore
ence of the fourth register atom which is not involved in
the dynamics of the first two atoms. The auxiliary atom
these rotations. In the case of a detuning of the inter-
is assumed to be distinguishable from the DFS atoms
actioncoefficientU the fidelity changesquite quickly if
ab
U /U differs from1, butis smaller than10−3 as longas and is confined in a sufficiently deep lattice such that it
ab
alwaysremainsatitssite. Iftheauxiliaryatomismoved
the detuning is less than 2%.
to the free site of the DFS string (cf. Fig. 7) and if a
The fidelity Eq. (27) and the phase difference φ be-
tween the logicalqubit for the faster scheme of a Rz(π)- register atom tunnels into the site of the auxiliary they
L interact with each other giving the Hamiltonian
rotationforsmall,residualinteractionsU/J (intheideal
case this interaction is vanishing) are shown in Fig. 10.
For this case the inaccuracy of the gate decreases with
decreasing U/J and is smaller than 2 10−3 for U/J Hˆ =Hˆ + Uqaˆ†aˆ qˆ†qˆ +Uqˆb†ˆb qˆ†qˆ , (29)
10−2. Theerrorcalculatedaccordingto×Eq.(27)isorde≈rs CPHASE σ 4 4 σ σ σ 4 4 σ σ
σ=0,1
of magnitudes smaller than the phase inaccuracy. X
The simulations of the Rx-rotations lead to similar
L
numbers. The fidelity of this process, determined by
where Hˆ is the Hamiltonian given by Eq. (15), qˆ is
σ
f 0 exp( iH t/~) 1 2, (28) the state-dependent annihilation operator of the auxil-
X ≡|h |L − X | iL| iary atom with q its internal state, and Uq the interac-
σ
is shown in Fig. 11 depending on the ratio U/J. As can tion strength which is independent of the internal state
be seen from this figure, the infidelity 1 f decreases of the DFS atoms. If the interaction Uq for state 0 of
− X 0 | iq
with increasing ratio of U/J and for a U/J 75 the theauxiliaryatomisverylargecomparedtothehopping
infidelity is smaller than 10−3. Any infidelities≈because strength, the third or fourth register atoms cannot en-
ofachangeoftherelativephasesbetweenthetwological ter the lattice site of the auxiliary atom and hence they
states areordersofmagnitudes smaller andplayno role. willremainintheirsites. Numericaltestsshowthatdur-
Figure 12showsthe results ofnumericalsimulationsif ing the gate operation this still holds for an interaction
various system parameters are detuned. For the case of strength of Uq/J 100, where J is the hopping term of
0 ≈
a nonidealhopping the deviation shouldbe smaller than the register atoms. Under these circumstances the prob-
1.5% to yield infidelities smaller than 3 10−3. A slight abilityoffindingthe registerinitsoriginalstateislarger
detuning in the interaction strength is×less critical, but than 1 10−3. If the auxiliary atom is in the state 1
− | iq
should nevertheless be smaller than 4% to achieve small and Uq = 0 the existence of the auxiliary atom can be
1
errors. ignored and we only need to solve the dynamics of two
registeratomsinthreelatticesites. Asimilarschemehas
been discussed in [69, 70], where an electromagnetically
C. Controlled phase gate inducedtransparency-likeconfigurationwasusedtocon-
trol the interaction strength. We will investigate in the
TheCPHASEgatecanberealizedbyacontrolledswap followingthe twolimiting casesofverylargeandvanish-
operation between the third and the fourth atom since, ing interaction strength between the register atoms.
10
(a) (b) (a) (b)
100 10−2 100 100
1− f 1−f
00 00
Infidelity10−4 1− f01 1−fph10−4 Infidelity1100−−36 1−f01 1 − fph1100−−36
101−80−4 U1q0 −/ 2J 100 101−60−4 U1q0 −/ 2J 100 101−90−4 1U0/−J2 100 101−90−4 1U0/−J2 100
1 1
FIG. 13: (a) Numerical results for the fidelities Eqs. (30) FIG. 14: (a) Fidelity after evolving the state for a time t =
and (31), where the auxiliary atom has a nonvanishing rest π~/√2J for the case of noninteracting register atoms. (b)
interactionwiththeregisteratoms. (b)Numericalresultsfor FidelityEq.(35)ofstate|1iL afterthetimeevolution,where
the fidelity Eq. (35) due to the phase difference, where the aperfectfidelityoftheatomswapoperationhasbeenassumed
auxiliary atom has a nonvanishing rest interaction with the and theerror is only dueto the phasedifference, see text.
register atoms. In both cases we took for the register atoms
U/J =100.
entirely contained in the DFS anymore. To study this
error we define the fidelity
1. Large interaction between the register atoms
5+4cosϕ
f = 1 (t) 1 2 = , (35)
ph |h |LU | iL| 9
For the case of large U/J we calculate the fidelities of
the swap operation, whichisshowninFig.13bdependingonUq/J. Theinfi-
1
delity 1 f istypically oneorderofmagnitude smaller
f01 = 0σ1 (t) 1σ0 2 , (30) than the−inpfihdelities shown in Fig. 13a.
|h |U | i|
f =f = 0σ0 (t) 0σ0 2 (31) The time which is needed to perform the swap gate
00 11
|h |U | i| hasbeencalculatednumerically. ForaratioofU/J =75
numerically,where tis the numericallyobtainedtime for and the same lattice parameters as for the Rz-rotations
L
the CPHASE gate and (t) = exp( iHˆ t/~) is we get t=11.4ms.
CPHASE
U −
the time evolution operator corresponding to Hamilto- NumericaltestsshowthatforthesameratioU/J =75
nian Eq. (29). The states ασβ , α,β = 0,1 denote the probability of finding the qubit outside the DFS is
the states of the third and |the fiourth DFS atom, α smaller than 5 10−4 during the whole gate operation
| i ×
and β , and σ =1 is the internalstate of the auxiliary time.
| i | i
atom,locatedinbetweenthetwoDFSatoms. Wechoose
J(a,b) =J and U =U =U.
a,b ab
In Fig. 13a the fidelities Eqs. (30) and (31) are shown 2. No interaction between DFS atoms
for a small but nonvanishing interaction Uq between the
1
auxiliary atom and the register atoms for a ratio U/J = A substantially shorter gate time can be achieved for
100. Ascanbeseentheinfidelitiesaresmallerthan10−3 the case of a vanishing interaction between the DFS
as long as the interactionfulfills Uq/J <0.05. A further atoms. In Appendix B it is shown that the qubits still
1
numerical observation is that the value of f00 (f01) for remain in the DFS during the gate operation. By ana-
very small Uq/J gets smaller with increasing U/J. lytically solvingthe time evolutionof the Hamiltonianit
1
The fidelities Eqs. (30) and (31) do not take into ac- can be shown (see also Appendix B) that after a time
counttheoccurrenceofphases. Ingeneral,thedynamics t=π~/√2J the dynamics yields the mapping
of the system will yield a phase,
ασβ βσα ,α,β =0,1, σ =1 (36)
ασβ eiϕαβ βσα (32) | i→| i
| i→ | i
which is a swap operation between the third and the
withα,β =0,1. SincetheinteractionsUσq donotdepend fourth DFS atom and thus a CPHASE gate. For the
ϕon00t=heϕs1t1ataendofathneonDvFanSisahtionmgsUw1qelehaadvsetoϕ0t1he=beϕh1a0vaionrd sgaamteetilmatetiicsegpivoetnenbtyiatl =as0e.3a3rlmiesr. with Vx = 7.7ER the
The results of numerical tests for a small rest inter-
(t) 0 =eiϕ10 0 (33)
U | iL | iL action between the register atoms are shown in Fig. 14.
(t) 1 =eiϕ10 1 [2(1100 + 0011 ) Figure 14a shows the infidelities derived form Eqs. (30)
U | iL 2√3 | i | i and (31) and Fig. 14b shows the infidelity derived from
eiϕ(01 + 10 )⊗2], (34) Eq. (35) all depending on U/J. The fidelities are better
− | i | i than 1 10−3 for interactions U/J < 0.05. The results
−
of the four DFS atoms, i.e., to the presence of a nonzero of numerical simulations for a nonvanishing rest inter-
phase ϕ ϕ ϕ . Thus the state (t) 1 is not action between the DFS atoms and a state 1 of the
≡ 00 − 10 U | iL | iq