- 1.Long range failure-tolerant entanglement distribution Ying
LiCentre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, Singapore 117543Sean D. Barrett
Blackett Laboratory and Institute for Mathematical
Sciences,Imperial College London, London SW7 2PG, United Kingdom
Thomas M. Stace School of Mathematics and Physics, University of
Queensland, Brisbane, QLD 4072, Australia andCentre for Quantum
Technologies, National University of Singapore, 3 Science Drive 2,
Singapore 117543Simon C. BenjaminarXiv:1209.4031v1 [quant-ph] 18
Sep 2012Department of Materials, University of Oxford, Parks Road,
Oxford OX1 3PH, UK andCentre for Quantum Technologies, National
University of Singapore, 3 Science Drive 2, Singapore 117543 We
introduce a protocol to distribute entanglement between remote
parties. Our protocol isbased on a chain of repeater stations, and
exploits topological encoding to tolerate very high levelsof
defects and errors. The repeater stations may employ probabilistic
entanglement operations whichusually fail; ours is the rst protocol
to explicitly allow for technologies of this kind. Given an
errorrate between stations in excess of 10%, arbitrarily long range
high delity entanglement distributionis possible even if the
heralded failure rate within the stations is as high as 99%,
providing thatunheralded errors are low (order 0.01%).PACS
numbers:Introduction. Distributing an entangled state among the
rate of distributing entanglement decreases only log- remote
quantum computers is one of the fundamental arithmically with the
communication distance. tasks of quantum information technologies.
It is crucialCluster states are resources of measurement-based for
quantum teleportation, quantum cryptography andquantum computing
[5], and long-range entanglement distributed quantum computing.
Using direct transmis-can be established in noisy cluster states
[6]. In this sion, the success probability of transmitting a qubit
andpaper, we propose a protocol of distributing entangle- the
delity of the resulting quantum state decrease expo-ment by
single-qubit measurements on a topologically nentially with
distance. Therefore, one needs quantumprotected cluster (TPC) state
[7] across the chain of re- repeaters to achieve long distance
entanglement [1, 2]. Apeater stations. The TPC state must rst be
grown good quantum repeater protocol should be fault-tolerantvia
operations within repeaters together with quantum and support a
high communication rate. In this paper,communication between pairs
of neighboring repeaters. we will propose a protocol to distribute
entanglement be-The operations within repeaters are expected to
have a tween two remote quantum computers. We consider noisemuch
better performance than communications between in quantum
communication channels, and of course errorsrepeaters (since the
latter may be over distances of kilo- generated by operations
within the repeaters. We assumemetres). We nd that the protocol is
valid if the probabil- that the repeater stations may employ
non-deterministicity of an error occurring in the communication
channel is entanglement operations (EOs): that is, a means of en-
lower than a threshold, which is 15% when errors induced
tanglement, even within the a single repeater, that oftenby
operations within repeaters are negligible. With errors fails but
the failures are heralded. In addition there isless than the
threshold, entanglement can be established of course a nite error
rate even for the operations thatbetween two remote logical qubits
encoded in two sepa- are deemed successful. Non-deterministic EOs
will occurrated graph states, which may be used for further infor-
within individual repeater stations if, for example, their mation
processing via the topological measurement based internal hardware
is based on networking small quantumquantum computing [7].
Alternatively one can also de- registers together optically, i.e.
qubits can be entangledcode each logical qubit to a physical qubit
via single-qubit by joint measurements on single photons emitted
frommeasurements. Although we describe only the two-party these
qubits rather than control of interactions [3, 4]. protocol here,
it should be straightforward to generalize Such an architecture may
be much easier to implement for distributing multi-party
entanglement. in a scalable way than monolithic architectures e.g.
large scale ion traps. Even with this assumption that EOs failIn
this protocol, the quality of the eventual entangle- both between
and within repeater stations, we nd that ment between logical
qubits is only limited by the numberof qubits in each repeater.
Therefore, our protocol eec-
2. 2 is grown across quantum repeater stations via probabilis-
Bob tic EOs and quantum communications between nearby stations. The
TPC state contains two parallel empty Alice(a)tubes, which
terminate in stations of Alice and Bob. Each empty tube is a void
in the TPC state, with anTPC state (d) elongated shape and shown as
a blue rectangular cuboid in Fig. 1(b). Once the TPC state is
generated, measure-Zplugments in the X basis are performed on all
qubits except(b)two parts of the TPC state located in stations of
Al- ice and Bob respectively [see Fig. 1(c)]. The two parts which
are to remain unmeasured are called plugs, and X (e) are connected
with empty tubes. Two empty tubes and two plugs form a closed loop.
There is one logical qubitdecoding(c)encoded in each plug. After
all other quits are measured, and the outcomes are communicated to
Alice and Bob, iithen these two logical qubits are entangled as one
of the Bell states (determined by measurement outcomes).The TPC
state is a cluster state of qubits located i(g)(f) on the a cubic
lattice [7]. There is one qubit on each face and edge of the
elementary cell [Fig. 1(d)]. ByFIG. 1: The scheme of quantum
entanglement distributionshifting the lattice, one can transfer
qubits on faces toprotocol based on topologically protected cluster
(TPC) state. edges, and vice versa. The new lattice is called the
dual(a) Alice and Bob can be entangled via a chain of
quantumlattice of the original primal lattice. The TPC
staterepeater stations, which are connected by optical quantumis
stabilized by K(c) = ac Xa bc Zb , where c iscommunication
channels. (b) Each station contains a slice an arbitrary primal
(dual) surface and c is the primalof the TPC state. The TPC state
contains two empty tubes(blue) without any qubit. (c) Once the TPC
is complete, all(dual) chain as the boundary of c. Qubits in the
set cqubits are measured in X basis except two parts of the TPC (c)
are located on faces (edges) composing the surfacestate (green) in
stations of Alice and Bob respectively; these (chain) c (c). The
logical qubit is encoded in a plugare called plugs and contain the
eventual encoded shared Bellas X = asection Xa and Z = bline Zb ,
where Xpair. (d) The elementary cell of the TPC state. Each logical
and Z are Pauli operators of the logical qubit. Here,qubit is
encoded as subgure (e) and can be decoded as subg- section is a
dual surface across the plug, and line is aure (f) (see text). (g)
Two surfaces propagating correlationsbetween two logical
qubits.primal chain on the surface of the plug and connecting two
empty tubes [Fig. 1(e)]. We consider two stabiliz- ers according to
the following surfaces: (i) ci is a pri- mal surface whose boundary
is enclosed by the tube-tively distills as well as distributes
entanglement. The plug loop, and (ii) cii is a closed dual surface
envelop-idea of using an error correction code with protected log-
ing one empty tube and crossing two plugs [Fig. 1(g)].ical qubits
for remote entanglement was rstly reportedThe two stabilizers are
K(ci ) = Z A Z B aci Xa andin Ref. [8], in which the
Calderbank-Shor-Steane code K(cii ) = X A X B ac Xa , where A, B
denote Alice andis employed. Subsequently 3D lattice-based
distributioniihas also been studied [9] and the extension to lower
di- Bob respectively, and cii denotes the part of the
sur-mensionality has been examined [10]. Recently, in aface cii
outside two plugs. After measurements in theprotocol for quantum
state transfer of a surface-code- X basis, one can replace Xa with
measurement out-encoded qubit, the eciency of quantum communication
comes. Then, we get two new stabilizers Z A Z B = 1is greatly
improved by removing the necessity of two-and X A X B = 1, i.e. the
two logical qubits are stabi-way communication [11]. Compared with
these proto- lized as one of Bell states. Here, the two signs
depend oncols, ours is the rst to consider a probabilistic
architec- measurement outcomes.ture within each repeater station,
so that the entangle-Besides two-party entanglement, we note that
ourment distribution can be ecient even if EOs are far fromscheme
can be directly generalized to multi-party entan-deterministic.
glement, e.g. three-party and four-party entanglement as Quantum
Repeaters based on Cluster States. Alice andshown in Ref. [12].Bob
are entangled via a chain of quantum repeater sta-Noise in quantum
communication channels and imper-tions. Two nearby repeaters are
connected by optical fections in operations will give rise to phase
errors onquantum communication channels [Fig. 1(a)] essen-the TPC
state. In order to eliminate errors from the Belltially a bundle of
optical bres that are used in parallel.state of two logical qubits,
we monitor errors on the TPCTo give an overview of the process:
Firstly, a TPC state state by parity check operators K(cc ), where
cc are min- 3. 3 we overcome the impact of high EO failure rates
whenroot we create the large scale TPC state. The structure of
these elementary graph states can be a star [13], a line [14], a
cross [15], or a tree [16, 17]. In this paper, we
Tree-structuregraph statetake the tree structure as an example, and
the scheme (a) can be adapted to other structures. The tree
structure accumulates fewer errors than other structures when the
Snowflake, local resource success probability of EOs is low [17,
18]. Tree-structure p1graph states can be generated by using parity
projections 312 434 (PPs) [3]. A PP on roots of two individual
trees canp2 fuse them into a double-size tree [Fig. 2(a)]. If all
PPsDumbbell, nonlocal resource are successful, after n steps, one
can grow a tree with (c) (b) 2n qubits from separated qubits, where
the integer n is called the generation of the tree.FIG. 2: Resource
graph states, i.e. building-blocks, for grow-Trees are fused into
two kinds of building-block graphing the topologically protected
cluster state. Each red dotted states. Snowake graph states are
prepared by fusingline denotes a parity projection (PP). (a) Tree
graph statescan be grown by PPs on roots of trees. (b) Four trees
can be four trees [Fig. 2(b)]. Each snowake will ultimatelyfused
into a snowake graph state as the following: fusing correspond to a
specic qubit on the TPC state. Eacheach pair of trees into a bigger
tree at rst; cutting two roots quarter of a snowake is used to
establish a connectionby measurements in Z basis; fusing them into
a snowakewith a neighboring snowake. We refer to the secondand
cutting the unwanted qubit. (c) Two trees in dierentkind of
building-block as a dumbbell. These are nonlocalquantum repeater
stations are fused into a dumbbell graphbuilding blocks connecting
two nearby quantum repeaterstate by a Bell measurement on photon-p1
and the photon- stations [Fig. 2(c)]. A dumbbell is formed by two
treesp2, each associated with a qubit in a dierent stations. Oneof
the photons (p2) will have travelled between stations. Thelocated
in dierent stations. For example, suppose thatBell measurement is
followed by a measurement in the Y ba- the basic qubits are
optically active atoms: then in ordersis on the qubit-1 and a
measurement in the X basis on the to prepare a dumbbell, we cause
each root qubit emitqubit-2, in order to get the desired dumbbell
graph state. a single photon as | j | j | pj , where j = 1, 2 de-
notes a root qubit, pj denotes the corresponding pho- tonic qubit,
= 0, 1 is the label the state in the compu-imum closed surfaces.
Usually, minimum closed surfaces tational basis and the photonic
qubit can be encoded inare surfaces of elementary cubes. However,
some qubits polarization, frequency [19] or time-bin [20]. One
pho-on the TPC state may be missing. The parity check op-ton is
transmitted from one station to another. After aerator of an
elementary cube with missing qubits can not Bell measurement on two
photons and single-qubit mea-be used to detect errors. Then, one
has to use productssurements on roots, we obtain the dumbbell graph
stateof parity check operators connected by missing qubits[12].to
form a new set of parity check operators [23]. Par-Making a
building-block graph state requires all oper-ity check operators
reveal the endpoints of error chains,ations to be successful, whose
probability may be quitewhere an error chain (ring) is a sequence
of phase er- small. Therefore, building-block graph states are
pro-rors. If the number of phase errors on the surface cc is duced
with a post selection strategy: if an operation isodd, the
existence of errors can be identied by K(cc ), heralded as failed,
the corresponding graph state is aban-which is called an error
syndrome. Errors are not actively doned with the qubits
reinitialized.corrected, rather parities of aci Xa and ac Xa ,
areOnce a sucient number of each resource (snowakesiimodied by
knowledge of the total number of error chains and dumbbells) have
been generated, we can assemblecrossing surfaces ci and cii
respectively. After the error them to create a suitable TPC state.
Snowakes are as-correction, only error rings encircling the
tube-plug loop,sembled by PPs on leaves, which are qubits on the
edge oferror chains connecting two empty tubes and error chains a
snowake (Fig. 3). Two snowakes in the same quan-connecting the loop
with the boundary of the TPC state tum repeater station can be
connected directly, while two[12], may contribute an error on
logical qubits. If noisesnowakes in dierent stations are connected
by bridg-and imperfections are less than a threshold, the probabil-
ing them with a dumbbell shared by these two stations.ity of an
error on logical qubits decreases exponentiallyThe number of leaf
qubits on each quarter of a snowakewith the minimum length of these
error rings and error is 2n1 . Therefore, the failure probability
of connectingn1chains [7].two snowakes in the same station is FL =
f 2 , and Cluster State Growth.- In order to grow the TPC state the
failure probability of connecting two snowakes inacross quantum
repeater stations, some building-blockdierent stations is FNL 2FL ,
where f is the basic fail-graph states should rst be prepared
within each repeaterure probability of EOs. After establishing
connectionsdevice. It is through the use of these building-blocks
thatbetween snowakes, all qubits except those at the center 4.
4Dumbbellstations are perfect (when heralded as successful);
thenonly joint qubits have errors, and these imperfect qubitsexist
in specic non-adjacent layers of the TPC state.53 4 6Then error
correction can be performed independentlyon each such layer. The
error threshold of a two dimen-sional layer is about 10% in the
limit of a perfectly con- 5 6nected lattice [26]. Moreover a
near-perfectly connectedlattice would indeed be achievable since,
given error freeEOs within repeaters, one could always grow
suciently 7big tree structures to make FL as low as desired.
There-7fore, with perfect operations, the condition of getting
acorrect correlation between two logical qubits faithfully TPC
stateis 2 /3 10%, i.e. the error threshold of communicationnoise is
t 15%.FIG. 3: The strategy of assembling resource graph statesinto
the full topologically protected cluster (TPC) state whichWith
imperfect operations, all qubits on the TPC statespans all quantum
repeater stations. (a) Snowakes within may aected by phase errors.
If the distribution of phasethe same station are connected directly
to each other by par- errors is uniform, i.e. all qubits may have a
phase errority projections (red dotted lines) on leaves. Two
snowakes with the same probability, the threshold of phase errorsin
dierent stations can be connected via a dumbbell whichis about 3%
for perfectly connected TPC state [25]. How-incorporates the
required nonlocal connection (dash line). (b)ever, in our case, the
TPC state grown by probabilisticAfter extraneous quits are removed,
ultimatelty the qubits atthe heart of each snowake survive as nodes
of the TPC state.EOs is unlikely to be perfectly connected and
there aremore errors on joint qubits than others. Our strategyis to
treat missing connections by transforming them toof each snowakes
are removed by appropriate single-qubit loss, by means of deleting
the qubits with missingqubit measurements, so that the surviving
qubits form connections using measurements in the Z basis. Then,the
TPC state. Here, the measurement pattern for re-the loss
probability of joint qubits is 5FL , and the lossmoving qubits can
be found in Ref. [12]. Since some probability of other qubits is
4FL . We determine errorsnowakes have failed to connect, this
implies some miss- thresholds for general cases numerically as
shown in Fig.ing connections on the TPC state. We presently
describe 4(a), using the method developed in Ref. [22,
23].simulations establishing that when connections are rarelyThe
error rate of imperfect operations must be lowermissing, i.e. FL
< 5%, then the cluster state is well con-than the threshold of
fault-tolerant quantum comput-nected: it is easy to nd surfaces
propagating correlations ing (FTQC). The threshold of FTQC on the
TPC statebetween two logical qubits, indeed this is guaranteed
inwith non-deterministic EOs (deterministic control-phasethe
scaling limit (as expected from percolation theory) gates) is about
2104 [18] (5103 [7]). By optimizing[21, 23]. the size of trees, (a
bigger tree can reduce missing con- As a footnote to this section
we note that the building-nections but generate more errors), we
have obtained theblock strategy is not always necessary. If the
failure thresholds of tolerable communication noise in the
pres-probability of EOs is low enough f < 5%, one may gen- ence
of nite error rates for internal EOs, see Fig. 4(b). Iferate the
TPC state directly, for example, by using con-the error rate of
operations is 104 , the threshold of com-trol phase gates [7],
where control-phase gates on twomunication noise is about 11% when
the success probabil-qubits located in dierent quantum repeater
stations canity of entangling operations is 1%. In contrast, by
usingbe simulated by consuming entanglement prepared via
control-phase gates to generate the TPC state directly,quantum
communication [24]. However here we are in- the threshold of
communication noise is still above 10%terested in the general case
where the failure probabilityeven if the error rate of operation is
2 103 , but themay be very high. success probability must be higher
than 98%. Noise, Imperfections and Error Correction.- BothFull
decoding.- A logical qubit can be decoded into anoise in quantum
communication channels and imper-physical qubit by measurements on
the correspondingfections in operations can give rise to errors on
the TPC plug, leaving just one qubit unmeasured. The residualstate.
We assume communication noise is depolarized,qubit carries the
quantum state of the logical qubit. Forand described by the
superoperator E = (1 )[1p2 ] + decoding, two (blue) pyramids inside
the plug, whose ([Xp2 ] + [Yp2 ] + [Zp2 ])/3 [see Fig. 2(c)]. We
call qubits apexes hold the residual qubit (red circle) and bases
con-with nonlocal connections joint qubits [gray circles in nect
tubes, are measured in the Z basis, while otherFig. 3]. Errors
induced by communication noise mayqubits are measured in the X
basis [see Fig. 1(f)]. Themake phase errors on corresponding joint
qubits (qubits residual qubit can acquire an error if there is an
error5 and 6) with a probability 2 /3 for each of them [12]. chain
connecting two pyramids. Therefore, the proba-Consider rst the case
that internal operations withinbility of an error on the residual
qubit is p + O(p3 ) [7], 5. 50.120.15 FL=0.00While preparing this
document we became aware of FL=0.010.15 a manuscript describing
closely related research: Ash-thresholds of pJ thresholds of
0.080.10 FL=0.040.10 ley Stephens, Jingjing Huang, Kae Nemoto and
William 0.05 0.90 0.95 1.00 J. Munro, Fault-tolerant quantum
communication with0.040.05rare-earth elements and superconducting
circuits.0.000.00 -3 0.0 0.2 0.40.60.8 10-2 10 -110 100 p/pJ(a)
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