Experimental demonstration of topological error correction

Experimental demonstration of topological error correction

Xing-Can Yao1 Tian-Xiong Wang1 Hao-Ze Chen1 Wei-Bo Gao1 Austin G Fowler2 Robert Raussendorf3

Zeng-Bing Chen1 Nai-Le Liu1 Chao-Yang Lu1 You-Jin Deng1 Yu-Ao Chen1 and Jian-Wei Pan1

1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern PhysicsUniversity of Science and Technology of China Hefei Anhui 230026 PR China

2CQC2T School of Physics University of Melbourne VIC 3010 Australia3Department of Physics and Astronomy University of British Columbia Vancouver BC V6T 1Z1 Canada

(Dated February 27 2012)

Scalable quantum computing can only be achieved if qubits are manipulated fault-tolerantlyTopological error correctionmdasha novel method which combines topological quantum computing andquantum error correctionmdashpossesses the highest known tolerable error rate for a local architectureThis scheme makes use of cluster states with topological properties and requires only nearest-neighbour interactions Here we report the first experimental demonstration of topological errorcorrection with an eight-photon cluster state It is shown that a correlation can be protectedagainst a single error on any qubit and when all qubits are simultaneously subjected to errorswith equal probability the effective error rate can be significantly reduced This demonstrates theviability of topological error correction Our work represents the first experimental effort to achievefault-tolerant quantum information processing by exploring the topological properties of quantumstates

Quantum computers exploit the laws of quantum me-chanics and can solve many problems exponentially moreefficiently than their classical counterparts [1ndash3] How-ever in the laboratory the ubiquitous decoherence makesit notoriously hard to achieve the required high degree ofquantum control To overcome this problem quantumerror correction (QEC) has been invented [4ndash6] Thecapstone result in QEC the so-called threshold theorem[7 8] states that as long as the error rate p per gate ina quantum computer is smaller than a threshold valuepc arbitrarily long and accurate quantum computationis efficiently possible Unfortunately most methods offault-tolerant quantum computing with high threshold(10minus4minus 10minus2) require strong and long-range interactions[7ndash9] and are thus difficult to implement Local archi-tectures are normally associated with much lower thresh-olds For traditional concatenated codes on a 2D latticeof qubits with nearest-neighbour gates the best thresh-old known to date [10] is 202times 10minus5

In such lattices it is advantageous to employ topolog-ical error correction (TEC) [12ndash15 S2] in the frameworkof topological cluster-state quantum computing Thisscheme makes use of the topological properties in three-dimensional (3D) cluster states which form an inherentlyerror-robust ldquofabricrdquo for computation Local measure-ments drive the computation and at the same time im-plement the error correction Active error correction andtopological methods are combined yielding a high errorthreshold [12 13] of 07ndash11 and tolerating loss rates[15] up to 249 This leaves room for the unavoidableimperfections of physical devices and makes TEC closeto the experimental state of the art The 3D architecturecan be further mapped onto a local setting in two spatialdimensions plus time [14] also with nearest-neighbour in-teractions only Two detailed architectures have alreadybeen proposed [16 17] Note that a distinct and also im-portant topological scheme has been proposed in which

quantum computation is driven by non-abelian anyons[18 19] and fault tolerance is achieved via passive stabi-lization afforded by a ground-state energy gap

Some simple QEC codes have been experimentallydemonstrated in nuclear magnetic resonance [20 21] iontraps [22 23] and optical systems [24 25] However theexperimental realization of topological QEC methods stillremains a challenging task The state-of-the-art technol-ogy for generating multipartite cluster state is up to sixphotons while great endeavor is still underway to createnon-ablelian anyons for the topological quantum comput-ing [18 19] Here we develop an ultra-bright entangled-photon source by utilizing an interferometric Bell-typesynthesizer Together with a noise-reduction interfer-ometer we generate a polarization-encoded eight-photoncluster state which is shown to possess the required topo-logical properties for TEC In accordance with the TECscheme we measure each photon (qubit) locally Errorsyndromes are constructed from the measurement out-comes and one topological quantum correlation is pro-tected We demonstrate (1) if only one physical qubitsuffers an error the noisy qubit can be located and cor-rected and (2) if all qubits are simultaneously subjectedto errors with equal probability the effective error rate issignificantly reduced by error correction Therefore wehave successfully carried out a proof-of-principle experi-ment that demonstrates the viability of Topological ErrorCorrectionmdasha central ingredient in topological cluster-state computing

Cluster states and quantum computing

In cluster-state quantum computing [26] projectiveone-qubit measurements replace unitary evolution as theelementary process driving a quantum computation Thecomputation begins with a highly entangled multi-qubit

arX

iv1

202

5459

v1 [

quan

t-ph

] 2

4 Fe

b 20

12

2

ba

t

FIG 1 Topological cluster states The elementary lattice cell Dashed lines represent the edges of the associated cellcomplex while solid lines are for the edges of the interaction graph Qubits live on the faces and the edges of the elementarycell b A larger topological cluster state of 5 times 5 times T cells Green dots represent local Z measurements effectively removingthese qubits from the cluster state and thereby creating a non-trivial topology capable of supporting a single correlation Reddots represent Z errors Red cells indicate CF = minus1 at the ends of error chains One axis of the cluster can be regarded assimulating the ldquocircuit timerdquo t The evolution of logical states from t1 to t2 is achieved by performing local X measurementson all physical qubits between t1 and t2

state the so-called cluster state |G〉 [27] which is speci-fied by an interaction graph G and can be created from aproduct state via the pairwise Ising interaction over theedges in G For each vertex i isin G one defines a stabilizeras Ki equiv Xi otimes

eijZj where the product is over all the in-

teraction edges eij connecting vertex i to its neighbour-ing vertex j As usual symbols Xi and Zj denote thebit- and phase-flip Pauli operators respectively actingon qubits i and j State |G〉 is the unique joint eigen-state of a complete set of stabilizers Ki Ki |G〉 = |G〉for all the vertices i isin G

Cluster states in d ge 3 dimensions are resourcesfor universal fault-tolerant quantum computing [12]Therein the TEC capabilitymdashshared with Kitaevrsquos toriccode [28 S2] and the color code [29]mdashis combined withthe capability to process quantum information

Topological error correction

Quantum error correction and fault-tolerant quantumcomputing are possible with cluster states whenever theunderlying interaction graph can be embedded in a 3Dcell structure known as a cell complex [30] which con-sists of volumes faces edges and vertices Qubits liveon the edges and faces of a cell complex The associ-ated interaction graph connects the qubit on each faceto the qubits on its surrounding edges via the inter-action edges Consider the elementary cell complex inFig 1a shown by the dashed lines it has 1 cubic vol-ume 6 square faces 12 edges and 8 vertices The in-teraction edges specified by the solid lines form an 18-

qubit cluster state |G18〉 There are 6 face stabilizersKf (f = 1 2 middot middot middot 6) It follows that multiplication ofthese stabilizers cancels out all Z operators in Kf andthus yields a unit expectation value 〈X1X2 middot middot middotX6〉 = 1This leads to a straightforward but important observa-tion that despite the X-measurement on each individualface-qubit having random outcome plusmn1 the product ofall the outcomes on any closed surface F is +1 Namelyany closed surface has the topological quantum correla-tion CF equiv 〈otimesfisinFXf 〉 = 1

A larger cell complex is displayed in Fig 1b whichencodes and propagates a logical qubit It consists of5times5timesT cells with T specifying a span of simulated timet A ldquodefectrdquo along the t direction (shown as the line ofgreen dots in Fig 1b) is first carved out via performinglocal Z measurements Then the topological quantumcorrelation CFD

= 1 on a defect-enclosing closed surfacecombined with the boundary is used to encode a logicalqubit The evolution of the logical state from t1 to t2 isachieved by local X measurements on all other physicalqubits between t1 and t2 (see Ref [31] for the details)Quantum computing requires a much larger cell complexand more defects where quantum algorithms are realizedby appropriate braiding-like manipulation of defects (asketch for the logical CNOT gate is shown in Appendix)

The quantum computation is possible precisely due tothe topological quantum correlation CFD

= 1 on defect-enclosing closed surfaces FD The TEC capability arisesfrom the Z2 homology a topological feature of a suffi-ciently large 3D cell complex (see Appendix) For a givendefect-enclosing closed surface FD there exist many ho-mologically equivalent closed surfaces that represent thesame topological correlation CFD

= 1 This redundancy

3

1 52 6 3 4

7

8

b

a

e7 e8zywv

s

f4f3f6f5f2f1

Right View

t

FIG 2 Cluster state |G8〉 and its cell complexa Interaction graph G8 of |G8〉 b The correspondingthree-dimensional cell complex with volumes v w y z facesf1 f2 f3 f4 f5 f6 edges e7 e8 and vertices s t The exte-rior and the center volume are not in the complex For betterillustration the cell complex is cut open and the front upquarter is removed cf ldquoright viewrdquo

leads to the topological protection of the correlation [12]Remarkably in TEC it is sufficient to deal with Z er-

rors because an X error has either no effect if immedi-ately before X measurements or is equivalent to multipleZ errors Finally as a measurement-based quantum com-putation corrections suggested by TEC are not appliedto the remaining cluster state but rather to the classicaloutcomes of X measurements

Simpler topological cluster state

The cell complex in Fig 1b encodes a propagating log-ical qubit via one topological correlation CFD

= 1 and isrobust against a local Z error Unfortunately it contains180 physical qubits per layer significantly beyond thereach of available techniques We design a simpler graphstate |G8〉 shown in Fig 2a to mimic the cell complex

TABLE I Location of a Z error in |G8〉 and the syndromesC12 = 〈X1X2〉 etc

Zerror C12 C25 C36 C34

1 -1 1 1 1

2 -1 -1 1 1

3 1 1 -1 -1

4 1 1 1 -1

5 1 -1 1 1

6 1 1 -1 1

of Fig 1bThe topological feature of |G8〉 can be seen via its

association with the 3D cell complex in Fig 2b whichconsists of 4 elementary volumes v w y z 6 facesf1 f2 f3 f4 f5 f6 2 edges e7 e8 and 2 verticess t All 6 faces have the same boundary e7 cup e8 andany two of them forms a closed surface F The center vol-ume is carved out resembling the defect in Fig 1b andthe to-be-protected topological correlation CFD

reads

CFDequiv 〈X5X6〉 = 1 (1)

In this simple cell complex the topological correlationCFD

= 1 is already multiply encoded represented by anyexpectation 〈XiXj〉 with i isin 1 2 5 and j isin 3 4 6Moreover there exist four other closed surfaces withoutenclosing the defect corresponding to the boundary ofvolumes v w y z respectively The ldquoredundantrdquo topo-logical correlations are

〈X1X2〉 = 〈X2X5〉 = 〈X3X6〉 = 〈X3X4〉 = 1 (2)

and can be used as error syndromes in TEC As shownin Table 1 a single Z error on any physical qubit can belocated and corrected

Therefore from the aspect of TEC capability the clus-ter state |G8〉 is analogous to the cell complex in Fig 1bThey protect one topological correlation and are robustagainst a single Z error albeit the cell complex in Fig 2bis too small to propagate a logical qubit (see Appendixfor detailed discussion)

Preparation of the eight-photon cluster state

In our experiment the desired eight-qubit cluster stateis created using spontaneous parametric down-conversionand linear optics The first step is to develop an ultra-bright and high-fidelity entangled-photon source Asshown in Fig 3a an ultraviolet mode-locked laser pulse(915 mW) passes through a β-barium borate (BBO) crys-tal generating a pair of polarization-entangled photonsin the state |φ〉 = (|HH〉+ |V V 〉)

radic2 By an inter-

ferometric Bell-state synthesizer [32] photons of differ-ent bandwidths (shown by red and blue dots in Fig 3arespectively) are guided through separate paths This

4

b

ca

a

b

a

b

HWPQWP

HWP

QWP

8nm

28nm

1

3

7

5

2

4

6

8

6rsquo

4rsquo

8rsquo

PBS1PBS2

PDBSrsquo

PDBSrsquo

PDBS

FIG 3 Experimental setup for the generation of the eight-photon cluster state and the demonstration oftopological error correction a Creation of ultra-bright entangled photon pairs An ultraviolet laser pulse passes through a2 mm nonlinear BBO crystal creating an entangled photon pair by parametric down conversion with ρ = 1

2(|Ho

a〉 |V eb 〉 〈V e

b | 〈Hoa |+

|V ea 〉 |Ho

b 〉 〈Hob | 〈V e

a |) where o and e indicate the polarization with respect to the V -polarized pump After both photons passthrough compensators including a 450 HWP and a 1 mm BBO crystal one of the photonsrsquo polarizations is rotated by another450 HWP Then we re-overlap the two photons on a PBS creating an entangled photon pair with |φab〉 = 1radic

2(|H〉 |H〉 +

eiϕ |V 〉 |V 〉) otimes |ea〉 |ob〉 b In order to create the desired cluster state we combine photons from path 6 and 8 at PDBS andlet each photon pass through another PDBSrsquo resulting a controlled-phase operation between photon 6 and 8 Meanwhilephoton 2 and photon 4 are interfered on PBS1 In the end photon 4rsquo and photon 6rsquo are overlapped on PBS2 Upon acoincidence detection we create the eight-photon cluster state (3) for topological error correction c Polarization analyzer foreach individual photon containing a QWP an HWP a PBS and two single-mode fibre-coupled single-photon detectors

disentangles the temporal from the polarization informa-tion In contrast to the conventional narrow-band fil-tering technique there is no photon-loss problem andthus an ultra-high brightness is achieved Four pairs ofsuch entangled photons are prepared and labelled as 1-2 3-4 5-6 and 7-8 in Fig 3b Then we generate twograph states each of four photons The first one is afour-photon GHZ state

(|Hotimes4〉1-4〉+ |V otimes4〉1-4

)radic

2 ob-tained by superposing photon 2 and photon 4 on a po-larizing beam-splitter (PBS1) which transmits H and re-flects V polarization Meanwhile photon 6 and photon 8are interfered on a polarization-dependent beam-splitter

(PDBS) and then separately pass through two PDBSsThe former has transmitting probabilities TH = 1 TV =13 and the latter have TH = 13 TV = 1 The combi-nation of these three PDBSs acts as a controlled-phasegate [33 34] With a success probability of 19 one hasthe twofold coincidence in path 6rsquo and 8rsquo yielding a four-photon cluster state [34] [|HH〉56 (|HH〉78 + |V V 〉78) +|V V 〉56 (|HH〉78 minus |V V 〉78)]2 Finally photon 4rsquo andphoton 6rsquo are superposed on PBS2 When eight pho-tons come out of the output ports simultaneously oneobtains an entangled eight-photon cluster state

5

01020

30

40

50

60

70

-1

-05

0

05

1

Measurem

-1

-05

0

05

1

Correlation

Initial

ment Settings

n Settings

State

Exp

ecta

tion

Valu

eE

xpec

tatio

n Va

lue

Eigh

t-fol

d co

inci

denc

es(8

0h)

a b

c

A0 A1 B0 B1 B2 B3 B4 B5

Z1Z2 Z2Z5 Z3Z6 Z3Z4

FIG 4 Experimental results for the created eight-photon cluster state a Measured eight-fold coincidence in |H〉|V 〉basis b The expectation values for different witness measurement settings From left to right the measurement settings areA0 = (|H〉〈H|otimes6 minus |V 〉〈V |otimes6)1minus6X7X8 A1 = (|H〉〈H|otimes6 minus |V 〉〈V |otimes6)1minus6Y7Y8 and Bi = Motimes6

i (|H〉〈H|otimes2 minus |V 〉〈V |otimes2)78 withi = 0 middot middot middot 5 The measurement of each setting takes 50 hours for the first two settings and 30 hours for the remainings cCorrelations for initial state without any engineered error The error bars represent one standard deviation deduced frompropagated poissonian counting statistics of the raw detection events

|ψ〉 =1

2

[|Hotimes6〉1-6 (|HH〉78 + |V V 〉78) + |V otimes6〉1-6 (|HH〉78 minus |V V 〉78)

] (3)

This is exactly the cluster state |G8〉 shown in Fig 2aunder Hadamard operationsHotimes8 on all qubits Note thatthe photons which are interfered on the PBSs or at thePDBS have the same bandwidth and a star topologyof the eight-photon interferometer leads to an effectivenoise-reduction

To ensure good spatial and temporal overlap the pho-tons are also spectrally filtered with ∆λFWHW = 8 nm for1-3-5-7 and ∆λFWHW = 28 nm for 2-4-6-8 and coupledby single-mode fibres We obtain an average two-foldcoincidence count of about 34 times 105 s and a visibilityof sim94 in the |H〉|V 〉 as well as in the |+〉|minus〉 ba-sis where |plusmn〉 = 1radic

2(|H〉 plusmn |V 〉) Fine adjustments of

the delays between the different paths are tuned to en-

sure that all the photons arrive at the PBSs and PDBSsimultaneously

Measurement is taken for each individual photon bya polarization analyzer which contains a combination ofa QWP a HWP and a PBS together with two single-mode fibre-coupled single-photon detectors in each out-put of the PBS (see Fig 3c) The complete set of the256 possible combinations of eight-photon coincidenceevents is registered by a home-made FPGA-based pro-grammable coincidence logic unit We obtain an eight-fold coincidence rate of 32 per hour Based on the mea-surements for the 256 possible polarization combinationsin the |H〉|V 〉 basis (Fig 4a) we obtain a signal-to-noiseratio of about 2001 defined as the ratio of the average of

6

-1

-05

0

05

1

Z

Error X1

Z1Z2Z2Z5 ZZ3Z6 Z3Z4

-1

-05

0

05

1

Z

Error X2

Z1Z2 Z2Z5ZZ3Z6 Z3Z4

Exp

ecta

tion

Valu

e

Exp

ecta

tion

Valu

e

Correlation Setting

-1

-05

0

05

1

Z

Error X5

Z1Z2Z2Z5

ZZ3Z6 Z3Z4

Correlation Setting

Exp

ecta

tion

Valu

e

-1

-05

0

05

1

Z

Error X6

Z1Z2 Z2Z5ZZ3Z6

Z3Z4

Correlation Setting

Exp

ecta

tion

Valu

e

Correlation Setting

-1

-05

0

05

1

Z

Error X4

Z1Z2 Z2Z5 ZZ3Z6Z3Z4

Correlation Setting

Exp

ecta

tion

Valu

e

-1

-05

0

05

1

Z

Error X3

Z1Z2 Z2Z5ZZ3Z6 Z3Z4

Correlation Setting

Exp

ecta

tion

Valu

e

FIG 5 Experimental results of syndrome correlations for topological error correction Only one qubit is subjectedto an X error in each sub-figure The measurement for each error setting takes about 80 hours The error bars represent onestandard deviation deduced from propagated poissonian counting statistics of the raw detection events

the desired components to that of the non-desired onesThis indicates the success of preparing the desired eight-photon cluster state

To more precisely characterize the cluster state weuse the entanglement-witness method to determine its fi-

delity For this purpose we construct a witness whichallows for the lower bound on the state fidelity and re-quires only eight measurement settings (see Appendix)

W8 =1

2minus (|ψ〉〈ψ| minus |ψprime〉〈ψprime|)

=1

2minus[

1

4

(|H〉〈H|otimes6 minus |V 〉〈V |otimes6

)1-6 otimes (X7X8 minusY7Y8) +

1

12

(5sumk=0

(minus1)kMotimes6k

)1-6

otimes(|H〉〈H|otimes2 minus |V 〉〈V |otimes2

)78

] (4)

where 〈ψprime|ψ〉 = 0 and Mk =[cos(kπ6 )X + sin(kπ6 )Y

] The

results are shown in Fig 4b which yields the witness〈W 〉 = minus0105 plusmn 0023 which is negative by 45 stan-dard deviations The state fidelity is F gt 1

2 minus 〈W 〉 =0605plusmn 0023 The presence of genuine eight-photon en-tanglement is confirmed

Experimental topological error correction

Given such a cluster state topological error correctionis implemented using a series of single-qubit measure-

ments and classical correction operations In the labora-tory operations are performed on state (3) differing from|G8〉 in Fig 2a by Hadamard operation Hotimes8 Thereforethe to-be-protected correlation 〈X5X6〉 in Eq (1) corre-sponds to 〈Z5Z6〉 in the experiment the same applies tothe syndrome correlations (2) Meanwhile X errors areengineered instead of Z errors

In the experiment the noisy quantum channels on po-larization qubits are engineered by one HWP sandwichedwith two QWPs which are set at 90 degrees By ran-domly setting the HWP axis to be oriented at plusmnθ withrespect to the horizontal direction the noisy quantum

7

0

01

02

03

04

05

06

0 01 02 03 04

UCUC

Error r4 05

ncorrecteorrectedTncorrecteorrectedE

rate of sing06 0

dTheoryTheorydExperimExperimen

gle qubit7 08

mentnt

09 1

Err

or ra

te o

f pro

tect

ed c

orre

latio

n

FIG 6 Experimental results of topological error cor-rection All physical qubits are simultaneously subject toan X error with equal probability ranging from 0 to 1 Theblue round dots (blue lines) represent the experimental (the-oretical) values of the error rate for the protected correlationwithout TEC and the red square dots (red lines) are for theerror rate with TEC The agreement between the experimen-tal and the theoretical results clearly demonstrates the via-bility of TEC The measurement of each data point takes 80hours The error bars represent one standard deviation de-duced from propagated poissonian counting statistics of theraw detection events

channel can be engineered with a bit-flip error probabil-ity of p = sin2(2θ)

We first study the case that only a single X error oc-curs on one of the six photons 1 middot middot middot 6 The syndromecorrelations are measured and the results are shown inFig 5 For comparison we also plot the correlationswithout any engineered error in Fig 4c Indeed one canprecisely locate the physical qubit undergoing an X error

We then consider the case that all the six photons aresimultaneously subject to a random X error with equalprobability 0 lt p lt 1 and study the rate of errors〈Z5Z6〉 = minus1 for the topological quantum correlation〈Z5Z6〉 Without error correction the error rate of cor-relation 〈Z5Z6〉 is P = 1 minus (1 minus p)2 minus p2 With errorcorrection the residual error becomes

P = 1minus[(1minus p)6 + p6

]minus[6p(1minus p)5 + 6(1minus p)p5

]minus[9p2(1minus p)4 + 9(1minus p)2p4

] (5)

For small p the residual error rate after error correctionis significantly reduced as compared to the unprotectedcase As shown in Fig 6 the experimental results are ingood agreement with these theoretical predictions Con-siderable improvement of the robustness of the 〈Z5Z6〉correlation can be seen both in theory and in practice

In the experiment the whole measurement takes about80 days This requires an ultra stability of our setupThe imperfections in the experiment are mainly due tothe undesired components in the |H〉|V 〉 basis arisingfrom higher-order emissions of entangled photons andthe imperfect photon overlapping at the PBSs and thePDBS In spite of these imperfections the viability ofTEC is clearly demonstrated in the experiment

Discussion

In the current work we have experimentally demon-strated TEC with an eight-photon cluster state Thisstate represents the current state-of-the-art for prepa-ration of cluster states in any qubit system and is of

particular interest in studying multipartite entanglementand quantum information processing The scalable con-struction of cluster states in the future will require fur-ther development of high-efficiency entanglement sourcesand single-photon detectors [35] Recent results haveshown that if the product of the number-resolving de-tector efficiency and the source efficiency is greater than23 efficient linear optical quantum computation is pos-sible [36] Solid technical progress towards this goal hasbeen made such as deterministic storable single-photonsources [37] and photon-number-resolving detectors [38]This work represents the first experimental demonstra-tion of TEC an important step towards fault-tolerantquantum computation In the scheme given sufficientqubits and physical error rates below 07ndash11 ar-bitrary quantum computations could be performed ar-bitrarily reliably The high threshold error rate is es-pecially remarkable given that only nearest neighbour-interactions are required Due to these advantages TECis especially well-suited for physical systems geometri-cally constrained to nearest-neighbour interactions suchas quantum dots [39] Josephson junction qubits [40] ion

8

traps [41] cold atoms in optical lattices [42] and pho-tonic modules [17] A quantum gate with an error ratebelow the threshold required in TEC is within reach ofcurrent experimental technology [43] It would be inter-esting in future work to exploit cluster states of reachablesize to implement topologically error-protected quantumalgorithms by local measurements

We acknowledge insightful discussions with M A

Martin-Delgado O Guhne We are grateful to X-HBao for his original idea of the ultra-bright entanglementand to C-Z Peng for his idea of reducing high orderemission We would also like to thank C Liu and SFolling for their help in designing the figures This workhas been supported by the NNSF of China the CAS theNational Fundamental Research Program (under GrantNo 2011CB921300) and NSERC

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[2] Grover L K Quantum mechanics helps in searching fora needle in a haystack Phys Rev Lett 79 325 (1997)

[3] Feynman R P Simulating physics with computers IntJ Theor Phys 21 467ndash488 (1982)

[4] Calderbank A R amp Shor P W Good quantum error-correcting codes exist Phys Rev A 54 1098ndash1105 (1996)

[5] Steane A M Error correcting codes in quantum theoryPhys Rev Lett 77 793ndash797 (1996)

[6] Gottesman D Theory of fault-tolerant quantum compu-tation Phys Rev A 57 127ndash137 (1998)

[7] Knill E Quantum computing with realistically noisy de-vices Nature 434 39ndash44 (2005)

[8] Aliferis P Gottesman D amp Preskill J Quantum accu-racy threshold for concatenated distance-3 code QuantInf Comput 6 97ndash165 (2006)

[9] Kitaev A Y Quantum computations Algorithms anderror correction Russ Math Surv 52 1191ndash1249 (1997)

[10] Spedalieri F amp Roychowdhury V P Latency in lo-cal two-dimensional fault-tolerant quantum computingQuant Inf Comput 9 666ndash682 (2009)

[11] Dennis E Landahl A Kitaev A amp Preskill J Topo-logical quantum memory J Math Phys 43 4452ndash4505(2002)

[12] Raussendorf R Harrington J amp Goyal K A fault-tolerant one-way quantum computer Ann Phys 3212242ndash2270 (2006)

[13] Wang D S Austin A G amp Hollenberg L C L Quan-tum computing with nearest neighbor interactions and er-ror rates over 1 Phys Rev A 83 R020302 (2011)

[14] Raussendorf R amp Harrington J Fault-tolerant quan-tum computation with high threshold in two dimensionsPhys Rev Lett 98 190504 (2007)

[15] Barrett S D amp Stace T M Fault tolerant quantumcomputation with very high threshold for loss errors PhysRev Lett 105 200502 (2010)

[16] Stock R amp James D F V A scalable high-speedmeasurement-based quantum computer using trappedions Phys Rev Lett 102 170501 (2009)

[17] Devitt S J et al Topological cluster state computationwith photons New J Phys 11 083032 (2009)

[18] Nayak C Simon S H Stern A Freedman M ampSarma S D Non-abelian anyons and topological quantumcomputation Rev Mod Phys 80 1083ndash1159 (2008)

[19] Wilczek F Fractional Statistics and Anyon Supercon-ductivity (World Scientific Singapore 1990)

[20] Cory D G et al Experimental quantum error correc-tion Phys Rev Lett 81 2152ndash2155 (1998)

[21] Knill E Laflamme R Martinez R amp Negrevergne CBenchmarking quantum computers The five-qubit errorcorrecting code Phys Rev Lett 86 5811ndash5814 (2001)

[22] Chiaverini J et al Realization of quantum error correc-tion Nature 432 602ndash605 (2004)

[23] Schindler P et al Experimental repetitive quantum er-ror correction Science 332 1059ndash1061 (2011)

[24] Lu C-Y et al Experimental quantum coding againstqubit loss error Proc Natl Acad Sci USA 105 11050ndash11054 (2008)

[25] Aoki T et al Quantum error correction beyond qubitsNature Physics 5 541ndash546 (2009)

[26] Raussendorf R amp Briegel H J A one-way quantumcomputer Phys Rev Lett 86 5188ndash5191 (2001)

[27] Schlingemann D amp Werner R F Quantum error-correcting codes associated with graphs Phys Rev A65 012308 (2001)

[28] Kitaev A Y Fault-tolerant quantum computation byanyons Ann Phys 303 2ndash30 (2003)

[29] Bombin H amp Martin-Delgado M A Topological quan-tum distillation Phys Rev Lett 97 180501 (2006)

[30] Hatcher A Algebraic Topology (Cambridge UniversityPress Cambridge UK 2002)

[31] Fowler A G amp Goyal K Topological cluster state quan-tum computing Quant Inf Comput 9 727ndash738 (2009)

[32] Yao X-C et al Observation of eight-photon entangle-ment arXiv 11056318v1 [quantndashph] (2011)

[33] Hofmann H F amp Takeuchi S Quantum phase gate forphotonic qubits using only beam splitters and postselec-tion Phys Rev A 66 024308 (2002)

[34] Kiesel N et al Experimental analysis of a four-qubitphoton cluster state Phys Rev Lett 95 210502 (2005)

[35] OrsquoBrien J L Optical quantum computing Science 3181567ndash1570 (2007)

[36] Varnava M Browne D E amp Rudolph T How goodmust single photon sources and detectors be for efficientlinear optical quantum computation Phys Rev Lett100 060502 (2008)

[37] Chen S et al Deterministic and storable single-photonsource based on quantum memory Phys Rev Lett 97173004 (2006)

[38] Kardynal B E Yuan Z L amp Shields A J Anavalanche-photodiode-based photon-number-resolving de-tector Nature Physics 2 425ndash428 (2008)

[39] Press D et al Complete quantum control of a singlequantum dot spin using ultrafast optical pulses Nature456 218ndash221 (2008)

[40] Hime T et al Solid-state qubits with current-controlledcoupling Science 314 1427ndash1429 (2006)

[41] Hensinger W K et al T-junction ion trap array fortwo-dimensional ion shuttling storage and manipulation

9

Appl Phys Lett 88 034101 (2006)[42] Jaksch D et al Entanglement of atoms via cold con-

trolled collisions Phys Rev Lett 82 1975ndash1978 (1999)[43] Benhelm J Kirchmair G Roos C F amp Blatt R

Towards fault-tolerant quantum computing with trappedions Nature physics 4 463ndash466 (2008)

Appendix

SI TOPOLOGICAL CLUSTER STATEQUANTUM COMPUTATION

a Cluster states and homology The topological fea-ture of error-correction with three-dimensional (3D) clus-ter states is homology which we shall illustrate in 2D forsimplicity Displayed in Fig S1 is a 2D plane with twopoint defects (bull) The boundary of a surface is definedas the sum of all the surrounding chains For instancethe boundary of the surface f (shown in blue) is the sumof e1 and e2 denoted as partf = e1 + e2 Because of thepresence of the point defects each of the three chains e1e2 and e3 is not sufficient to be the whole boundary of asurface Analogously the boundary of a chain is definedas the sum of its endpoints Since the three chains are cy-cles they have no boundarymdashie parte1 = parte2 = parte3 = 0The chain e2 can be smoothly transformed into e1 andvice versa In other words e1 and e2 differ only by theboundary of a surface e2 = e1 + partf We say that e1and e2 are homologically equivalent In contrast e3 isinequivalent to e1 or e2 due to the defect on the right-hand side The homology in higher dimensions is definedin an analogous way In 3D the boundary of a volume isthe sum of all its surrounding surfaces A closed surfaceF is said to have no boundary ie partF = 0 A simpleexample is the surface of a sphere Two surfaces F andF prime are homologically equivalent if they differ only by theboundary of a volume V F prime = F plusmn partV

In topological cluster state computation the error cor-rection scheme only involves local measurements in theX basis with outcomes λ = plusmn1 Computational resultsare represented by correlations R(F ) =

prodaisinF λa of these

outcomes on a closed surface FmdashpartF = 0 As in any en-coding error-resilience is brought about by redundancyA given bit of the computational result is inferred notonly from a particular surface F but from any one ina huge homology equivalence class This arises becausetwo homologically equivalent surfaces F and F prime have thesame correlation R(F ) = R(F prime) in the absence of errors[S1] As a result one has R(partV ) = 1 for every volumeV An outcome of -1 then indicates the occurrence of anerror in the volume V and thus R(partV ) can be used as er-ror syndromes We obtain one bit of such error syndromeper lattice cell cf Fig 1a

The errors have a geometrical interpretation tooThey correspond to 1-chains e [S2] Again homologybecomes relevant Two homologically equivalent errorchains e and eprime have the same effect on computation

FIG S1 Illustration of homological equivalence intwo dimensions Here is a 2D plane with two point defects(bull) All three chains e1 e2 and e3 are cycles and thus haveno boundary They are furthermore nontrivial each of themis insufficient to be the whole boundary of a surface The cy-cles e1 and e2 are said to be homologically equivalent becausethey differ only by the boundary partf of the surface f In con-trast e3 is not equivalent to e1 or e2 In other words e1 ande2 can collapse into each other by smooth deformation whilee3 cannot be smoothly transformed into e2 or e1 because ofthe presence of the right-hand-side defect In the topologicalcluster state quantum computation errors are represented bychains two homologically equivalent error chains have thesame effect on computation

In topological error correction with cluster statesthe computational results and the syndromes are con-tained in correlations among outcomes of local X-measurements Detecting and correcting only phase flipsof physical qubits is thus sufficient to correct arbitraryerrors Nevertheless both bit flip and phase flip errorsare present at the level of logical operations The qubitsin a 3D cluster state live on the faces and edges of theassociated lattice Logical phase errors are caused by er-roneous measurement of face qubits and logical spin fliperrors are by erroneous measurement of edge qubits Forexample the 8-qubit cluster state |G8〉 considered in thisexperiment has the correlation 〈G8|X2otimesX2prime |G8〉 = 1 inaddition to the four correlations used as error syndromesfor face qubits It can be derived from a dual complex[S1] and provides one bit of (dual) syndrome for the edgequbits of L8

b Topologically protected quantum gates Topolog-ically protected quantum gates are performed by mea-suring certain regions of qubits in the Z basis whicheffectively removes them The remaining cluster whosequbits are to be measured in the X and X plusmn Y basisthereby attains a non-trivial topology in which fault-tolerant quantum gates can be encoded Fig S2 shows amacroscopic view of a 3D sub-cluster for the realization ofa topologically protected CNOT gate [S3 S4] Only thetopology of the cluster matters individual lattice cells arenot resolved The cluster qubits in the line-like regions Dare measured in the Z-basis the remaining cluster qubitsin the X-basis

10

FIG S2 Topological gates with 3D cluster states Thesub-cluster C (indicated by the wire frame) realizes an en-coded CNOT gate between control c and target t The re-gions D are missing from the cluster Each line-like suchregion supports the world-line of one encoded qubit One ofthe four homologically non-trivial correlation surfaces for theencoded CNOT is shown in orange

The fault-tolerance of measurement-based quantumcomputation with a 3D cluster state can be understoodby mapping it to a Kitaev surface code propagating intime [S3] In this picture a 3D cluster state consistsof many linked toric code surfaces plus extra qubits forcode stabilizer measurement entangled with these sur-faces The local measurements in each slice have theeffect of teleporting the encoded state to the subsequentcode surface The code surfaces can support many en-coded qubits because they have boundary Encoded gatesare implemented by changing the boundary conditionswith time This process is illustrated in Fig S2 for theCNOT gate Pieces of boundary in the code surface arecreated by the intersection of the line-like regions D withsurfaces of ldquoconstant timerdquo The 1-chains displayed in

red represent encoded Pauli operators X at a given in-stant of simulated time When propagating forward aninitial operator Xc is converted into XcotimesXt as requiredby conjugation under CNOT

c Further Reading For the interested reader weadd a few references The topological error-correctioncapability in 3D cluster states is for the purpose of estab-lishing long-range entanglement in the presence of noisediscussed in [S5] How to perform universal fault-tolerantquantum computation with 3D cluster states is describedin [S1] and in terms of stabilizers in [S6] In [S3] amapping from three spatial dimensions to two spatial di-mensions plus time is provided and the fault-tolerancethreshold is improved to 07 for both the three andthe two-dimensional version The 2D scheme is describedsolely in terms of the toric code in [S7]SII CHARACTERIZATION OF THE 8-QUBIT

CLUSTER STATE

In order to characterize the generated 8-qubit clusterstate we use entanglement witnesses to verify its genuinemultipartite entanglement [S8] If W is an observablewhich has a positive expectation value on all biseparablestates and a negative expectation value on the generatedentangled state we call this observable an entanglementwitness With the method introduced in Ref [S9] thewitness is constructed as

W =1

2minus |ψ〉 〈ψ|+ |ψprime〉 〈ψprime| (S1)

where

|ψprime〉 =1

2

[|H〉otimes6 (|V V 〉 minus |HH〉)

+ |V 〉otimes6 (|HH〉+ |V V 〉)]

is an orthogonal state of |ψ〉 that is 〈ψ|ψprime〉 = 0

Then the witness is decomposed into a number of localvon Neumann (or projective) measurements

W =1

2minus (|ψ〉〈ψ| minus |ψprime〉〈ψprime|)

=1

2minus 1

2

[(|H〉〈H|otimes6 minus |V 〉〈V |otimes6)1minus6 otimes (|H〉〈V |otimes2 + |V 〉〈H|otimes2)78

+(|H〉〈V |otimes6 + |V 〉〈H|otimes6)1minus6 otimes (|H〉〈H|otimes2 minus |V 〉〈V |otimes2)78]

=1

2minus[

1

4

(|H〉〈H|otimes6 minus |V 〉〈V |otimes6

)1minus6 otimes (X7X8 minusY7Y8) +

1

12

(5sumk=0

(minus1)kMotimes6k

)1minus6

otimes(|H〉〈H|otimes2 minus |V 〉〈V |otimes2

)78

(S2)

11

where Mk =[cos(kπ6 )X + sin(kπ6 )Y

] The experimental

results are shown in Fig 4b in the main text which yieldsthe witness 〈W 〉 = minus0105plusmn 0023 which is negative by45 standard deviations

[S1] Raussendorf R Harrington J Goyal K A fault-tolerant one-way quantum computer Ann Phys 3212242-2270 (2006)

[S2] Dennis E Landahl A Kitaev A amp Preskill J Topo-logical quantum memory J Math Phys 43 44524505(2002)

[S3] Raussendorf R Harrington J Fault-tolerant quantumcomputation with high threshold in two dimensions PhysRev Lett 98 190504 (2007)

[S4] Raussendorf R Harrington J Goyal K Topologicalfault-tolerance in cluster state quantum computation NewJ Phys 9 199 (2007)

[S5] Raussendorf R Bravyi S amp Harrington J Long-rangequantum entanglement in noisy cluster states Phys RevA 71 062313 (2005)

[S6] Fowler A G Goyal K Topological clus-ter state quantum computing Preprint at〈httparxivorgabs08053202〉 (2008)

[S7] Fowler A G Stephens A M Groszkowski P Highthreshold universal quantum computation on the sur-face code Preprint at 〈httparxivorgabs08030272〉(2008)

[S8] Bourennane M Eibl M Kurtsiefer C et al Experi-mental detection of multipartite entanglement using wit-ness operators Phys Rev Lett 92 087902 (2004)

[S9] Guhne O Lu C Y Gao W B Pan J W Toolboxfor entanglement detection and fidelity estimation PhysRev A 76 030305 (2007)