aa r X i v : . [ qu a n t - ph ] J a n Adiabatic Cluster State Quantum Computing
Dave Bacon
1, 2 and Steven T. Flammia Department of Computer Science & Engineering, University of Washington, Seattle, WA 98195 Department of Physics, University of Washington, Seattle, WA 98195 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada (Dated: October 22, 2018)Models of quantum computation are important because they change the physical requirements forachieving universal quantum computation (QC). For example, one-way QC requires the preparationof an entangled “cluster” state followed by adaptive measurement on this state, a set of requirementswhich is different from the standard quantum circuit model. Here we introduce a model based onone-way QC but without measurements (except for the final readout), instead using adiabatic defor-mation of a Hamiltonian whose initial ground state is the cluster state. This opens the possibilityto use the copious results from one-way QC to build more feasible adiabatic schemes.
Computers that can exploit the laws of quantum the-ory can, in principle, outperform today’s classical com-puters. For example, quantum computers can efficientlyfactor [1], something classical computers are thought in-capable of doing. Motivated by this fact, a vast amountof ongoing research focuses on figuring out exactly howto build a quantum computer. In addition to differ-ent physical mediums for implementing QC, numerousdifferent models for how to achieve QC have been pro-posed. While to date each of these models providesthe same computational power, they differ substantiallyon the requirements they put on the physical hard-ware. The most widely used model of QC is the quan-tum circuit model, but other models include one-way (ormeasurement-based) QC [2], holonomic QC [3], univer-sal adiabatic QC [4], and topological QC [5]. Here wepropose a new model of computing which combines ideasfrom all of these models. In particular we demonstratehow one can perform one-way QC adiabatically.One-way QC [2] is a method for QC in which one cre-ates a specific, fixed entangled state of a quantum many-body system and then computes via a series of local mea-surements on the subsystems. The choice of measure-ments correspond to unitary gates enacted in the QCand these measurements are adaptive: that is, the ex-act measurement being executed depends on the previ-ous measurement results. One set of states which canbe used for one-way QC are the class of so-called clus-ter states [6]. Cluster states are defined for any graph,though not all graphs allow for universal one-way QC.A cluster state can be defined as a stabilizer code state.Equivalently, there is a Hamiltonian with at most ( d +1)-qubit interactions, where d is the maximum degree of thegraph, whose ground state is the cluster state (one canreplace this Hamiltonian with another involving only 2-qubit interactions while retaining the cluster state as anapproximate ground state [7].) Thus one could imag-ine engineering a physical system with this Hamiltonian,cooling the system to its ground state, and then doingmeasurements that enact the cluster state QC. Here weshow that instead of performing these measurements one can instead simply adiabatically turn on appropriate lo-cal fields while turning off portions of the cluster statein order to perform the QC. Thus we can dispense withmeasurements in the one-way model (except, of course,for the final readout) and instead use adiabatic evolutionsto enact one-way QC. This model provides many of theadvantages of adiabatic control; in particular it retainsrobustness to deformations of the specific adiabatic pathtraversed during the open-loop holonomic evolution [8]. Adiabatic dragging.
The main tools we use in this pa-per are adiabatic changes in a Hamiltonian. Supposeinitially we have a system with Hamiltonian H i and thesystem is in an energy eigenstate. Then we evolve thesystem under a time-varying Hamiltonian over a timeperiod 0 ≤ t ≤ T as H ( t ) = f ( s ) H i + g ( s ) H f where f (0) = g (1) = 1, f (1) = g (0) = 0 and s = tT is a scaledtime. If we vary this evolution smoothly and there areno level crossings, then it is always possible to choosea T large enough such that at the end of this evolutionwe will be in the eigenstate of H f which is continuouslyconnected to the initially prepared eigenstate. In partic-ular, if we choose T to be on the order of the minimumenergy gap between the instantaneous eigenstate of H ( t )and the nearest eigenstates, then with high probabilityat the end of the above evolution the system will be inthe connected eigenstate of the final Hamiltonian [9]. Wewill call such a setup and evolution an adiabatic dragging .Recently, adiabatic dragging between Hamiltonians withenergy eigenstates that are degenerate and are quantumerror-correcting codeword states has emerged as a pow-erful primitive for building a quantum computer [10–12].Here we extend these ideas to one-way QC.
1D degenerate cluster-state model.
Begin by consid-ering a line of n qubits and a degenerate variation on theone-dimensional cluster state. In particular define thefollowing n − S i = [ Z ] i [ X ] i +1 [ Z ] i +2 , ≤ i ≤ n − , S n − = [ Z ] n − [ X ] n , where X and Z are the corresponding Pauli operatorsand we use the notation [ P ] i to denote the operator P acting on the i th physical qubit. These are n − n operators usually used to define a cluster state [6].Define now the stabilizer code corresponding to these op-erators as the common +1 eigenstates of all of the S i , i.e. | ψ i such that S i | ψ i = | ψ i . By standard results in thetheory of stabilizer codes [13], this code space is two di-mensional (encodes a qubit.) We can define the logicaloperators for this encoded qubit as¯ X = [ X ] [ Z ] and ¯ Z = [ Z ] . (1)Now consider the Hamiltonian H = − ∆ n − X i =1 S i . (2)Since the S i all commute and have eigenvalues ±
1, theground state subspace of this Hamiltonian is the +1 com-mon eigenstate of the S i ’s or, in other words, the encodedqubit defined above. Note that quantum information inthe degenerate ground state can be accessed by measur-ing or manipulating the encoded Pauli operators whichare themselves localized on the first two qubits.Now suppose that we adiabatically turn on a local fieldalong the − [ X ] direction while turning off the S termin H , which anticommutes with [ X ] . In particular con-sider adiabatic dragging from H to H + ∆( S − [ X ] ).Notice that while ¯ X commutes with [ X ] , ¯ Z does notcommute with [ X ] . However because we are in the +1eigenspace of each S i , instead of defining the logical ¯ Z as we have done above in Eq. (1) we could also definethe encoded Z as ¯ Z ′ = ¯ ZS = [ X ] [ Z ] . If we do this,then the encoded qubit commutes with the terms we areturning on and off ( S and [ X ] .) Thus the quantum in-formation in this encoded subspace is not touched. How-ever since S anticommutes with [ X ] , the informationin S is changed. To see how this evolution proceeds, wecan consider a code in which we promote S into an en-coded Pauli Z operator and [ X ] is its conjugate encoded X operator. The adiabatic evolution is then simply be-tween these the two encoded Pauli operators (i.e. froman encoded − ∆ ¯ Z a to an encoded − ∆ ¯ X a where a denotesthis newly defined encoded qubit.) Such an evolution hasno level crossing and an energy gap for reasonable adia-batic interpolations which is proportional to ∆. Thus atthe end of this evolution we will be in the +1 eigenstateof [ X ] along with all the remaining S i . In other wordwe are in the stabilizer code with stabilizer generators[ X ] , S , S , . . . , S n − . The information in the degener-ate subspace, which originally was represented via the en-coded operators ¯ X = [ X ] [ Z ] and ¯ Z = [ Z ] is now rep-resented by ¯ X ′ = [ X ] [ Z ] and ¯ Z ′ = [ X ] [ Z ] . However,since we are in the +1 eigenstate of [ X ] this is equivalentto the encoded operator ¯ X ′′ = [ Z ] and ¯ Z ′′ = [ X ] [ Z ] .In other words the information which was originally en-coded in the first two qubits, after the above adiabaticdragging, will be in the second and third qubit. Using thesame logical Pauli encoding (logical X is [ X ] i [ Z ] i +1 and logical Z is [ Z ] i ) we see that a Hadamard gate has beenapplied to this information. Thus, by turning on a [ X ] term on the first qubit while turning off the term in theHamiltonian with which it anti-commuted, we have effec-tively moved this information one step down the line, andapplied a Hadamard gate to the quantum information.Proceeding inductively, if we first adiabatically turn on[ X ] , then [ X ] , etc, while turning off the correspondinganticommuting term in the original Hamiltonian we willend up with the qubit which was originally localized toone end of the line moved to the other end of the line,along with a sequence of Hamadard gates applied to thisqubit. Throughout this piecewise evolution the energygap will remain constant because each successive adia-batic dragging acts independently. If we proceed to turnon each [ X ] i all the way up to the ( n − X is mapped to [ X ] n and ¯ Z is mappedto [ Z ] n if the chain is odd length and ¯ X is mapped to[ Z ] n and ¯ Z is mapped to [ X ] n otherwise — these dif-ferences arising from whether an even or odd number ofHadamards have been applied to the encoded qubit. Single qubit gates.
We now show how to modify theabove setup such that in addition to propagating a singlequbit of information down the one dimensional system,we also apply gates other than the Hadamard gate to thequbit. This scheme is motivated directly by the one-wayQC model where instead of measuring the qubit along the X direction to propagate the information, we measurealong a rotated direction, M ( θ ) = cos( θ ) X + sin( θ ) Y .Importantly, however, our scheme proceeds without adap-tive operations . Consider mimicking the above scheme,but instead of turning on successive − ∆[ X ] i s while turn-ing off the appropriate anticommuting terms in H (the − ∆[ Z ] i [ X ] i +1 [ Z ] i +2 terms) we instead turn on successive − ∆ M i terms where M i = [ M ( − θ i )] i is a set of rotatedlocal fields, 1 ≤ i ≤ n −
1. We claim that this will take thequbit localized to one end of the line and propagate it tothe other end of the line while applying a gate dependenton the choice of θ i .To analyze this scheme it is easier to work in a frameof reference in which the i th qubit has been rotated by U ( θ i ) = exp( − iθ i [ Z ] i / θ = 0,which we will now assume. Consider again n qubits on aline and define now the rotated stabilizer code operators: T i = [ Z ] i [ X U i +1 ] i +1 [ Z ] i +2 , ≤ i ≤ n − ,T n − = [ Z ] n − [ X ] n , (3)where we use the superscript to denote conjugation, P U = U P U † , and U i = U ( θ i ). Note that this con-jugation does not change the fact that these operatorscommute and square to identity, and therefore we canagain define a codespace as the joint +1 eigenspace ofthese operators. Let H be the initial Hamiltonian forour system as in Eq. (2), but now with the rotatedstabilizer operators T i substituted for S i . Again, ini-tially we can define the information in the degener-ate subspace as localized to the first two qubits with¯ X = [ X ] [ Z ] and ¯ Z = [ Z ] . Now imagine adiabati-cally dragging H to H + ∆( T − [ X ] ), then draggingto H + ∆( T + T − [ X ] − [ X ] ), etc. We claim that atthe end of this scheme we will end up with the quantuminformation in ¯ X and ¯ Z propagated to the last qubit witha gate dependent on θ i applied to this information.To see this, we proceed in three steps. First wewill show that using the rotated stabilizer operators itis possible to write the logical qubit in a form whereeach X i (except i = n ) commutes with this informa-tion. Define the following operators for α, β ∈ { X, Y, Z } :¯ α i = P β ( P α,βi [ β ] i ) C i,i +1 where C i,i +1 is the controlledphase gate between the i and ( i + 1)st qubits exceptwhen i = n in which case we define C n,n +1 = I . Weclaim that these new Pauli operators are, under the ro-tated stabilizer code generated by the T i ’s, equivalent tothe logical operators ¯ X = [ X ] [ Z ] , ¯ Y = [ Y ] [ Z ] , and¯ Z = [ Z ] , with the condition that the P α,β ’s are a sum ofproducts of [ X ] j operators for j < i . This can be proveninductively. The base case corresponds to P α,β = δ α,β I where ¯ X = ¯ X and ¯ Z = ¯ Z . Now assume the hypothesisis true for the i th operators. Examine, for example, ¯ X i and expand the controlled-phase:¯ X i = P X,Xi [ X ] i [ Z ] i +1 + P X,Yi [ Y ] i [ Z ] i +1 + P X,Zi [ Z ] i . (4)Recall that the T i operators act as identity on thecodespace and thus can be inserted into this sum in anymanner to yield any equivalent operator (over the code.)Left multiplying ¯ X i by T i for the last two terms yields¯ X i = P X,Xi [ X ] i [ Z ] i +1 + P X,Zi [ X U i +1 ] i +1 [ Z ] i +2 − iP X,Yi [ X ] i [ X U i +1 ] i +1 [ Z ] i +1 [ Z ] i +2 (5)Expanding out X U i +1 , we find that P X,Xi +1 = cos( θ i +1 ) P X,Zi + sin( θ i +1 )[ X ] i P X,Yi P X,Yi +1 = sin( θ i +1 ) P X,Zi − cos( θ i +1 )[ X ] i P X,Yi P X,Zi +1 = [ X ] i P X,Xi (6)Similar relations hold for ¯ Y i +1 and ¯ Z i +1 with the impor-tant property that the new P α,βi +1 s are functions of theprevious P α,βi s and [ X ] i s. This proves our statement.But these expressions also prove much more. In partic-ular if we restrict the above equivalence to the +1 sub-space of [ X ] i , then we see (when we calculate out allnine new P α,βi +1 ’s) that the relationship between the ¯ α i and ¯ α i +1 is ¯ α i +1 = ¯ α U i +1 Hi . In other words, with this re-striction, the effect on the encoded quantum informationin this new form is as if the gate U i +1 H has been ap-plied to the quantum information. Further note that in the procedure we have described for adiabatically drag-ging the initial Hamiltonian, we are always turning off a − ∆ T i while turning on a − ∆[ X ] i . Then not only does[ α ] i +1 commute with these terms (because the P α,βi +1 ismade up entirely of a product of [ X ] j ’s with j < i + 1),and thus is untouched by the evolution, but by an ar-gument identical to the untwisted Hamiltonian case, weend each such dragging in the +1 eigenvalue of − ∆[ X ] i .Thus we end up exactly in the subspace where the gate U i +1 H has been applied and the quantum informationshifted one site down the chain for each such adiabaticdragging. The final effect for the turning on all n − X ] i in order is that the sequence of gates H Q i = n − ( U i +1 H )is applied to the quantum information.In recap, we have shown that by starting with a Hamil-tonian which is a negative sum of twisted stabilizer op-erators T i and then turning off the T i ’s while turning onthe [ X ] i ’s sequentially, we have enacted a gate which de-pends on the angles θ i . This is equivalent to using thestandard cluster state Hamiltonian from Eq. (2) with theunrotated S i stabilizer operators as the initial Hamilto-nian and using rotated magentic fields [ M ( − θ i )] i for thepiecewise final Hamiltonians. Note that we did not workin a rotating frame for the final qubit and therefore theinformation ends up exactly on the last qubit of this evo-lution. Throughout this piecewise evolution the energygap is constant (independent of the length of the chain.)The gates enacted are universal for single qubit gates. State preparation.
In the previous section we enactedgates on the degenerate ground state of a Hamiltonian.We now show how it is possible to prepare quantum infor-mation in a particular state, with the Hamiltonian non-degenerate, and then propagate the information downthe line while turning the Hamiltonian into one with adegenerate ground state where this encoded informationlives. Consider, for example, our original Hamiltonianin Eq. (2) but now with the full cluster state Hamilto-nian H ′ = H − ∆ S where S = [ X ] [ Z ] . The groundstate of H ′ is now not degenerate and corresponds, inour previous picture of H to being in the +1 eigen-state of ¯ X . Consider first adiabatically dragging H ′ to H ′ +∆( S − [ X ] ). At the end of this evolution we will bein the +1 eigenspace of [ X ] as before. Since we startedin the +1 eigenspace of ¯ X we will be in the +1 eigenspaceof ¯ X ′ = [ Z ] . Next adiabatically drag the Hamiltonian to H ′ + ∆( S + S + S − [ X ] − [ X ] ). Notice that we haveto turn off two stabilizer generators while turning on asingle field. This implies that we must increase the degen-eracy of the ground state. We will see that this seconddragging, despite increasing the degeneracy, ends withthe system in the +1 eigenstate of the ¯ X ′′ = [ X ] [ Z ] .To see this note that while both S and S do notcommute with [ X ] , S S does. Thus the eigenvalueof S S is preserved while turning on [ X ] . If we thenrewrite S + S as S ( I + S S ), then if we are in the − S S then this term vanishes, but ifwe are in the +1 eigenspace then in this space the op-erator effectively acts as 2 S (or equivalently 2 S ). Wecan then consider the code where we promote S to anencoded Z operator and [ X ] to an encoded X opera-tor, and then at the end of the evolution we will be inthe +1 eigenstate of [ X ] , and we are also in the +1eigenstate of S S (due to this operator commuting with[ X ] .) Translating this into the coding language, we arein the +1 eigenstate of a stabilizer code with genera-tors [ X ] , [ X ] , [ X ] [ Z ] , S , . . . , S n , which is equivalentto saying that we are in the +1 eigenstate of the 1Dcluster state with n − X at one end of this chain. Ifwe wish to apply gates to this information, we can pro-ceed as above by applying rotated local fields or rotatingthe stabilizer Hamiltonian. It is important to realize thatthe above evolution has gone from a non-degenerate toa degenerate ground state, so that the energy gap van-ishes. However over the subspaces defined by the con-served quantity S S the energy gap is constant and thusthe adiabatic theorem holds. In fact, the same situationoccurs in the creation of anyons in topological QC [14]. Two-qubit gates.
Let us now show how to apply two-qubit gates. The idea, just as in one-way QC, is to usea Hamiltonian which has a coupling between two chainswhich support single qubits. To see how this works letus analyze a cluster state Hamiltonian with a degenerateground subspace and a single coupling between two en-coded qubits. Consider the six-qubit initial Hamiltonian H = − ∆([ Z ] ,a [ X ] ,a [ Z ] ,a [ Z ] ,b +[ Z ] ,a [ X ] ,a )+( a ↔ b )where the encoded qubits will be associated with a and b and ( a ↔ b ) denotes same term with the a and b la-bels reversed. This Hamiltonian is degenerate, but nowthere are two qubits of degeneracy, corresponding to log-ical operators ¯ X γ = [ X ] ,γ [ Z ] ,γ and ¯ Z γ = [ Z ] ,γ with γ ∈ { a, b } . Now suppose that we turn on − ∆([ X ] ,a +[ X ] ,b +[ X ] ,a +[ X ] ,b ) while turning off H (we could pro-ceed by turning each of these on separately and achievesimilar results.) Using the four stabilizer terms in theHamiltonian above we can rewrite the encoded opera-tors as ¯ X ′ γ = [ X ] ,γ [ X ] ,γ and ¯ Z ′ γ = [ X ] ,γ [ Z ] ,γ [ X ] , ¬ γ where ¬ a = b and ¬ b = a . Using an argument simi-lar to the single-qubit gates, we will end up in the +1eigenstate of the X i,γ operators, i ∈ { , } . Over thiseigenspace, the logical operators become ¯ X f = [ X ] ,γ and ¯ Z f = [ Z ] ,γ [ X ] , ¯ γ . This is equivalent to performinga Hadamard on each encoded qubit, a controlled phasegate between these qubits, and then a Hadamard on eachqubit again. Adiabatic Cluster State QC (ACSQC).
We now seehow to build a quantum computer using piecewise adia-batic evolutions from a Hamiltonian whose ground stateis a cluster state to a Hamiltonian consisting of local fields (we note that this initial state can also be piece-wise adiabatically prepared [15].) Consider a quantumcircuit made up of gates from a universal gate set suchas { HU ( π ) , H, ( H ⊗ H ) C i,j } (other sets are also possi-ble) along with the preparation in the +1 eigenstate ofthe Pauli X operator. Then one can map the graph ofthis circuit onto a cluster state graph using the aboveelements in such a way that one can also prescribe localfields which, when turned on piecewise, enact the quan-tum circuit (or equivalently one can us a twisted clusterstate Hamiltonian and local fields all along X .) Conclusion.
We have shown how to perform one-wayQC on a cluster state using only piecewise adiabatic evo-lutions. This scheme shares many of the traits of therecently introduced primitive of adiabatic gate telepor-tation [12]: it has a robustness to the adiabatic path,for example. Further, as in [12] we can use perturbationtheory gadgets [7] to implement this entire scheme us-ing only two-qubit interactions instead of the four-qubitinteractions we have presented; it would be interestingto make this calculation explicit. Our model shows thenovelty of starting with a global entangled ground stateand then piecewise turning on local fields to do QC. Wehave also shown how it is possible to use cluster statesand their parent Hamiltonians to perform QC withoutresorting to adaptive measurements. ACSQC thus opensup a new way to adapt the numerous results of one-wayQC to viable adiabatic architectures.While preparing this manuscript we learned of similarresults for single qubit circuits using the AKLT state [16].DB was supported by the NSF under grants 0803478,0829937, and 0916400 and by DARPA under QuESTgrant FA-9550-09-1-0044. STF and research at Perime-ter are supported by the Government of Canada throughIndustry Canada and by the Province of Ontario throughthe Ministry of Research & Innovation. [1] P. W. Shor, in
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