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Toward a more economical cluster state quantum computation M. S. Tame1, M. Paternostro1, M. S. Kim1, V. Vedral2,3,4 1School of Mathematics and Physics, The Queen’s University, Belfast, BT7 1NN, UK 2Institute of Experimental Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria 3The Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria 4The School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK (Dated: February 1, 2008) Weassess theeffectsofanintrinsicmodelforimperfections inclusterstatesbyintroducingnoisy cluster states and characterizing their role in the one-way model for quantum computation. The action ofindividualdephasingchannelsonclusterqubitsisalsostudied. Weshowthattheeffectof non-idealities is limited by using small clusters, which requires compact schemes for computation. In light of this, we address an experimentally realizable four-qubit linear cluster which simulates a 5 controlled-NOT (CNOT). 0 0 PACSnumbers: 03.67.Lx,03.67.Mn,42.50.Dv 2 n The one-way quantum computer is a model for quan- proposal overcomes the expensive schemes for gate sim- a J tum computation (QC) exploiting multipartite entan- ulation proposed so far [2] and significantly reduces the 1 gled resources, the cluster states, and adaptive single- requirements for experimental implementations. 1 qubit measurements [1]. Quantum information process- The model- Given a set of qubits occupying the sites ing(QIP)isperformedinthismodelusingalargeenough of a square lattice , a cluster state φ is a pure en- {κ} C 2 C | i cluster state and appropriatemeasurements [2]. The ba- tangled state characterized by the eigenvalue equations v 6 sic features of one-way QC have very recently been ex- K(a) φ{κ} C = ( 1)κa φ{κ} C. Here K(a) = σx(a) b | i − | i ⊗ 5 perimentally demonstrated in [3]. Despite the stimulat- σ(b∈nbgh(a)∩C) are correlation operators forming a com- z 1 ing possibilities offered by this model, cluster state QC plete set of commuting observables for the qubits in 2 is still an open field of research from both a theoreti- |C| [2]. Theset κ ,whichcompletely specifiesthe cluster 1 cal and practical viewpoint. On one hand, we have wit- C { } 4 state φ{κ} C, contains the values κa 0,1 a , nessed the first attempts of combining cluster state and | i ∈ { } ∀ ∈ C 0 while σ are the x and z-Pauli matrices respectively. x,z / linearopticsinordertoincreasetheefficiencyofexisting Cluster states φ with κ = 0 ( a ) are gener- h C a all-optical QC schemes [4]. On the other hand, it is im- | i ∀ ∈ C p ated by preparing a register of qubits within in the portant to develop the model by studying the effects of C - state + , where = (1/√2)(0 1 ) , and a∈C a a a nt imperfections on the performances of one-way QC. This applyi⊗ng th|eitransforma|±tiion S(C) = | i±| i Sab a is the central aim of this work. So far, it is not clear with γ = 1 , γ = (1,0)T,(0,Q1)aT,b∈Ca|bn−da∈γγd = u if the cluster state model, exposed to sources of noise, 1 { } 2 { } 3 (1,0,0)T,(0,1,0)T,(0,0,1)T for the respective dimen- q wouldperformbetterthanthestandardquantumcircuit { } : sionofthe cluster. EachSab is a controlledπ-phase gate v model. Very recently, there have been some intriguing Xi investigations in this sense [5]. We introduce a realistic SaHbo=w|e0viearh0t|h⊗e r11e(abl)iz+at|i1oinaho1f|t⊗heσz(Sb)a.b gates can be inher- model for imperfect cluster state generation and study r ently imperfect. For example, an optical-lattice loaded a how this affects both the intrinsic properties of cluster by a bunch of neutral atoms, where the two-qubit inter- statesandthebasicingredientsincomputation. Wealso actionsareviacontrolledcollisions,hasbeensuggestedto address the effect of environmental decoherence due to embodyaone-wayQC[7]. Theefficiencyofthegatesde- dephasing channels individually affecting the qubits in pendsonthecontroloverthestrengthoftheinteractions one- and two-dimensional configurations. In this con- (which can fluctuate over the sites of the lattice, gener- text, an informative initial step is the study of the state ating inhomogeneities) and the interaction time. These fidelity of cluster states [6]. We show that in order to parameters fix the amount of the controlled phase im- limit the spoiling effects of non-idealities, one must deal posedby eachinteraction. In addition, the initial filling- with small clusters of just a few qubits. This imposes fractionpersiteinthelatticeinfluencestheperformances severe contraints on the use of this model and paves the of the gates [7]. If the degree of control is not optimal way toward research of more compact protocols for gate (which may be the case when a large number of qubits simulation and QIP. We find such a possibility, address- ing a CNOT simulated through a simple four-qubit lin- is considered), imperfect (inhomogeneous) interactions throughout the physical lattice are settled. This can be ear cluster and outline an all-optical setup where it can addressed by considering the imperfect operations be implemented. Our study contributes to the analysis otefnmceoroef erecaolnisotmicicoanles-cwhaeymeQsIPforancldushteigrhbliagshetds QthCe.eOxiusr- SDab =|0iah0|⊗11(b)+|1iah1|⊗(cid:0)|0ibh0|−eiθa|1ibh1|(cid:1), (1) which add unwanted phases θ to the desired value π. a 2 Applied to the initial state of a N-qubit register (pre- φ . Genuine multipartite entanglement is shared be- | iCN pared via the idealized protocol), they generate a noisy tween the subparties of ideal cluster states, where the cluster state φ D . The use of this set of operations entanglementis encodedin the state as a whole [1]. Any | iCN profoundly modifies the structure and properties of a reducedbipartitestate,obtainedbytracingallthequbits N-qubit cluster with respect to the ideal φ . Ex- but an arbitrary pair, is separable as it does not violate | iCN plicitly, a noisy linear cluster state becomes φ D = thenecessaryandsufficientPeres-Horodeckicriterionfor | iCN 2o−f tNh/e2Pj-tzhiQbiNjn=a−1r1y(−diegiiθtj)ozfjztjh+e1|iznitie,gwerhzerie(izj i[s0,t2hNe va1lu])e. aseltpearratbhiilsitryesouflta. mFioxredexqaumbpitle,sttaatkee[8a].linHeoawrecvleurstSerDabo’sf ∈ − N = 3, the ideal state is locally equivalent to a GHZ In order to characterize the effect of this kind of state. However, two unwanted phases are embedded in non-ideality, an immediate benchmark is provided by the corresponding noisy state φ D. The partial trace the fidelity between ideal and noisy cluster states. | iC3 of φ D φ over the third qubit gives a bipartite state 2T−hNe PovzieQrlaNjp=−11feNiθjzj=zj+1.CTNhhφer|φeiaDCrNe2NleatdersmstointfhNisex=- w(kh=i|chi0C,v31hio,.l|.a)teasndtheθP1e=resπ-.HTorhoidseicskaicchriatrearicotnerfiostricθ2sh6=arkeπd pression, with the fidelity FN = fN 2. However, our as- by all the two-q∀ubit6 states obtained by tracing qubits | | sumptionisthatthecontrolonthe phasesintroducedby 3 to N in a φ D . In order to quantify the entangle- the qubit-qubit interaction is only limited. Thus, there ment of the r|ediuCcNed density matrices ρ (i,j [1,N]), ij is a lack of knowledge about the values of θj’s in a noisy we use the concurrence, C = max 0,2αij ∈Tr(ρ˜ ) , cluster state, which means each of them must be aver- where αij αij αij ij αij are{the e1ige−nvalueisj o}f agedoveranappropriateprobabilitydistribution. Weset 1 ≥ 2 ≥ 3 ≥ 4 a range rj within which each phase can take values and ρ˜ij =qρij(σyi ⊗σyj)ρ∗ij(σyi ⊗σyj)(ρ∗ij isthecomplexcon- introduce the probability distribution p(θ ). The aver- jugateofρ inthecomputationalbasis)[9]. Wefindthat j ij age overlapf¯ for the noisy linear cluster state becomes onlynearest-neighborbipartiteentanglementissettledin N f¯N =2−NPziQNj=−11{R|rj|p(θj)eiθjdθj}zjzj+1,where|rj| a symmetric way along an arbitrarily long noisy cluster is the width of the range of variation of θj and p(θj) de- (e.g. C12 =C(N−1)N, irrespective of N), while any non- pends on the specific physical model used in the cluster nearestneighbor entanglement is absent due to random- stategeneration. However,thenatureofthefluctuations ness, more pronounced in pairs of non-nearest neighbor of the phases is characterized by the way in which the qubits. Indeed the corresponding density matrices de- interactions among the elements of a lattice are realized pend,ingeneral,onmoreunwantedphasesthanthoseof and no universal model can be found. In Fig. 1 (a) we nearestneighborones. Atθj =π( j)noentanglementis ∀ provide an example of F¯ for a flat p(θ ) distribution, found, as in this case Sij 11ij. We have explicitly con- N j D ≡ forthe caseoflinearclusters. Alargedeviationfromthe sidered the average concurrence obtained by assuming ideal state is found and we have checked that this qual- Gaussianfluctuationsofeachθj,aroundθj =0andwith itative behavior holds irrespective of the model for the a standard deviation σ. We find that as σ increases (up unwanted phase distribution [10]. This analysis can be toσ =1),thefragilequantumcorrelationsofthebridging extended to two-dimensional cluster states where analo- pairs (i,i+1)withi [2,N 2]disappear,breakingthe ∈ − gous qualitative results can be found. quantum channel connecting qubit 1 to qubit N. This is due to the fact that the qubits in these pairs are ex- The structure of the quantum correlations has been posed to more randomness than those in the pairs (1,2) found to be profoundly different from what is found in and (N 1,N). We are left with the entangled mixed − states of these extremal pairs which can mutually share (a) (b) only classical correlations. By increasing the random- Fidelity Fidelity ness, even this entanglement will disappear, eventually. λ This analysis can be extended to the case of arbitrary 0.8 0.5 1 1.5 2 2.5 3 N, despite the difficulties in finding the reduced density 0.8 0.6 matrices of large cluster states. Thus, the multipartite 0.6 0.4 entanglement is reduced to the benefit of bipartite cor- 0.4 0.2 relations which may be very fragile against fluctuations N 0.2 0 10 20 30 40 50 in the unwanted phases, possibly leading to a complete entanglement breaking effect. FIG. 1: (a): F¯N against |rj|=λ for p(θj)=λ−1. From top In addition to studying imperfect generation of clus- tobottomN =3→10;(b): Dephasingeffectsonmultiqubit states. ThecurvesfromtoptobottomrepresentN-qubitW, ter states, it is also necessary to understand effects of GHZ, linear cluster and N ×N cluster states. For conve- environmental decoherence. This is important for phys- nience,therescaleddephasingtimeisΓ=0.062(correspond- ical realizations, as the accuracy of a gate simulated ingto FClin =0.5). Similar behaviors are found for different on a cluster state interacting with an environment may 25 valuesof Γ. be spoiled. We consider decoherence due to individ- 3 ual dephasing channels affecting each qubit in the clus- the set κ = 0 can be reduced to a three-qubit one, { } { } ter, a model which is relevant in practical situations of simply by measuring 3 and 4 in the σ eigenbasis. This x qubits exposed to locally fluctuating potentials. Here, gives a cluster state |φ{κ′}iC3 with {κ′} = {0,sx4,sx3}. the off-diagonal elements of a single-qubit density ma- Here, sx = 0(sx = 1) corresponds to qubits i = 3,4 be- i i trix ρd decay as e−Γ, with Γ the rescaled dephasing ing in + ( ). On the other hand, measurements in | i |−i time. For a single qubit prepared in + , the fidelity the σ eigenbasis remove a qubit from a cluster too, but z | i is F = +ρd + = 1(1+e−Γ). For larger registers, it also break any intra-cluster connection bridged by the h | | i 2 is relevant to compare the behavior of cluster states un- qubit [2]. While this strategy does not affect gate per- derdephasingwithothermultiqubitstatessuchasGHZ formances in the ideal one-way QC, when the model in andW states. ForN-qubitGHZ andW states(N 3), Eq. (1) is contemplated, the removal of every redundant ≥ the state fidelities are FGHZ =(1+e−ΓN)/2 and FW = qubitdeterminesthespreadofthenoise(intermsofran- N N (1+(N 1)e−2Γ)/N, while for N-qubit (N 2) linear dom phases) from the removed qubits to the remainder − ≥ clusterstatesFClin =2−N N B(N,h)e−Γh, withthe ofthecluster. Werefertothiseffectastheinheritanceof N Ph=0 binomialcoefficientB(N,h). TheexpressionforaN N noise by the survivor qubits. In turn, this results in ad- × clusterisfoundforN N2. Thesefunctionsareplotted ditional spoiling mechanisms reducing the fidelity of the → against N and shown in Fig. 1 (b), where one can see gate being simulated. The conclusions of our study do that when state fidelity is used as a benchmark, linear notqualitativelydependonthespecificgateweconsider. cluster states are very fragile against individual dephas- Thus, to give evidence of these effects, we go directly to ing. the case of a CNOT. Noisy cluster state QC- To analyze the noise model in In order to perform a CNOT between a control qubit computational protocols, we briefly review the workings c = a0 +b1 and a target qubit t = c0 +d1 , in in of the one-way QC. If a unitary operation Ug has to be t|heischem| ei in |[2i] uses 15 qubits, wh|erei (C|NOiT) c|oni- performedonaninput state|ψiniofn logicalqubits, the sists of measurements in the σx,y eigenbaseMs as shown in ideais to use acluster inaparticularphysicalconfigura- Fig. 2 (a). After the measurements, decoding operators itnionaCs(egct)i.oTnhei(ngp)uotfstthateecilsuesntecrodceodnsinisttihnegσozfenigeqnubbaitssis. U˜Σc,t†({si}) = σzγzc,t ⊗σxγxc,t are applied to qubits 7 and CI 15 (which embody the logical controland target), where This section is then entangled to the rest of the cluster γc,t depend on the outcomes s of the individual mea- throughtheidealentanglingoperations. Ameasurement x,z i surements [2]. A φ D state is needed in the noisy gate pattern (g), i.e. a set of single-qubit measurements | iC15 M simulation and to construct it, we take small subclus- along particular directions in the Bloch sphere, is per- ters which are then mutually entangled. For instance, formed on all the qubits of the cluster except a part of the subclusters φ D and φ D are entangled as it which embodies the output register CO(g). The input S4,5 φ D φ D| i1=,2,3ψ,4D |φiD5,6,7+ χ D η D and output logical state in the simulation of g are re- D | i1,2,3,4| i5,6,7 | i1,2,3,4| i5,6,7 | i1,2,3,4| i5,6,7− lated via |ψouti= UgUΣ({si})|ψini, where UΣ({si}) is a eiθ4|χiD1,2,3,4|µiD5,6,7. Here, |ψiD (|ηiD) is the part of a local byproduct operator which acts on the output log- subcluster state with the last (first) qubit in 0 , while | i ical qubits and depends on the set of outcomes ( s ) χ D (µ D) is the part with the last (first) qubit in 1 obtained after the measurements of (g) [2]. { i} a|nid th|e ilabels of the qubits are explicitly indicated. B| yi It is not obvious how a particularMQC protocol is af- applyingthepatternshowninFig.2(a),thegatefidelity fected by the exposure of a cluster state to a dephasing canbeevaluated. Toillustratetheeffectofimperfections, channel and it is the specific nature of the one-way QC weaddressthecasewherequbitsmeasuredintheσx (σy) which makes this point non-trivial. Considering that in eigenbasisareallfoundin + (+ y 0 +i1 )andthe | i | i ∝| i | i order to perform a computation, the qubits in a cluster requireddecodingoperatorisU˜Σ†({0})=σz(7)⊗11(15). To are measured and thus removed from the dynamics of take into account the phase randomness, we have aver- the system, one would conjecture that in this way, they aged the gate fidelity over Gaussian distributions cen- nolongercontributetotheexpecteddecreaseofthe gate teredonθ =0andwithastandarddeviationσ foreach j fidelitiesduetodephasing[10]. InthecontextofQCpro- phase in φ D . The results for σ [0.1,1] are shown in | iC15 ∈ tocolswithnoisyclusterstates,hereweassesstheeffects Fig. 2 (c) (solid curve) where the decay of the fidelity of redundant qubits not functional to the specific sim- against the increasing randomness is evident. The input ulation to be performed. These qubits can be removed states are normalized and we take a = c = 0.5. Similar from the cluster configuration by measuring them in an behaviors are observed for other choices of a and c. To appropriate basis. In the ideal case, if we measure in seenoise inheritanceeffects,wemodifytheconfiguration the σ eigenbasis,we effectively obtain a reduced cluster of Fig. 2 (a) by adding a bridging qubit between 8 and x state, where the measured qubits have no influence [2]. 12, thus considering a state φ D . This qubit is redun- | iC16 In general, the new cluster state will satisfy eigenvalue dantandisremovedviaaσ -measurement. The15-qubit x equations with a new set κ′ . For example, a five-qubit cluster state is retrievedvia localoperations on qubit 12 { } linearcluster(qubitslabelledfrom1to5)associatedwith (the removal is equivalent to a Hadamard gate between 4 qubits16and12). Whenthegatefidelityiscalculated,a ters in the input state. In fact, single-qubit gates can be faster decreaseagainstσ is observed(Fig. 2 (c), bottom done viaa two-qubitcluster andone measurementalong curve). The same effect is found if the information flow an appropriate direction [10]. This circumstantial evi- through a linear cluster state is studied [2]. In this case, dence suggests that our scheme uses the smallest qubit the longer the noisy linear cluster across which informa- register needed for a two-qubit gate and is thus optimal. tion is transferred (thus requiring many measurements), The four-qubit CNOT can be realized in an optical the worse the transfer fidelity [10]. This is important setup, requiring two pure entangled states (encoded in evidence of the effect of measurements on noisy cluster photonic polarizations) and an entangling gate. A pure states. Theeliminationofredundantqubitsandthemea- state of arbitrary entanglement can be produced using surementpatternspreadtheθj inthenoisyclusterstate, the entanglement between two field modes generated affectingtheQIPprotocol. Ifpurificationproceduresare by concatenating Type-I parametric-down-conversion not considered (usually expensive in terms of resources), (PDC) processes [11]. In this scheme, the polarization a key method in counteractingboth the noise models we of the pump field sets the entanglement at the output have considered is the use of small cluster states. of the PDC process and encodes arbitrary target and Indeed,thedropinfidelityseenforthesquashed-Iclus- control input states in pairs of output modes (i.e. pairs ter and our discussion about information flow legitimize 1+2 and 3+4) without postselection. By adapting the some doubts about the convenience of many-qubit con- scheme [12], modes 2 and 3 canbe entangledthroughan figurationsinthe one-wayQCandstimulate researchfor effectivecontrolledπ-phasegate. Thisprotocolresultsin more compact protocols, in order to bypass noise inher- the four-qubit cluster state we have addressed [10]. itance or the effect of dephasing in the dynamics of a Remarks- We have shown that realistic imperfections cluster. The scheme for a CNOT in Fig. 2 (b) proves in the generation and processing of cluster states affects the existence of such economical configurations. It con- the model for one-way QC. To counteract these effects, sists of a four-qubit linear cluster and a measurement the dimension of a cluster has to be minimized. In this pattern made by the σx measurement of qubits 1 and 3 context, we have demonstrated an experimentally real- (encoding the input target and control state). In order izable four-qubit CNOT which uses the minimum num- to demonstrate the gate simulation, we assume perfect ber of qubits for a cluster state based CNOT. Our pro- entangling operations Sab, so that the four-qubit cluster posaldemonstratesthepossibilityofdesigningmorecom- stateis ψ 1,2 η 3,4+ψ 1,2 µ 3,4+χ 1,2 η 3,4 χ 1,2 µ 3,4 pact schemes for cluster state QIP. This theoretically | i | i | i | i | i | i −| i | i with ψ 1,2 (a00 + b10 )1,2, χ 1,2 (a01 1,2 challenges the way in which cluster state-QC has been | i ∝ | i | i | i ∝ | i − b11 )1,2, η 3,4 c(00 + 01 )3,4 and µ 3,4 d(10 thought about so far and allows for more controllable | i | i ∝ | i | i | i ∝ | i− 11 )3,4. Measuring qubits 1 and 3 onto 1,3, qubits experimental implementations. | i |±i 2 and 4 are left in a state locally equivalent to the out- WeacknowledgesupportbyEPSRC,DELandIRCEP. put state of a CNOT (in the σ eigenbasis) [13]. c=4,t=2 x MST thanks the E. Schrödinger Institute. When Sab Sab, the cluster state becomes noisy and → D the gate fidelity is affected by the distributions for θ . j However, the small number of qubits limits any spoiling effect and improves the averagegate fidelity (Fig. 2 (c), top curve). This reinforces the idea that more economi- [1] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, calschemesforclusterstateQCexist,wherethenumber 5188 (2001). ofqubits involvedcorrespondsto the number ofparame- [2] R. Raussendorf et al.,Phys. Rev.A 68, 022312 (2003). [3] P. Walther , K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter,V.Vedral,M.Aspelmeyer,and A.Zeilinger, (c) Nature (toappear) (2005). [4] M.A.Nielsen,Phys.Rev.Lett.93,040503(2004);D.E. GateFidelity σ Browne and T. Rudolph,quant-ph/0405157. 0.9 0.2 0.4 0.6 0.8 1 [5] M. A.Nielsen and C. M. Dawson, quant-ph/0405134. [6] W. Du¨r and H. J. Briegel, Phys. Rev. Lett. 92, 180403 0.8 (2004). 0.7 [7] M.Greineretal.,Nature415,39(2002);J.Garcia-Ripoll 0.6 and J. I. Cirac, New J. Phys. 5, 76 (2003). 0.5 [8] M. Horodecki et al., Phys. Lett. A 223, 1 (1996); A. 0.4 Peres, Phys.Rev. Lett.77, 1413 (1996). [9] W. K. Wootters, Phys. Rev.Lett. 80, 2245 (1998). FIG. 2: (a): The squashed-I configuration [2]. (b): Scheme [10] M. S.Tame et al.,in preparation. foraCNOTwithafour-qubitcluster. (c): Averagefidelityof [11] A. G. White et al.,Phys.Rev. Lett.83, 3103 (1999). a CNOT against the σ of Gaussian distributions for each θj. [12] S. Gasparoni et al.,Phys. Rev.Lett. 92, 020504 (2004). Fromtoptobottomweshowtheresultsforpanel(b),(a)and [13] The decoding operators are U˜Σ†(s1,s3) = [σx(2)]s1 ⊗ thesquashed-I with an additional bridge-qubitrespectively. [σx(4)]s1⊕s3, with ⊕ the logical XOR.