Cluster states are entangled multipartite states which enable to do universal
quantum computation with local measurements only. We show that these states
have a very simple interpretation in terms of valence bond solids, which allows
to understand their entanglement properties in a transparent way. This allows
to bridge the gap between the differences of the measurement-based proposals
for quantum computing, and we will discuss several features and possible
extensions

D. Schlingemann

E. Knill

F. Verstraete

J. I. Cirac

M. Nielsen

R. Raussendorf

English

arXiv

Crossref

Valence-bond states for quantum computation

Verstraete, F.

Cirac, J.

MPG.PuRe

We propose a way of universal quantum computation by doing joint measurements on distributed singlets. We show how these joint measurements become local measurements when the singlets are interpreted as the virtual components of a large valence-bond state. This proves the equivalence of the cluster-state-based quantum computational model and the teleportation-based model, and we discuss several features and possible extensions. We show that all stabilizer states have a very simple interpretation in terms of valence-bond solids, which allows to understand their entanglement properties in a transparent way

Verstraete, Frank

Cirac, JI

Ghent University Academic Bibliography

arXiv:quant-ph/0311130v1  19 Nov 2003Valence Bond Solids for Quantum ComputationF. Verstraete and J.I. CiracMax-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany.(Dated: August 21, 2018)Cluster states are entangled multipartite states which enable to do universal quantum computationwith local measurements only. We show that these states have a very simple interpretation in termsof valence bond solids, which allows to understand their entanglement properties in a transparentway. This allows to bridge the gap between the differences of the measurement-based proposals forquantum computing, and we will discuss several features and possible extensions.PACS numbers: 75.10.Pq, 03.67.Mn, 03.65.Ud, 03.67.-aThe concept of teleportation [1] plays a crucial rolein the understanding of entangled quantum systems. Itdoes not only allow us to use entangled states as perfectquantum channels, but also to implement non-local uni-tary operations [2]. Based on this idea it was shown thatuniversal quantum computation can be achieved if onecan prepare a separable initial state and implement jointtwo-qubit measurements [3, 4, 5, 6]. In the same spirit,but somehow orthogonal to these schemes, Raussendorfand Briegel [7] showed that universal quantum computa-tion is possible by doing local measurements on the qubitsin a highly entangled so–called cluster state [8]. Thesestudies highlighted the central role of entanglement forquantum computation [9, 10]. However, the structureof general multiparticle entanglement is, for the momentbeing, still very poorly understood, and it is somehowmysterious that the cluster states enable for universalquantum computation. In this note, we show that thestructure of entanglement in cluster states is particularlysimple and can be well understood by looking at it as aso–called valence bond solid with only nearest-neighborbonds. This enables us to show that the one-way com-puter [7] essentially works in an equivalent way as theother measurement-based proposals for quantum compu-tation.Let us start by recalling how universal quantum com-putation can be performed using measurements only ona collection of maximally entangled states of 2 qubits.Schematically, we will present the logical qubits as theleftmost particles. Logical gates can be implemented byintroducing extra pairs of maximally entangled states|H〉 = |00〉 + |01〉 + |10〉 − |11〉 of two qubits (everyother maximally entangled would also be good), and do-ing Bell measurements on halves of these |H〉 and thelogical qubits. The new logical qubits are now in theother parts of the maximally entangled state. It is wellknown that a universal set of gates [26] is given by ar-bitrary local unitary transformations and the phase gateUph = |00〉〈00| + |01〉〈01| + |10〉〈10| − |11〉〈11|. A localunitary transformation U on qubit i can be implementedby doing a Bell measurement in the basis|α〉 = (U †σα ⊗ 1 )|H〉, α = 0, 1, 2, 3 (1)where σα denote the Pauli matrices (including σ0 = 1 );see Figure 1A. This implements the unitary gate σαUØin ØoutØin1Øin2Øout2Øout1(A)(B)| ñHáa|áa|áb|FIG. 1: (A) Implementation of a 1-qubit gate by measuringin the 2-qubit basis |α〉 (1). The edges connected by the linedenote the maximally entangled state |H〉. (B) Implementa-tion of a 2-qubit gate by 3-qubit measurements in the basis|α〉 and |β〉 (2).which is U up to an extra multiplication with a Paulioperator (α is conditioned by the measurement outcome);this extra Pauli operator however does not harm [27].Similarly, the phase gate Uph can be implemented byadding three extra pairs of maximally entangled states|H〉 as depicted in Figure 1B. Suppose two three-qubitmeasurements are done (see Fig. 1B) in the completebases{|α〉} = {|β〉} = (σx)i ⊗ (σx)j ⊗ 1 (|0〉|0〉|0〉 ± |1〉|1〉|1〉)(2)with i, j ∈ {0, 1} and |±〉 = (|0〉± |1〉)/√2. It can readilybe checked that this implements the gate (H ⊗ H)Uphwith H the Hadamard gate H = |+〉〈0| + |−〉〈1|, upto a harmless extra multiplication with Pauli operators.Together with the possibility of implementing local uni-taries, this proves that universal quantum computationcan be done by doing only Bell measurements on two orthree qubits.Let us summarize the three ingredients needed for be-ing able to implement quantum computing along the linessketched: 1/ it must be possible to create ancillary sin-glets; 2/ two- and three-qubit measurements of the form(1,2) can be implemented between halves of these extrasinglets and the logical qubits.Inspired by the AKLT-valence bond solids (VBS)[11, 12, 13], playing a central role in condensed mat-ter physics, it is now interesting to investigate whetherit is possible to interpret the two (or three) qubits on2FIG. 2: Representation of a Valence Bond Solid. The edgesconnected with a dotted line denote a singlet, while the circlesdenote a projection P of all qubits inside it with Hilbert spaceH⊗n2to a single qubit H̃2. In the present paper, P is alwaysof the form P = |0̃〉〈00 . . . 0|+ |1̃〉〈11 . . . 1|.which the measurements have to be implemented as vir-tual qubits representing one physical qubit. As willbecome clear, this will exactly lead to the concept ofthe one-way computer [7]. As depicted in Figure 2,VBS are constructed by distributing singlets |H〉 madefrom virtual qubits between different sites, followed bya local projection of the virtual qubits on a smallerdimensional subspace H̃ encoding the physical qubit.In the case of the 2-D lattice in Figure 2 e.g., the 4qubits can be projected on a one qubit subspace bythe operator P4 = |0̃〉〈0000| + |1̃〉〈1111| (here 4 spec-ifies that there are 4 virtual qubits; Pn is defined asPn = |0̃〉〈00 . . . 0| + |1̃〉〈11 . . . 1| with n virtual qubits).The idea is thus to interpret one of the virtual qubits asthe logical one, and the other virtual ones as the onesenabling teleportation.Let us see whether the two listed requirements for mea-surement based quantum computation can be fulfilled.1/ Consider the VBS of Figure 2. To implement avirtual 2-qubit (3-qubit) measurement, we want that thephysical qubit is only made up of 2 (3) virtual qubits(on which we want to implement a Bell measurement),and hence that 2 (1) virtual qubit(s) effectively disap-pear. Suppose the physical qubit (Hilbert space H̃) isobtained by projecting the virtual ones by the projec-tor P4 = |0̃〉〈0000| + |1̃〉〈1111|. Measuring 2 (1) neigh-boring physical qubit(s) in the |0̃〉, |1̃〉-basis effectivelyreplaces 2 (1) virtual qubit(s) with |+〉 or |−〉 depend-ing on the measurement outcome. But it holds thatP4|+〉 = P3 ≡ |0̃〉〈000| + |1̃〉〈111| and P3|−〉 = σ̃zP3,and similarly P3|+〉 = P2. Therefore the bonds in a VBScan be broken at will by measuring neighboring physicalqubits, the virtual qubits effectively disappear and theprojector Pn changes into Pn−1.2/ Let us now investigate whether virtual measure-ments such as (1,meas2) can be implemented by local11_22_| ñHP P1 2FIG. 3: Implementing a global unitary transformation onqubits 1 and 2 by doing local projections P1 and P2 on themand a maximally entangled state |H〉.measurements on physical qubits. This will be possible ifthe projector Pn has full support on a subset of rays cor-responding to these multiqubit measurements. Clearly,this is the case for the phase gate if the projector is P3, asa measurement in the physical basis |0̃〉 ± |1̃〉 effectivelycorresponds to the measurement of the virtual qubits inthe basis |000〉±|111〉 which belongs to the optimal basis(2). In the case of the single qubit gates, things are abit more subtle. Suppose the projector is P2. Only mea-surements in bases of the form |0̃〉±exp(iξ)|1̃〉 correspondto Bell measurements, and its effect is to implement theunitaryU = (σx)k 1√2(1 exp(−iξ)1 − exp(−iξ))on the (virtual) logical qubit (k = 0, 1 depending on themeasurement outcome). It can however easily be checkedthat a sequence of 4 such unitaries can generate everyspecified unitary ∈ SU(2) [28] (this again holds up toleft multiplication with Pauli matrices). Therefore thesecond requirement is also fulfilled.By measuring the qubits from left to right on a lat-tice, the (virtual) logical qubits travel from left to right,yielding quantum computation using single-qubit mea-surements only. This shows that 2-D valence bond solidscan be used to do a quantum computation. This modelof computation exactly coincides with the one-way com-puter of Rausendorf and Briegel; indeed, we will nextshow that the cluster state is exactly the VBS used inthe above construction.The cluster states are a subset of the so–called stabi-lizer states [14], which are defined by specifying a com-plete set of commuting observables Oi, where each Oi is atensor product of the Pauli matrices σ0 = 1 2, σx, σy, σz .The stabilizer states are the common eigenstates of theseoperators. Let us show that any stabilizer state can beinterpreted as a valence bond state. Stabilizer states canefficiently be prepared from a completely separable stateby applying appropriate 2-qubit unitary operations toit (see e.g. [15]). The reason that stabilizer states arevery simple and manageable to work with is due to thefact all these 2-qubit unitary operations can be chosento commute with each other. This can readily be seenby looking at the normal form for stabilizer states [16],and then making use of properties of Pauli operators suchas U(σx ⊗ σz)U † = σx ⊗ I with U = diag[1, 1, 1,−1] to3diagonalize all operators Oi. The trick is now to imple-ment these commuting 2-qubit unitary transformationsby a teleportation-like principle that consists of addingvirtual singlets, and then doing appropriate projections[2, 17]. More specifically, consider the two qubits 1 and2 in Figure 3; an extra singlet |H〉1̄2̄ is added, and thenany unitary transformation between 1 and 2 can be sim-ulated by projecting the two-qubit spaces labelled by 1, 1̄(2, 2̄) onto the qubits 1 (2 with appropriate projectors P1(P2 (P1, P2 are 2 × 4 matrices). Iterating this scheme,one readily sees that every stabilizer state can be inter-preted as a VBS, possibly with bonds extending over allsites [29]. In the case of the cluster states however, onlyunitaries between the nearest neighbors have to be im-plemented, and hence a simple VBS as depicted in Figure2 is obtained.As an example, let us explicitly construct the valencebond states corresponding to arbitrary cluster states [30].To each cluster state, one can associate a graph pa-rameterized by its adjacency matrix Γ. The numberof qubits on each site in the VBS is of course equal tothe number of bonds of the given site, and is equal tothe number of vertices emanating from a given physi-cal qubit. The bonds are maximally entangled states|H〉 = |00〉+ |01〉+ |10〉−|11〉, and the projectors on eachsite are all of the form P = |0̃〉〈00 . . . 0| + |1̃〉〈11 . . . 1|.This simple construction describes all possible clusterstates.This VBS interpretation of cluster states makes theirnice and appealing properties very explicit. The factthat e.g. a singlet can be created between two arbitraryqubits by doing appropriate local measurement on theother ones can readily be understood by the concept ofentanglement swapping [20]. The entropy of a block ofspins can readily be seen to be given by the number ofbonds emanating from it (i.e. proportional to the area ofthe surface of the block). The fact that the sensitivity tonoise of a cluster state does not scale with the numberof (physical) qubits [21], is of due to the fact that it iseffectively made up by local singlet pairs. This insightalso enables to construct distillation protocols for clus-ter states by translating bipartite distillation protocolsto the valence bond picture [22].On the other hand, the description of valence bondstates in terms of stabilizer states is also interesting fromthe point a view of condensed matter theory. It is e.g.well-known that operations of the Clifford group actingon a stabilizer state can easily be simulated efficientlyclassically. This implies that evolutions generated by theClifford group on VBS-states can be simulated efficiently,and correlation functions of products of Pauli operatorscan easily calculated. On the other hand, the possibil-ity to do quantum computation with VBS using localmeasurements only proves that the complexity-class forcalculating general expectation values on 2-D VBS is thesame as the complexity class of quantum computating.The present study also opens the question whetherthere exist ground states of (gapped) Hamiltonians in-volving only 2-body short-range interactions on a latticethat would enable to implement the presented measure-ment scheme (this is not the case for cluster states). Such2-D Valence Bond Solids indeed exist for higher spins(e.g. spin 3/2), and it is trivial do devise a toy modelfor which this holds. Consider e.g. a hexagonal latticewith spin-7/2 particles at each vertex. To each parti-cle corresponds a 8-dimensional Hilbert space, which wecan interpret as a system of 3 qubits. We associate eachoutgoing edge to one of these qubits, and associate theHamiltonian ~S~S+31 to two of these qubits connected byan edge. One readily sees that the ground state on such ahexagonal lattice with this 2-body local Hamiltonian willbe unique, and that the teleportation scheme can be im-plemented perfectly on it. Note that the cluster state isvery similar to that construction, but there the 3 qubitsare interpreted as virtual qubits and a smart projectionwas used to reduce the dimension of the effective Hilbertspace.More interestingly, the trick used to implement 2-qubitunitary gates by introducing a virtual singlet followed bya projection - this is the way cluster states can be gen-erated from completely separable ones - can also be ex-tended to the case where the unitaries do not commutewith each other. Indeed, the cluster state can be madein the lab if an Ising interaction can be implemented onneighboring qubits [23]. However, in some experimentalsetups, it is not always possible to implement such com-muting gates, as is the case e.g. for quantum dots [24]:here one is essentially restricted to implement 2-qubitgates generated by the Heisenberg interaction, which cer-tainly do not commute when acting on neighboring spins.However, if one can apply these unitary gates sequen-tially (i.e. one has control over the sites on which oneimplements the gate), then it is also possible to constructvalence bond solids that are suitable for quantum com-putation.The present results also prove that the valence bondsolid picture is very useful for understanding multipartiteentanglement. Indeed, VBS are particularly interestingfrom the point a view of quantum information theory,as the simple and elegant tools developed for bipartitequantum systems can be applied to it (see e.g. [13]).Moreover, one can readily see that the VBS form a densesubset of all possible quantum states if one allows thebonds to extend beyond nearest neighbors, if the singletsare replaced by higher dimensional maximally entangledstates and if the projectors can be chosen arbitrary (e.g.in the case of 3 qubits, every state can be made by con-sidering 2 singlets and projecting 2 qubits of them onto aqubit space [25]). It would be very interesting to developa general theory of multiparticle entanglement based onthis VBS-picture, where one could construct entangle-ment measures that quantify the valence bond resourcesneeded to describe the state. This will be reported else-where.In conclusion, we have identified the entanglementproperties of the cluster states that are responsible for4the possibility of universal quantum computation. Themain insight was given by the fact that the structure ofentanglement in these states is essentially bipartite andcan be understood in terms of valence bonds. This al-lowed to prove the equivalence of the one-way computerwith teleportation-based computation schemes, and toclarify the special features of the cluster states.AcknowledgmentsWe acknowledge M. Mart́in–Delgado for interestingdiscussions about VBS.[1] C.H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).[2] M. A. Nielsen and I. L. Chuang, Phys. Rev. Lett. 79, 321(1997).[3] D. Gottesman and I. Chuang, Nature 402, 390 (1999).[4] E. Knill, R. Laflamme and G. Milburn, Nature 409, 26(2001).[5] M.A. Nielsen, quant-ph/0108020.[6] D. Leung, quant-ph/0111122 and quant-ph/0310189.[7] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86,5188 (2001); Quant. Inf. Comp. 6, 443 (2002).[8] H. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910(2001).[9] R. Jozsa and N. Linden, quant-ph/0201143.[10] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).[11] I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, Com-mun. Math. Phys. 115, 477 (1988); Phys. Rev. Lett. 59,799-802 (1987)[12] M. Fannes, B. Nachtergaele and R.F. Werner, Comm.Math. Phys. 144, 443 (1992).[13] F. Verstraete, M.A. Mart́in–Delgado and J.I. Cirac,quant-ph/0311087.[14] D. Gottesman, Phys.Rev. A 54, 1862 (1996).[15] M. Nielsen and I. Chuang, Quantum computationand quantum information, Cambridge University Press,Cambridge, U.K (2000).[16] D. Gottesman, Caltech Ph.D. Thesis, quant-ph/9705052.[17] J.I. Cirac, W. Dür, B. Kraus and M. Lewenstein, Phys.Rev. Lett. 86, 544 (2001).[18] D. Schlingemann and R.F. Werner, Phys. Rev. A 65,012308 (2002).[19] M. Hein, J. Eisert and H.J. Briegel, quant-ph/0307130;M. Van den Nest, J. Dehaene and B. De Moor,quant-ph/0308151; J. Dehaene and B. De Moor, Phys.Rev. A 68, 042318 (2003).[20] M. Zukowski et al., Phys. Rev. Lett. 71, 4287 (1993).[21] W. Dür and H.-J. Briegel, quant-ph/0307180.[22] W. Dür, H. Aschauer and H.-J. Briegel, Phys. Rev. Lett.91, 107903 (2003).[23] O. Mandel et al., Nature 425, 937 (2003).[24] D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120(1998).[25] A. Miyake and F. Verstraete, quant-ph/0307067.[26] Note that universal quantum computation can beachieved by only implementing gates between neighbor-ing qubits.[27] The extra left multiplication with Pauli operators doesnot harm: a later 1-qubit operation V can be chosen tobe conditioned on the outcome (i.e. implementing V σαinstead of V ). Furthermore, right multiplication of the 2-qubit phase gate Uph with Pauli operators is equivalent toleft multiplication of it with different ones. Therefore theextra Pauli operators can be pushed through the quan-tum circuit without affecting the computation.[28] See e.g. the way 1-qubit unitary gates are implementedin the one-way computer [7].[29] It would be interesting in this respect to find a normalform for stabilizer states that minimizes this number ofbonds. This however seems to be a difficult problem [19].[30] The construction given also works for the set of so–calledgraph states [18], which is, up to local unitaries, equiva-lent to the set of stabilizer states.

Cirac, J. Ignacio

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