Strategies for measurement-based quantum computation with cluster states transformed by stochastic local operations and classical communication
SStrategies for measurement-based quantum computation with cluster statestransformed by stochastic local operations and classical communication
Adam G. D’Souza and David L. Feder
Department of Physics and Astronomy and Institute for Quantum Information Science,University of Calgary, Calgary, Alberta, Canada (Dated: November 2, 2018)We examine cluster states transformed by stochastic local operations and classical communication,as a resource for deterministic universal computation driven strictly by projective measurements.We identify circumstances under which such states in one dimension constitute resources for random-length single-qubit rotations, in one case quasi-deterministically (N − U − N states) and in anotherprobabilistically (B − U − B states). In contrast to the cluster states, the N − U − N states exhibitspin correlation functions that decay exponentially with distance, while the B − U − B states can bearbitrarily locally pure. A two-dimensional square N − U − N lattice is a universal resource for quasi-deterministic measurement-based quantum computation. Measurements on cubic B − U − B statesyield two-dimensional cluster states with bond defects, whose connectivity exceeds the percolationthreshold for a critical value of the local purity.
I. INTRODUCTION
In the Measurement-Based Quantum Computation(MBQC) model [1, 2], one starts with a highly entan-gled many-qubit quantum state called a resource state,and processes logical information via single-qubit mea-surements on the physical qubits of the resource state.In order to compensate for the randomness of the mea-surement outcomes, the bases in which measurements areperformed must be conditioned on the outcomes of previ-ous measurements. Proceeding in this way, one can tele-port logical quantum information situated on one partof the state to another part, but having been subjectedto some desired unitary transformation. If the basic uni-tary transformations that can be applied via single-qubitmeasurements on the resource state generate a set that isdense in SU(2), then the resource is said to be universal(in the terminology of Ref. [3], this is the notion of CQ-universality, and such a resource state would be called auniversal state preparator).The archetypal family of resource states known tobe universal for efficient MBQC is the so-called clusterstate [1, 2]. This state is special in several ways: all spincorrelation functions are strictly nearest-neighbor [4, 5],the localizable entanglement between any pair of qubitsis maximal [4, 5], it is the only state (up to local uni-taries) on small system sizes that saturates the Tsallisand Renyi entropies of entanglement [6], it cannot be thenon-degenerate ground state of a two-body spin Hamil-tonian [7, 8], and so on. One might expect that one ormore of these properties would be necessarily satisfiedby any universal resource state. This has turned outnot to be the case; several authors in recent years haveidentified resources that differ materially from the clusterstates [4, 5, 9–14].The newly discovered richness in the landscape of re-sources notwithstanding, the property of universality isexceedingly rare; not only must a family of universal re-source states saturate various measures of entanglementin the thermodynamic limit [3, 15], but the entangle- ment with respect to other measures must not be toohigh [16, 17]. Therefore, it is highly unlikely that a ran-dom pure state will be universal, so a search for newresources must be heavily constrained in order to have areasonable chance of success.Recently, a number of new resources [4, 5, 10, 18] havebeen proven to be universal by means of reduction to aknown resource state [19]. The reduction strategies ofinterest are those composed purely of local operations,possibly augmented by classical communication. Theyare typically stochastic, in the sense that the known re-source state is smaller than the original state. In otherwords, these resources all appear to be within the equiv-alence class of the cluster states under Stochastic Lo-cal Operations and Classical Communication (SLOCC).The SLOCC-equivalence class of an n -qubit pure stateis known to be its orbit under GL (2 , C ) ⊗ n , the group of n -fold tensor products of two-by-two invertible matricesover the complex numbers. In other words, two n -qubitstates | ψ (cid:105) and | φ (cid:105) are equivalent under SLOCC if andonly if | ψ (cid:105) = S (1) ⊗ S ⊗ · · · ⊗ S n | φ (cid:105) (1)where the { S i } ∈ GL (2 , C ) are invertible, two-by-twocomplex matrices.A natural question thus arises: are all universal MBQCresources SLOCC-equivalent to the cluster states? Moreprecisely, is any n -qubit element of a family of universalresource states SLOCC-equivalent to an n -qubit clusterstate?In this paper, we tackle a related question, namely:what states in the SLOCC-equivalence class of the two-dimensional cluster states are universal for MBQC? It isclear, by construction, that each state in this class canbe stochastically reduced to a cluster state, but what isnot clear is whether it is possible to compute directly onthe image of a cluster state under some invertible, lo-cal map. We show that there is a restricted subclass ofinvertible local transformations, strictly including the lo-cal unitaries, whose image is a set of quasi-deterministic a r X i v : . [ qu a n t - ph ] A ug resources for MBQC, where in general the computationis of random length and ‘repeat-until-success’ strategiesmust be employed (c.f. Refs. [5, 18]). In particular, weidentify two types of SLOCC operators whose action canin certain cases preserve the usefulness of the cluster stateas a resource. The first type, which we call N-type opera-tors, comprises those operators that preserve the relativenorms of the computational basis states. The secondtype, called B-type operators, are those that preservetheir orthogonality (i.e. are in a sense basis-preserving).In particular, we show that when N-type operators acton alternating qubits in a 1D cluster state, the state re-mains a quasi-deterministic resource for single-qubit ro-tations. We refer to such 1D states as N − U − N chains. Incontrast to the cluster state, the number of measurementsrequired to implement an arbitrary single-qubit rotationwith an N − U − N chain is random rather than fixed.Furthermore, the state exhibits non-zero spin-spin cor-relations that decay exponentially with distance. Theseproperties are shared by other resources previously ap-pearing in the literature [4, 10, 20, 20], notably thosebased on the so-called AKLT model [21, 22]. We alsohow 1D N − U − N chains can be coupled together toproduce a quasi-deterministic 2D resource for universalMBQC.Next, we show that when B-type operators act onalternating qubits in a 1D cluster state, the result isin general a probabilistic resource for single-qubit rota-tions. We call these states B − U − B chains. We findthat under a restricted subset of B-type operators, thethree-dimensional analogs of B − U − B chains constitutequasi-deterministic resources for MBQC under strictlyprojective measurements. A similar result was exhibitedin Ref. [15], in which a 2D cluster state deformed byB-type operators was shown to be reducible to a perco-lated 2D cluster state [23] by the action of three-elementPOVMs. The B − U − B states have the interesting prop-erty that each qubit can be arbitrarily locally pure, oralternatively that an individual qubit can be arbitrarilyweakly entangled with the rest of the state, as measuredby the von Neumann entropy of entanglement. Like thecluster states, they also exhibit vanishing long-range cor-relations, with no spin-spin correlation functions beyondsecond-nearest-neighbor surviving.The structure of the paper is as follows: in Section II,we briefly review the theory of measurement-based quan-tum computation using cluster states, and introduce thevarious definitions and notation used in the technical partof this paper. In Section III, we describe the effects ofinvertible local operators acting on a cluster state on theclass of linear transformations that can be logically imple-mented via adaptive single-qubit measurements on thisnew state, and outline some strategies for dealing withthese effects. In Section IV, we provide explicit examplesof some structures of SLOCC-transformed cluster statesthat are universal for either probabilistic or determinis-tic single-qubit rotations or full MBQC. Finally, in Sec-tion V, we discuss the relationship of our resource states with previously known quasi-deterministic resources, andoutline the prospects of identifying hitherto unknown re-source states by this method.
II. BACKGROUND An n -qubit cluster state can be defined in terms of thestabilizer formalism [24] as the unique n -qubit pure state | Cl n (cid:105) satisfying the n conditionsX i (cid:79) j ∈N ( i ) Z j | Cl n (cid:105) = | Cl n (cid:105) , (2)where i ∈ { , . . . , n } labels a qubit, N ( i ) denotes thespatial neighbourhood of qubit i , and X i and Z i denotethe standard single-qubit Pauli operators, given in thecomputational basis byX = (cid:20) (cid:21) ;Z = (cid:20) − (cid:21) , acting on qubit i . Alternatively, the cluster state can beidentified as the result of a dynamical process in which1. n qubits are initialized in the state | + (cid:105) ⊗ n , where | + (cid:105) ≡ √ ( | (cid:105) + | (cid:105) ) is the +1-eigenstate of X;2. CZ entangling gates, whose action on the compu-tational basis states is given byCZ i,j | x, y (cid:105) i,j = ( − x · y | x, y (cid:105) i,j , (3)are applied between each pair of neighbouringqubits ( i, j ). Thus, | Cl n (cid:105) = (cid:89) (cid:104) i,j (cid:105) CZ i,j | + (cid:105) ⊗ n , (4)where (cid:104) i, j (cid:105) indicates that i and j label neighbour-ing qubits.For notational convenience, define a global entanglingoperation on a lattice, G k,l := l − (cid:89) j = k CZ j,j +1 , (5)to be the tensor product of CZ gates acting betweenall nearest-neighbour pairs of vertices on a line with la-bels between k and l . Now consider a modified one-dimensional n -qubit cluster state, | Cl Dn (cid:105) (cid:48) = G ,n | ψ (cid:105) | + (cid:105) ⊗ n − ,...,n . (6)where the first qubit was encoded in some general purestate | ψ (cid:105) before the global entangling operation G ,n wasperformed. The effect of projectively measuring the firstqubit, the one on which | ψ (cid:105) was initially encoded, is toteleport the quantum information corresponding to thestate | ψ (cid:105) to the next qubit, subject to some linear trans-formation depending upon the basis and outcome of themeasurement. To see this, assume that | ψ (cid:105) = a | (cid:105) + b | (cid:105) .We then find that | Cl Dn (cid:105) (cid:48) = G ,n CZ , ( a | (cid:105) , + b | (cid:105) , ) | + (cid:105) ⊗ n − ,...,n = G ,n ( a | (cid:105) , + b | −(cid:105) , ) | + (cid:105) ⊗ n − ,...,n = G ,n ( a | (cid:105) I + b | (cid:105) Z ) | + (cid:105) ⊗ n − ,...,n . (7)Projecting the first qubit via an arbitrary rank-1 pro-jector | m (cid:105)(cid:104) m | , the state of the system (neglecting theprojected qubit and overall normalization) becomes | Φ (cid:105) = G ,n ( a (cid:104) m | (cid:105) I + b (cid:104) m | (cid:105) Z ) | + (cid:105) ⊗ n − ,...,n = G ,n ( a (cid:104) m | (cid:105)| + (cid:105) + b (cid:104) m | (cid:105)|−(cid:105) ) | + (cid:105) ⊗ n − ,...,n = G ,n H ( a (cid:104) m | (cid:105)| (cid:105) + b (cid:104) m | (cid:105)| (cid:105) ) | + (cid:105) ⊗ n − ,...,n ∝ G ,n H ( (cid:104) m | + (cid:105) I + (cid:104) m |−(cid:105) Z ) | ψ (cid:105) | + (cid:105) ⊗ n − ,...,n , where the Hadamard operator H i = (X i + Z i ) / √
2. Inother words, the quantum information has been tele-ported to the second qubit through the linear transfor-mation M = H ( (cid:104) m | + (cid:105) I + (cid:104) m |−(cid:105) Z) . (8)Without loss of generality, the single-qubit state actingas the projector can be written as | m ( ξ, φ ) (cid:105) = cos ξ | + (cid:105) + e iφ sin ξ |−(cid:105) (9)for some 0 ≤ ξ < π , − π ≤ φ < π . Thus, the lineartransformation through which the quantum informationis teleported can be written asM = H (cid:18) cos ξ e iφ sin ξ (cid:19) . (10)In the special case that φ = ± π , corresponding to | m (cid:105) lying on the x − y plane of the Bloch sphere, this transfor-mation becomes the (familiar from cluster state MBQC)unitary transformation HR z [ ± ξ ]. Thus, there is an entiresingle-parameter family of unitary gates through whichthe initial state | ψ (cid:105) can be teleported, each correspondingto a projection of the first qubit on to some state lyingin the x − y plane. This family is universal for single-qubit rotations: via four projections, corresponding to ξ = 0 , ξ , ξ , ξ respectively, one teleports the transfor-mationU ( ξ , ξ , ξ ) = HR z [ ξ ] HR z [ ξ ] HR z [ ξ ] HR z [0]= R x [ ξ ] R z [ ξ ] R x [ ξ ] , (11)which is an arbitrary single-qubit unitary decomposed interms of Euler angles. To this point, we have not discussed how to com-pensate for the randomness associated with the mea-surements. If one were to drive the gate teleportationdescribed above via projective measurements, measure-ments must be made in an orthonormal basis containing | m ( ξ, φ = π ) (cid:105) . For a single qubit, this basis would be B ( ξ, φ ) = (cid:8) | m ( ξ, φ ) (cid:105) , | m ⊥ ( ξ, φ ) (cid:105) (cid:9) , where | m ( ξ, φ ) (cid:105) = cos ξ | + (cid:105) + e iφ sin ξ |−(cid:105) ; (12) | m ⊥ ( ξ, φ ) (cid:105) = sin ξ | + (cid:105) − e iφ cos ξ |−(cid:105) . (13)From Eq. (8), the teleported gate associated with theapplication of the projector | m ⊥ (cid:105)(cid:104) m ⊥ | on the first qubitwould be M ⊥ = H (cid:20) sin ξ − e iφ cos ξ (cid:21) ≡ XH (cid:20) cos ξ − e − iφ sin ξ (cid:21) , where in the last step we have made use of the identityXH = HZ, and have dropped an unimportant overallphase. Once again considering the special case that φ = ± π , this reduces toM ⊥ = XHR z [ ± ξ ] = XM . The teleported gate can be summarized succinctly asfollows: measuring in the basis (cid:8) | m (cid:105) , | m ⊥ (cid:105) (cid:9) defined byEqs. (12,13) with φ = π , and denoting the measurementoutcome by m = 0 for state | m (cid:105) and m = 1 for | m ⊥ (cid:105) ,then the teleported gate is X m HR z [ ξ ]. The operator Xcan be thought of as a byproduct operator that occursas a result of obtaining measurement outcome 1.If the aim is to teleport the operator U ( ξ , ξ , ξ ) de-fined in Eq. (11), then apparently one runs into a problemshould a measurement outcome of 1 be obtained for anyof the four measurements needed to teleport this gate. Infact the X byproduct operators can be pushed throughthe rotations because R z [ ξ ] X = XR z [ − ξ ]. Suppose thenthat one performs four projective measurements on a one-dimensional cluster state, with the i th measurement be-ing a projective measurement of qubit i in the orthonor-mal basis B ( θ i , π ), θ = 0 and the measurement outcomedenoted m i ∈ { , } . The quantum information orig-inally situated on qubit 1 before the global entanglingoperation is then teleported through the gateM = X m HR z [ θ ] X m HR z [ θ ] X m HR z [ θ ] X m H= X m Z m X m Z m R x (cid:104) ( − m + m θ (cid:105) × R z [( − m θ ] R x [( − m θ ] . (14)Comparing Eq. (11) and Eq. (14), the choices θ =( − m ξ , θ = ( − m ξ , and θ = ( − m + m ξ makethe implemented teleported gateM = X m Z m X m Z m U ( ξ , ξ , ξ ) . (15)Thus, any gate can be implemented by conditioning eachof the last three measurement bases on the results of pre-vious measurements, up to an overall Pauli byproductoperation. The byproduct is of no concern, as Z has noeffect on computational basis states while X merely swapsthem; this means that the effect of the byproduct can betaken into account simply by appropriate reinterpreta-tion of the final measurement outcomes of the circuit,contingent on the intermediate measurement outcomes.An equivalent description of this universal gate telepor-tation can be obtained within the Matrix-Product State(MPS) representation [4, 5] of the one-dimensional clus-ter state: | Cl n (cid:105) = (cid:88) (cid:126)i A [1] [ i ] A [2] [ i ] . . . A [ n ] [ i n ] | i i . . . i n (cid:105) , (16)where (cid:126)i is an n-bit string and the site matrices (cid:8) A [ j ] [ i j ] (cid:9) are all two-by-two, except for the boundaries; the (cid:8) A [1] [ i ] (cid:9) are row vectors and the (cid:8) A [ n ] [ i n ] (cid:9) are col-umn vectors. The site matrices are not unique, but itis particularly convenient if they are chosen to satisfythe relation (cid:80) i j A [ j ] [ i j ] A [ j ] † [ i j ] = I for each j , corre-sponding to the canonical form of the MPS [25]. For theleft and right boundaries one obtains A [1] [0] = √ (cid:104) + | , A [1] [1] = √ (cid:104)−| , A [ n ] [0] = | (cid:105) , and A [ n ] [1] = | (cid:105) ; forthe bulk sites 1 < j < n they are A [ j ] [0] = √ H and A [ j ] [1] = √ HZ = √ XH. The ‘always-on’ operator His teleported on each measurement of a qubit, and theX gate serves as the byproduct operator. In general, anMPS state is a universal resource for measurement-basedsingle-qubit gate teleportation (a ‘computational wire’)if the bulk site matrices can be chosen to be propor-tional to unitaries [18]. In this case they can be writ-ten as A [ j ] [0] = √ W and A [ j ] [1] = √ W R z ( φ ) with W ∈ SU (2) and φ ∈ R .In the context of the calculations presented in thenext two sections, it is worth pointing out that there aretwo special features associated with the projective single-qubit measurements on one-dimensional cluster statespresented above. The first is that the linear transfor-mation M on the quantum state | ψ (cid:105) is guaranteed to beunitary; in practice, this means that the effect of sucha measurement is not dependent on the input state | ψ (cid:105) .If the linear transformation is not unitary (for exampleprojections of the local system outside the x − y plane,as discussed below), then there is an equivalent unitarytransformation resulting in the same final state vector;however, the equivalent unitary will depend on | ψ (cid:105) . Thesecond feature is that the byproduct operator resultingfrom measurement outcome 1 is always X. This is bene-ficial as X operators can be pushed through R z operatorswith an easily characterized effect, as discussed above.These properties can be summarized as follows: • Property IA: the teleported gate is in general ofthe form HR z [ ξ ] where ξ ∈ R , i.e. a unitary gate corresponding to a z -axis rotation by a real angle,followed by a Hadamard gate. • Property IIA: the byproduct operator is alwaysX ≡ R x [ π ].The above two features do not hold for single-qubitprojective measurements outside the x − y plane. In gen-eral, • Property IB: the teleported gate is in general ofthe form HR z [ ξ ] where ξ ∈ C , i.e. a non-unitarygate corresponding to a z -axis rotation by a com-plex angle, followed by a Hadamard gate. • Property IIB: the byproduct operator is in gen-eral R x [ η ] where η (cid:54) = π in general, i.e. an x -axisrotation not corresponding simply to Pauli X.Another way to view this is that in x − y plane, onealways teleports HR z [ ξ + δ m, π ] with ξ ∈ R , whereasin any other plane, one teleports HR z [ ξ + δ m, (cid:15) ] where ξ ∈ R corresponds to the angle of the z-rotation in thedesired gate, and (cid:15) ∈ C is a complex error that occurs onmeasurement outcome 1.Single-qubit gates alone are not sufficient for universalcomputation; at least one multiqubit entangling gate isrequired as well. In the cluster state model, multiqubitgates are accomplished via measurement patterns on 2Dstructures. Logical qubits are processed by horizontal1D wires, while entangling operations are mediated byvertical links between them. An entangling gate that islocally equivalent to controlled-NOT can be achieved bymeasuring a link qubit in the Y basis [1, 2].As with the case of single-qubit rotations, local Paulibyproduct operators may exist as well, depending on themeasurement outcomes. As before, these byproduct op-erators are of no concern computationally. Thus there ex-ists for cluster states a measurement pattern that deter-ministically implements a two-qubit unitary entanglinggate, with any byproducts that occur being of the localPauli type. It is not immediately clear that this will bethe case for states other than the cluster state; in gen-eral, the teleported two-qubit gate may be non-unitaryand the byproduct may be non-local. Any candidate re-source state for MBQC must be shown to be amenable toa measurement pattern implementing some suitable two-qubit entangling gate. In Section III, we describe some1D structures that are resources for single-qubit rotationsand then in the examples of Section IV, we demonstratehow to perform entangling gates with natural 2D or 3Dextensions of the 1D structures, and how to compensatethe randomness associated the measurements. III. PROJECTIVE MEASUREMENTS ONSLOCC-TRANSFORMED CLUSTER STATES
As discussed in the previous section, the distinguishingfeature of MBQC with regular cluster states is that thereexists a plane of the Bloch sphere onto which successive,adaptive, single-qubit projective measurements drive anarbitrary computation that is deterministic and of fixedlength. No matter which single-qubit gate is desired, itwill be implemented with certainty up to an unimpor-tant Pauli byproduct with four measurements. For anSLOCC-transformed cluster state, it is not obvious thatthere exists any such plane: in general it is not possible tosimultaneously satisfy both Properties IA and IIA (or forthe latter any another convenient Clifford gate). A nat-ural question to ask is then: under what circumstancescan either property IA or IIA be satisfied by itself? Andif only one property is satisfied, does there remain a de-terministic protocol for universal quantum computation?Sec. III A and Sec. III B discuss the circumstances underwhich it is possible to independently satisfy Property IAand IIA, respectively.
A. Strategy I: Guaranteed Unitary Evolution
1. Derivation of N-type Operators
For convenience, define S k,l := l (cid:79) j = k S ( j ) j (17)where S ( j ) ∈ GL (2 , C ). From Eq. (7), it is clear thatthe SLOCC-transformed cluster state encoding quantuminformation can be written in the form S ,n | Cl n (cid:105) (cid:48) = S ,n G ,n (cid:16) aS (1)1 | (cid:105) I + bS (1)1 | (cid:105) Z (cid:17) | + (cid:105) ⊗ n − ,...,n . (18)Following the procedure discussed in Sec. II, applying theprojector | m (cid:105)(cid:104) m | to the first qubit yields the resultingstate on the remaining qubits: | Φ (cid:105) = S ,n G ,n (cid:16) a (cid:104) m | S (1)1 | (cid:105) I + b (cid:104) m | S (1)1 | (cid:105) Z (cid:17) | + (cid:105) ⊗ n − ,...,n = S ,n G ,n H M | ψ (cid:105) | + (cid:105) ⊗ n − ,...,n , whereM = H (cid:20) (cid:104) m | S (1) | + (cid:105)√ + (cid:104) m | S (1) |−(cid:105)√ (cid:21) . (19)The only way for this to correspond to a unitary gate isif 1 √ (cid:104) m | S (1) | + (cid:105) = e iα cos ξ √ (cid:104) m | S (1) |−(cid:105) = − ie iα sin ξ , where 0 ≤ ξ < π , and therefore S (1) † | m (cid:105) = √ e − iα (cid:20) cos ξ | + (cid:105) + i sin ξ |−(cid:105) (cid:21) , (20) or equivalently | m (cid:105) = √ (cid:16) S (1) † (cid:17) − e − iα (cid:20) cos ξ | + (cid:105) + i sin ξ |−(cid:105) (cid:21) = e − iα (cid:16) S (1) † (cid:17) − R z [ ξ ] | + (cid:105) . (21)Eq. (21) is the condition on the state | m (cid:105) such thatmeasurement outcome 0 yields a unitary teleported gate.Note that there is a family of states characterized by asingle parameter ξ fulfilling this condition, not includingthe unimportant overall phase α .To ensure that the measurement yields a unitary tele-ported gate independent of the measurement outcome, asimilar condition must follow for the orthogonal comple-ment | m ⊥ (cid:105) . Orthogonality requires (cid:104) m ⊥ | ∝ (cid:104)−| R z [ − ξ ] S (1) † , (22)and therefore | m ⊥ (cid:105) = cS (1) R z [ ξ ] |−(cid:105) (23)for some constant c ∈ C . Repeating the procedure thatled to Eq. (20), but with | m ⊥ (cid:105) instead of | m (cid:105) , one obtains S (1) † | m ⊥ (cid:105) = √ e − iβ (cid:20) cos ξ | + (cid:105) + i sin ξ |−(cid:105) (cid:21) . (24)Substituting Eq. (23) into Eq. (24) yields √ e − iβ (cid:20) cos ξ | + (cid:105) + i sin ξ |−(cid:105) (cid:21) = cS (1) † S (1) R z [ ξ ] |−(cid:105) = cS (1) † S (1) (cid:20) cos (cid:18) ξ (cid:19) |−(cid:105) − i sin (cid:18) ξ (cid:19) | + (cid:105) (cid:21) . (25)Rewriting Eq. (25) in the computational basis results inthe expression | (cid:105) + e − iξ | (cid:105) = c (cid:48) S (1) † S (1) (cid:0) | (cid:105) − e iξ | (cid:105) (cid:1) (26)for a suitably defined constant c (cid:48) . Defining T k := S ( k ) † S ( k ) and T i,jk := (cid:104) i | T k | j (cid:105) , we can see from Eq. (26)that c (cid:48) (cid:16) T , − e iξ T , (cid:17) = 1; (27) c (cid:48) (cid:16) T , − e iξ T , (cid:17) = e − iξ . (28)From here, it is easily deduced that (cid:12)(cid:12)(cid:12) T , − e iξ T , (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) T , − e − iξ T , (cid:12)(cid:12)(cid:12) . (29)Making use of the Hermiticity of T k , simple algebra yields T , = T , . (30)Eq. (30) above has a very simple geometric interpreta-tion: it means that S (1) must preserve the relative normof the computational basis states. There is no require-ment for S (1) to preserve their orthogonality, however,which means that S (1) is allowed to differ quite drasti-cally from a unitary transformation; in fact, it can bemade arbitrarily close to singular, as the relative anglebetween the computational basis states under the trans-formation by S (1) can be vanishingly small. We will referto operators obeying this norm-preservation restrictionas N-type operators. Definition III.1. A GL (2 , C ) operator S satisfying (cid:104) | S † S | (cid:105) = (cid:104) | S † S | (cid:105) is called an N -type operator. The singular value decomposition is helpful for charac-terizing N-type operators. An arbitrary SLOCC operator S can be written in terms of its singular value decompo-sition as S = U D V, where U is an arbitrary two-qubitunitary, D is a positive-definite diagonal matrix D = κ (cid:20) cos θ
00 sin θ (cid:21) , (31)where 0 < θ < π , and V is an arbitrary unitary ma-trix parametrized via the Euler decomposition as V =HR z [ α ] R x [ β ] R z [ γ ], 0 ≤ α, β, γ < π , and any globalphase has been absorbed into U. It is then straightfor-ward to determine that (cid:104) | S † S | (cid:105) = 12 κ (1 + cos 2 θ sin α sin β ) ; (32) (cid:104) | S † S | (cid:105) = 12 κ (1 − cos 2 θ sin α sin β ) . (33)So, if S is an N-type operator, we must have θ = π (inwhich case S is proportional to a unitary), α = 0 or β = 0. The case where β = 0 still allows us to assume α = 0 without loss of generality. Doing so, the R x [ β ]operator can be commuted past the H to turn into a z -rotation, and then absorbed into U. Thus we have thefollowing characterization of N-type operators. Lemma III.2.
Every N -type operator S must eitherbe proportional to a unitary operator, or of the form S = U D V , where U is an arbitrary two-by-two unitaryoperator, D is defined as in Eq. (31) and V = HR z [ γ ] with ≤ γ < π . It is straightforward to obtain an expression for thebyproduct angle µ (cid:48) in the case of measurement outcome1 in terms of the parameters θ , γ and ξ (recall that this isthe degree of freedom in the measurement basis). UsingEq. (19), it can easily be checked that the teleportedgate when | m (cid:105) ∝ (cid:0) S † (cid:1) − R z [ − ξ ] H | (cid:105) is M = HR z [ ξ ],and when | m (cid:105) ∝ S R z [ − ξ ] H | (cid:105) is M ⊥ = R x [ µ (cid:48) ] HR z [ ξ ],where the byproduct angle µ (cid:48) obeystan µ (cid:48) − cos 2 θ cos ( γ − ξ )cos 2 θ sin ( γ − ξ ) . (34)The probabilities of the two measurement outcomes canalso be easily calculated in terms of the same parameters, and are found to be p (0) = 12 (1 + cos 2 θ cos 2 ξ ) ; (35) p (1) = 12 (1 − cos 2 θ cos 2 ξ ) . (36)This differs from the case of gate teleportation with aperfect cluster state, where the two measurement prob-abilities are always exactly . That said, the expectedprobability of obtaining a byproduct here, averaged overall ξ , is (cid:104) p (1) (cid:105) ξ = 12 , (37)irrespective of θ . We can thus generically expect to ob-tain an unwanted byproduct operator that we must com-pensate on half of our single-qubit measurements. Thispoint will be discussed in Sec. IV.
2. Properties of N-Transformed Cluster States
Cluster states locally transformed by N-type operatorscan exhibit remarkably different properties from perfectcluster states. Nevertheless, as will be shown later inSec. IV B, they can under some circumstances serve asuniversal resources for MBQC of random length.Consider for the moment the Schmidt decompositionof an n -qubit cluster state on some set of qubits V withrespect to a bipartition separating qubit k from the rest: | Cl n (cid:105) V = 1 √ (cid:0) | (cid:105) k | Cl n − (cid:105) V\ k + | (cid:105) k Z N ( k ) | Cl n − (cid:105) V\ k (cid:1) , (38)where | Cl n − (cid:105) V\ k refers to the cluster state resulting fromdeleting qubit k and Z N ( k ) is the tensor product of Z op-erators acting on all the neighbors of k . This can bechecked by verifying that this state satisfies the stabi-lizer conditions (2). The Schmidt basis for the multiqubitcomponent can be further decomposed if desired by thesame technique. The equality of the Schmidt coefficientsin Eq. (38) demonstrates that any individual qubit in acluster state has a maximally mixed local reduced densitymatrix, or in other words that it is maximally entangledwith the rest of the cluster with respect to the von Neu-mann entanglement entropy. Likewise, exactly one ebitof entanglement is shared across any bipartition of thecluster state.One effect of N-type operators is to change the lo-cal reduced density matrices of individual qubits withinthe state. In the canonical representation, the sitematrices of the MPS representation [cf. Eq. (16)] are A [ j ] [0] = √ W = HR z (cid:2) − ( γ ( j ) + 2 θ ( j ) ) (cid:3) and A [ j ] [1] = √ W R z (cid:2) θ ( j ) (cid:3) , which are both unitary. This immedi-ately implies that the channel having these matrices asKraus operators is unital, and like the ordinary clusterstate one ebit of entanglement is shared across any bipar-tition. That said, the entanglement between any givenqubit and the rest of the system need not be unity.Consider for example the local reduced density matrixof a qubit adjacent to an endpoint of a 1D cluster statewith n qubits, numbered 1 to n from left to right. Sin-gling out first qubit 2 and then qubit 3, this state can bewritten as | Cl Dn (cid:105) ...n = 1 √ | (cid:105) | + (cid:105) I + | (cid:105) |−(cid:105) Z ) | Cl Dn − (cid:105) ...n . (39)Now consider the action of an N-type operator, N (2) = D HR z [ γ ] on qubit 2, where the leading U operator isdropped because it can be absorbed into the measure-ment basis. It is easy to check that N (2) | Cl Dn (cid:105) ...n = 1 √ (cid:0) cos θ | (cid:105) | Φ (cid:105) − i sin θ | (cid:105) | Φ ⊥ (cid:105) (cid:1) , (40)where | Φ (cid:105) and | Φ ⊥ (cid:105) are γ -dependent states for qubits1 , , . . . , n such that (cid:104) Φ | Φ ⊥ (cid:105) = 0. Thus, Eq. (40) remainsa Schmidt decomposition. The local reduced density ma-trix of qubit 2 is ρ = (cid:20) cos θ
00 sin θ (cid:21) , revealing that qubit 2 is no longer maximally entangledwith the rest of the state. A similar calculation can beperformed for qubits further from the boundary, withqualitatively similar results.Another property of these states is the long-range be-havior of two-point correlation functions, those of theform C i,j ( A , B ) := (cid:104)A i B j (cid:105) − (cid:104)A i (cid:105)(cid:104)B j (cid:105) for some oper-ators A and B . Two-point correlation functions oflarge 1D cluster states with periodic boundary condi-tions can be efficiently calculated using the Matrix Prod-uct State (MPS) representation. For an n -qubit ring,calculation of the correlation functions amounts to tak-ing traces of products of n × N ( i ) acting on qubit i . For this state, calculationsshow that all two-point Pauli correlation functions van-ish except for the second-nearest-neighbour correlationfunction C i − ,i +1 ( Z, Z ) = cos 2 θ sin γ . This is in con-trast to the perfect cluster state with periodic boundaryconditions, for which all two-point correlation functionsidentically vanish.As another example, the relevance of which will be-come clear in Sec. IV A, consider a ring with an evennumber of qubits, with N acting on every alternate qubit;say, the ones with even labels. In this case, the magni-tude of the same two-point correlation function betweenodd-numbered qubits decays exponentially: | C , j +1 ( Z, Z ) | ∼ exp (cid:18) − jL (cid:19) . (41)The length scale L depends on θ and γ . The same corre-lation function between pairs of qubits with at least oneeven label is zero. The numerically obtained behavior of (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236)(cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242)(cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Π Π
20 3 Π Π Π Π
20 7 Π Π Π Γ L (cid:244) Θ (cid:61) Π (cid:242) Θ (cid:61) Π (cid:236) Θ (cid:61) Π (cid:224) Θ (cid:61) Π (cid:230) Θ (cid:61) Π FIG. 1: (Color online) Correlation length scale L associatedwith the correlation function | C , j +1 ( Z, Z ) | ∼ exp (cid:0) − jL (cid:1) fora N − U − N ring with all N-type operators identical, on a ringof 1000 qubits, as a function of the parameters of γ and θ , theparameters of N . The length scale increases as N approachesthe singular limit, i.e. as θ gets close to 0 or π/ L for a ring of 1000 qubits is shown in Fig. 1 as a func-tion of γ for several values of θ between 0 and π/
4. Ascan be seen from the figure, the length scale increaseswith decreasing θ over this range, i.e. as the N-type op-erators approach the singular limit θ = 0. The lengthscale is symmetric about θ = π/ π/ γ with θ held constant, the cor-relation function is convex and non-negative in γ overthe interval from 0 to π/ π/ π/ ≤ γ < π , the magnitudes behave the same wayas in the previous interval, but the signs alternate. The γ -behavior is periodic with period π . We note in passingthat these non-zero correlation functions provides a lowerbound for the localizable entanglement [26] between thatpair of qubits in the state via projective measurements,with respect to the concurrence [27].Note that a number of resources for MBQC withnon-vanishing long-range correlation functions have beenpointed out in the literature [5, 10, 13, 14, 20], basedon the so-called spin-1 AKLT model [21, 22]. Thesestates are quasi-deterministic resources, in the sense thatmeasurement-based computations using these states canbe made arbitrarily likely to succeed, either by reduc-tion of the resource state to a deterministic resource orby a repeat-until-success strategy with each elementarygate requiring a random number of measurements. InSecs. IV A and IV B, we describe resource states calledN − U − N states that are based on cluster states trans-formed by N-type operators; these states share the prop-erties of quasi-determinism and non-vanishing long-rangecorrelations.
B. Strategy II: Guaranteed Pauli Byproduct
1. Derivation of B-Type Operators
Another possible strategy is to attempt to ensure thatthe byproduct operator is guaranteed to be Pauli-X,whether or not the teleported linear transformation isunitary. The advantage of this approach is that X hasnice commutation properties through rotation operatorsabout the z -axis, whether they be by real or complex an-gles, leading to the hope that the randomness inherentin the measurement process can be easily compensated.When projectively measuring in the orthonormal basis (cid:8) | m (cid:105) , | m ⊥ (cid:105) (cid:9) , the two possible operations that can occurare M = H (cid:20) (cid:104) m | S (1) | + (cid:105)√ (cid:104) m | S (1) |−(cid:105)√ (cid:21) ; (42)M ⊥ = H (cid:20) (cid:104) m ⊥ | S (1) | + (cid:105)√ (cid:104) m ⊥ | S (1) |−(cid:105)√ (cid:21) (43)= XH (cid:20) (cid:104) m ⊥ | S (1) |−(cid:105)√ (cid:104) m ⊥ | S (1) | + (cid:105)√ (cid:21) . (44)Since I and Z are linearly independent, it follows thatfor the byproduct to be guaranteed to be proportional toPauli-X, one must have (cid:104) m | S | + (cid:105) = c (cid:104) m ⊥ | S |−(cid:105) ; (45) (cid:104) m | S |−(cid:105) = c (cid:104) m ⊥ | S | + (cid:105) , (46)or equivalently, (cid:104) m | S | (cid:105) = c (cid:104) m ⊥ | S | (cid:105) ; (47) (cid:104) m | S | (cid:105) = − c (cid:104) m ⊥ | S | (cid:105) , (48)for some non-zero constant c ∈ C . Suppose S = U D Vwhere U is an arbitrary single-qubit unitary, D is de-fined in Eq. (31), and V = R z [ β ] R x [ γ ] R z [ δ ]. Furthersuppose that | m (cid:105) = U U (cid:48) | (cid:105) and | m ⊥ (cid:105) = U U (cid:48) | (cid:105) , with U (cid:48) = R z [ β (cid:48) ] R x [ γ (cid:48) ] R z [ δ (cid:48) ]. The reason for the appear-ance of U in the definitions of | m (cid:105) and | m ⊥ (cid:105) is to com-pensate for the appearance of U in the singular valuedecomposition of S . The only effect of the R z [ δ (cid:48) ] opera-tion is to multiply the teleported gate by a global phase,so we can choose δ (cid:48) = 0 in what follows without loss ofgenerality (it remains a free parameter for the appliedunitary U (cid:48) ). Having done so, Eqs. (47-48) can be rewrit-ten as (cid:104) | Q | (cid:105) = c (cid:104) | Q | (cid:105) ; (49) (cid:104) | Q | (cid:105) = − c (cid:104) | Q | (cid:105) , (50)where we have defined Q := ( U (cid:48) ) † DV (51)= R x [ − γ (cid:48) ] R z [ − β (cid:48) ] D R z [ β ] R x [ γ ] R z [ δ ]= R x [ − γ (cid:48) ] R z [ β − β (cid:48) ] D R x [ γ ] R z [ δ ]:= R x [ − γ (cid:48) ] R z [ b ] D R x [ γ ] R z [ δ ] ; (52) in the last line above we have defined b := β − β (cid:48) . In theexpression above, γ (cid:48) and b are free parameters, while D , γ and δ are determined by the SLOCC operator S .Return now to the constraints, Eqs, (49-50). Denoting Q ij := (cid:104) i | Q | j (cid:105) , one finds that Q /Q = − Q /Q .Note that neither Q nor Q can be zero; if either werezero, then the constraints would force Q and therefore S to be singular, which by assumption is not the case. Thisin turn means thatDet( Q ) = 2 Q Q . (53)From the definition of Q , Eq. (52),Det( Q ) = sin 2 θ ;2 Q Q = sin γ (cid:48) sin γ (cos γ − i cos 2 θ sin b )+ sin 2 θ (1 + cos γ cos γ (cid:48) ) . (54)Substituting the above expressions into Eq. (53) andequating real and imaginary parts gives us the two con-ditions sin γ (cid:48) sin γ sin b cos 2 θ = 0; (55)cos γ (sin γ (cid:48) sin γ + cos γ (cid:48) sin 2 θ ) = 0 . (56)Recall that since S is invertible, we cannot have sin 2 θ =0 and since S is non-unitary, we cannot have cos 2 θ = 0.The only ways to satisfy Eq. (55) are if sin γ (cid:48) sin γ = 0 orsin b = 0. In the first case, Eq. (56) immediately impliesthat cos γ (cid:48) cos γ = 0, leaving b as a free parameter forour measurement basis. In the second case, γ (cid:48) is fixed interms of θ and γ , leaving no freedom in the measurementbasis we are using. Furthermore, if we choose sin γ (cid:48) = 0,i.e. γ (cid:48) ∈ { , π } , then the measurement basis we are usingis restricted to being the computational basis acted on by U (completely specified by S ); again, no freedom. There-fore, the only solutions available to us that leave freedomin the measurement basis, and thus the teleported gate,are γ ∈ { , π } and γ (cid:48) ∈ (cid:8) π , π (cid:9) . Note that S † S = (cid:20) γ cos 2 θ − ie iδ sin γ cos 2 θie − iδ sin γ cos 2 θ − cos γ cos 2 θ (cid:21) . (57)Thus, demanding that the SLOCC operators allow for aguaranteed Pauli by-product, assuming the SLOCC op-erator is not unitary and thus cos 2 θ (cid:54) = 0, is equivalentto demanding that S † S be diagonal. Geometrically, thismeans that S must preserve the overlap of the computa-tional basis states (the transformed computational basisis still orthogonal). We will refer to this kind of basis-preserving operator as a B-type operator. Definition III.3. A GL (2 , C ) operator S satisfying (cid:104) | S † S | (cid:105) = (cid:104) | S † S | (cid:105) = 0 is called a B -type operator. A B-type operator can therefore be writtenB = (cid:40) U D R z [ β ]R z [ δ ] , γ = 0U D R z [ β ]XR z [ δ ] , γ = π, ignoring overall phases. The two possibilities above canbe simplified and collapsed into one. First, note thatR z [ β ] can be commuted past D and absorbed into U.Next, note that X D X is itself a diagonal matrix thatresults from swapping the diagonal entries of D . Thismeans that the case where γ = π can be written insteadas U (cid:48) D (cid:48) R z [ δ ], where U (cid:48) = UR z [ β ] X and D (cid:48) = X D X. Ofcourse, R z [ δ ] can also be absorbed into U (cid:48) ; thus, a simpleand completely general expression for a B-type operatoris B = U D. (58)The diagonal matrix D in the singular value decomposi-tion can be expressed as D ∝ diag (cos( θ ) , sin( θ )) = (cid:112) sin( θ ) cos( θ )R z [ i ln cot( θ )] , so that the B-type operator becomesB ∝ UR z [ i ln cot( θ )] . (59)Because the unitary U can be absorbed directly into themeasurement basis, one can interpret B-type operatorsas z -rotations by an imaginary angle, the value of whichis related to the ratio of the singular values.When the local operator is B-type, the single-parameter family of measurement bases satisfies γ (cid:48) ∈ (cid:8) π , π (cid:9) , and β (cid:48) ∈ [0 , π ) is a free parameter. Whenthis family of bases is used, the byproduct operator as-sociated with measurement outcome 1 is always Z (up toa global phase). The teleported linear transformation isno longer unitary, however; it takes the form of a rota-tion about the z -axis of the Bloch sphere by a complexangle, followed by a Hadamard operation. The real partof the angle is completely specified by the choice of mea-surement basis, via the free parameter β (cid:48) . The imaginarypart is purely a function of the ratio of the singular valuesof the local GL (2 , C ) operator. Denoting the measure-ment outcome corresponding to γ (cid:48) = π by m = 0 andthat for γ (cid:48) = π by m = 1, the teleported gate is given(up to a global phase) by M = X m HR z [ β (cid:48) + i ln cot θ ] . (60)
2. Properties of B-Transformed Cluster States
Interpreting the B-type operators as z -rotations byimaginary angles provides a simple insight into the natureof B-transformed cluster states. The R z operator com-mutes with all CZ gates, so one can push it all the waythrough to the | + (cid:105) states in the definition of the clusterstate, Eq. (4). Because R z ( ξ ) | + (cid:105) is an arbitrary single-qubit state, B-transformed cluster states are equivalentto applying CZ gates between qubits in arbitrary states(not including computational basis states, which wouldrequire singular B operators). One might assume that B-transformed cluster statesare equivalent to weighted cluster states [28–32], but thisis not in fact the case. Weighted graph states are definedas (cid:81) (cid:104) i,j (cid:105) CP( ϕ ) i,j | + (cid:105) ⊗ n , where the controlled-phase en-tangling gate is CP( ϕ ) = diag (cid:0) , , , e i ϕ (cid:1) ; the cluster-state edge weights are then given by w ij = ϕ ij . Con-sider the simplest counter-example of a three-qubit lin-ear cluster state with the central qubit transformed by aB-type operator B = D R z [ γ ] with D = diag(cos θ, sin θ ).The eigenvalues of the local reduced density matrices areall (cid:8) (1 ± cos 2 θ ) (cid:9) . On the other hand, for a three-qubit 1D weighted graph state with edge weights ϕ and ϕ , the eigenvalues of the reduced density ma-trix are (cid:0) ± cos ϕ (cid:1) , (cid:0) ± cos ϕ cos ϕ (cid:1) , and (cid:0) ± cos ϕ (cid:1) for qubits 1 through 3, respectively. Ifthe weighted graph and the B-transformed cluster areLU-equivalent, there must be some choice of ϕ and ϕ such that the spectra of the reduced density ma-trices are the same in both cases. For qubits 1 and3 this implies φ = φ = 4 θ . For qubit 2 one ob-tains (cid:0) ± cos 4 θ (cid:1) . This matches the correspond-ing spectrum for the B-transformed cluster only when θ = ± π , ϕ = ± π , in which case both states are LU-equivalent to a perfect cluster.Cluster states locally transformed by B-type opera-tors also exhibit different properties from perfect clus-ter states. As with N-type operators, B-type operatorschange the local reduced density matrices of individualqubits within the state, as described in the followinglemma. Lemma III.4.
Let | Cl n (cid:105) V be an n -qubit cluster stateon the set of qubits V , with some subset Q ⊆ V actedupon by B -type operators. In particular, suppose that foreach qubit i ∈ Q , the B -type operator acting is given by B ( i ) = D ( i ) with D ( i ) = √ (cid:0) cos θ ( i ) , sin θ ( i ) (cid:1) . Then,the local reduced density matrix for any qubit k ∈ V isgiven by ρ k = cos θ ( k ) | (cid:105)(cid:104) | + sin θ ( k ) | (cid:105)(cid:104) | +
12 sin 2 θ ( k ) (cid:89) j ∈N ( k ) cos 2 θ ( j ) | (cid:105)(cid:104) | + h . c . , where θ ( k ) := π if k / ∈ Q . The lemma is easily proved by taking advantage of theexpression (38) for the Schmidt decomposition of a clus-ter state with one subsystem being qubit k alone, andthen calculating ρ k directly. The calculation is done byexpressing the cluster state as the action of controlled-Zgates acting on the product state | + (cid:105) ⊗ n , and then usingthe fact that the D ( i ) and controlled-Z gates are mutu-ally commuting. A consequence of this lemma is thatthe reduced density matrix of a given qubit is maximallymixed if and only if the qubit itself and at least one ofits neighbors are untouched by B-type operators. In gen-eral, qubits within B-transformed cluster states are notmaximally entangled with the rest of the state; in fact,0they can be arbitrarily weakly entangled (with respect tothe von Neumann entanglement entropy).Consider now a ring of an even number of qubits, withidentical B-type operators specified by θ (2 k ) = θ act-ing on the qubits with even labels. The significance ofsuch a state will become clear in Example IV D. Two-point correlation functions can be calculated exactly forthis state. The result is that the nearest-neighbor cor-relation functions C k, k ± (Z , X) = cos 2 θ and the next-nearest-neighbor correlation functions C k, k ± (Z , Z) = C k − , k +1 (X , X) = cos (2 θ ) are the only ones that arenon-zero, while all the other two-point Pauli correlationfunctions are identically zero. Again, this differs fromthe perfect cluster state, where all correlation functionsare zero.Exact calculations on chains of up to 7 qubitswith identical (but arbitrary) B-type operators B j = D R z (cid:2) γ (2 j ) (cid:3) acting on even qubits 2 j reveal that thenon-zero Schmidt coefficients corresponding to any bi-partition of the chain into two contiguous halves are { cos θ, sin θ } . The von Neumann entropy of entangle-ment is equal to the Shannon entropy of this list, andis generally less than one ebit. The MPS representationbears out this observation. In the canonical form the sitematrices for the boundary qubits are A [1] [0] = 1 √ θ (2) (cid:104) | + sin θ (2) (cid:104) | ); (61) A [1] [1] = 1 √ θ (2) (cid:104) | − sin θ (2) (cid:104) | ); (62) A [ n ] [0] = | + (cid:105) ; A [ n ] [1] = −|−(cid:105) , (63)those for the bulk even sites are A [2 j ] [0] = (cid:20) e − iγ (2 j ) (cid:21) ; (64) A [2 j ] [1] = (cid:20) e iγ (2 j ) (cid:21) , (65)and those for the bulk odd sites are A [2 j − [0] = (cid:20) cos θ (2 j ) sin θ (2 j ) cos θ (2 j ) sin θ (2 j ) (cid:21) ; (66) A [2 j − [1] = (cid:20) − cos θ (2 j ) sin θ (2 j ) cos θ (2 j ) − sin θ (2 j ) (cid:21) . (67)It’s very easy to verify that the channels induced by thematrices on the odd sites are not unital in the sense givenin Ref. [18], so a B − U − B chain is not a quantum wire.Such a 1D state would appear not to be capable of reli-ably processing a single qubit. This is true, but a simplemodification of the geometry from one to two dimensionsyields a usable resource for random length computation.This will be elaborated upon in Example IV D.
IV. RANDOM LENGTH COMPUTATION
Neither Strategy I nor Strategy II discussed in the pre-vious section directly offers a way to perform determin- N (1) U (2) N (3) U (4) N (5) FIG. 2: N − U − N state, a one-dimensional structure thatcan be used for deterministic random-length single-qubit ro-tations. istic single-qubit rotations. For Strategy I, it is unclearhow to compensate for a byproduct operator R x [ η ] where η (cid:54) = π , as such a byproduct operator does not possess con-venient commutation properties with the H and R z op-erations. Similarly, for Strategy II, it is unclear whethersome number of non-unitary teleported gates can be com-bined to form a desired unitary.Another perspective on the strategies is that a sin-gle measurement with outcome 1 teleports the gateHR z [ ξ + (cid:15) ], where ξ ∈ C is some angle associated withthe always-on operation HR z [ ξ ] (in the terminology ofRef. [18]) and (cid:15) ∈ C is a possibly complex error associ-ated with the byproduct. To correct this error in princi-ple requires two additional measurement steps. The firstmeasurement step should teleport the gate HR z [0] ≡ H,which would cancel the previously applied Hadamardgate; a possible X byproduct operator might result de-pending on the measurement outcome. On the sec-ond measurement step one would attempt to teleportHR z [ − (cid:15) ] or HR z [ (cid:15) ] depending on the previous measure-ment outcome, thus cancelling the original error (cid:15) .This procedure is only possible if the measurement im-mediately after first incurring an error cannot itself gen-erate any further error (cid:15) (cid:48) . One way to guarantee sucha circumstance is to impose that every alternate S i op-erator is in fact unitary. Thus there must exist a classof states that are a strict subset of SLOCC-transformedcluster states, which constitute resources for random-length universal gate teleportation. Likewise, a subsetof SLOCC-transformed cluster states in two dimensionsmust be universal resources for MBQC. The remainderof this section is devoted to various explicit examples. A. Deterministic single-qubit rotations: N − U − N state Consider a one-dimensional state of the form | R (cid:105) = N (1)1 ⊗ U (2)2 ⊗ N (3)3 ⊗ U (4)4 · · · ⊗ N ( n ) n | Cl n (cid:105) , (68)where the (cid:8) N ( i ) (cid:9) are N-type operators, and the (cid:8) U ( j ) (cid:9) are local unitaries (c.f. Fig. 2). The goal is to teleportthe single-qubit unitaryU ( ζ, η, ξ ) = R x [ ζ ] R z [ η ] R x [ ξ ] . (69)The first step is to use Strategy I to attempt a telepor-tation of HR z [0], analogously to the scheme with theperfect 1D cluster state. For the correct choice of basisthe measurement outcome m = 0 corresponds to suc-cess. One can then immediately measure qubit 2 in a1basis that teleports X m HR z [ ξ ], and then use StrategyI to attempt the teleportation of HR z [( − m η ] startingon qubit 3.If the first measurement outcome is instead m = 1then one instead teleports HR z [ (cid:15) ] with (cid:15) ∈ R . Thiserror must be immediately corrected, because the nextdesired rotation is around an orthogonal axis. Happily,there is a local unitary U (2) acting on the next qubitin the chain. The Hadamard operator that effects thenow-undesired transformation of the rotation axes canbe eliminated by teleporting another one (H = I). Thisis accomplished by measuring the next qubit in the basis (cid:8) U (2) | + (cid:105) , U (2) |−(cid:105) (cid:9) . Labelling the measurement outcome m , the teleported gates areX m HHR z [ (cid:15) ] ≡ X m R z [ (cid:15) ] . (70)The measurement basis for qubit 3 is then chosen suchthat measurement outcome m = 0 results in the gateHR z (cid:104) ( − m +1 (cid:15) (cid:105) being teleported. In this case, theoverall unitary becomesHR z (cid:104) ( − m +1 (cid:15) (cid:105) X m R z [ (cid:15) ] = Z m HR z [ − (cid:15) ] R z [ (cid:15) ]= Z m H . At this point one has successfully teleported a Hadamardgate and an unimportant Pauli byproduct. The nextmeasurement on a qubit with an even label can teleportthe desired HR z [ ξ ] gate without error. One then at-tempts to teleport HR z [ η ] by measuring qubit 5, usingStrategy I, etc.The procedure corresponds to the following steps:1. Measure qubit 1 with outcome m in the basis (cid:110) (N (1) † ) − R z [ − ξ ] H | (cid:105) , N (1) R z [ − ξ ] H | (cid:105) (cid:111) ; (71)2. If m = 0, then success;3. If m = 1 then one has effectively teleported thegate R x (cid:2) (cid:15) (1) (cid:3) HR z [ ξ ], where (cid:15) (1) = ± θ (1) cos ξ ± cos 2 θ (1) sin ξ + π. (72)Note that (cid:15) (1) = 0 when N (1) = U (1) ( θ (1) = π/ m inthe basis (cid:8) U (2) X | (cid:105) , U (2) X | (cid:105) (cid:9) ;4. Measure qubit 3 with outcome m in the basis (cid:8) (N (3) † ) − R z [ χ ] H | (cid:105) , N (3) R z [ χ ] H | (cid:105) (cid:9) , where χ =( − m (cid:15) (1) ;5. Repeat steps 3 and 4 on successive qubits 2 k and2 k + 1 until outcome 1 is achieved on an odd qubit,using U (2) → U (2 k ) , N (3) → N (2 k +1) , m → m k , (cid:15) (1) → (cid:15) (2 k − . The key point of this example is that as for any measure-ment on an odd-numbered qubit that yields the ‘correct’outcome m i = 0, one will have succeeded in implement-ing part of the desired single-qubit rotation. Further-more, any errors resulting from outcomes m i = 1 arecorrectible by making further measurements. This thusconstitutes a repeat-until-success strategy, and gives riseto a quasi-deterministic random-length single-qubit rota-tion. The likely reason for this one-dimensional state tobe capable of processing a logical qubit is that the leftand right parts of the state share an ebit of entanglementwith respect to any cut, as mentioned in Sec. III A 2. B. Deterministic Universal MBQC: 2D N − U − N State
For universal MBQC, a two-dimensional resource stateis required. The precise geometry of the two-dimensionalstate on which MBQC occurs is determined by the spe-cific circuit to be implemented. Ideally one would startwith a state defined on a convenient and simple geometry,and then ‘carve’ the desired shape out by deleting certainqubits. For cluster-state MBQC, for example, one carvesthe required state out of a rectangular lattice by projec-tively measuring the unwanted qubits in the computa-tional basis. The goal is to yield isolated one-dimensionalwires, each of which represents a logical qubit, with linksonly existing between wires in places where an entanglinggate between logical qubits is needed.Consider now a regular two-dimensional lattice com-posed of N − U − N states, as depicted in Fig. 3. As inthe usual cluster state, logical qubits are processed byalternating horizontal wires composed of physical qubits,and entangling gates by vertical chains connecting them.Unlike the cluster case, however, the procedure for im-plementing single-qubit rotations with N − U − N statesis of random length, so it is impossible to decide in ad-vance where the desired links between wires will occur.The computational cluster state then must be carved ‘onthe fly.’Suppose that the quantum information encoding twological qubits resides on (yet unmeasured) N-transformedphysical qubits on two different wires. If an entanglinggate between logical qubits is not desired at the next step,then the link between the wires can be first severed bymeasuring the intervening U-transformed chain qubit inthe computational basis. An example of the method todecouple qubits 1 and 3 is shown in Fig. 3, where qubit2 is measured in the computational basis. This has theeffect of teleporting a Z gate to each of the logical qubits ifthe measurement outcome is m = 1. Other than takinginto account the possible existence of these byproductoperators, the computation subsequently proceeds as inthe one-dimensional case discussed above.The desired entangling gate is implemented as follows.At the time that an entangling gate is needed, the localpart of the resource state looks like two 1D N − U − NN NN NN NNUU UU UUUU UU UU UU123FIG. 3: 2D N − U − N state, universal for quasi-deterministicMBQC. The qubits labeled 1, 2 and 3 can be used to imple-ment an entangling gate, which involves measuring qubit 2 inthe Y basis. Alternatively, if an entangling gate is not desiredhere, qubits 1 and 3 can be decoupled by measuring qubit 2in the Z basis.
N states, each coupled via CZ operations to an ancillainitially in the state | + (cid:105) and subsequently acted on by anarbitrary U. In Fig. 3, the entangling link is representedby the vertical N − U − N chain labeled by qubits 1, 2, and3. The local part of the state is mathematically describedas | R (cid:105) = N (1)1 U (2)2 N (3)3 CZ , CZ , | c + t (cid:105) , (73)where the states | c (cid:105) and | t (cid:105) could be thought of as controland target states respectively for some entangling gate.Now, qubits 1 and 3 are measured in the usual Strategy Ibasis (71) with ξ ( i ) = 0, while qubit 2 is measured in theeigenbasis of the Pauli operator Y, suitably rotated byU (2) . This procedure teleports the state initially situatedon qubits 1 and 3 through an entangling gate G , = R x (cid:104) µ (1) (cid:105) m X m + m H × R x (cid:104) µ (3) (cid:105) m X m + m H M , (74)to qubits 4 and 5, with M , = | (cid:105)(cid:104) | , + i | (cid:105)(cid:104) | , + i | (cid:105)(cid:104) | , + | (cid:105)(cid:104) | , . (75)Here, the (cid:8) µ ( i ) (cid:9) are the standard Strategy I byproductangles, Eq. (34) or (72). This entangling operation isrelated to CZ viaCZ , ≡ X X R z [ π/ R z [ π/ M , X X , (76)and so G i,j together with single-qubit operators forms auniversal set of gates. C. Probabilistic single-qubit rotations: B − U − B state Next consider a one-dimensional state of the form | R (cid:105) = B (1)1 ⊗ U (2)2 ⊗ B (3)3 ⊗ U (4)4 · · ·⊗ B (2 n +1)2 n +1 | Cl n +1 (cid:105) , (77) B (1) U (2) B (3) U (4) B (5) FIG. 4: B − U − B state, a one-dimensional structure that canbe used for probabilistic random-length single-qubit rotations. where the (cid:8) B (2 i +1) (cid:9) are B-type operators, and the (cid:8) U (2 i ) (cid:9) are once again local unitaries (c.f. Fig. 4). Thisstructure ensures that none of the bonds present in thestructure is perfect; no particle has maximal entropy ofentanglement with the rest of the state. This fact is aconsequence of Lemma III.4; there is no qubit unaffectedby B-type operators whose neighborhood contains anyunaffected qubits.All single-qubit measurements for the odd-numberedqubits now correspond to Strategy II, in whichthe byproduct operator is always X if it oc-curs. All even-numbered qubits are measured in the (cid:8) U (2 i ) | + (cid:105) , U (2 i ) |−(cid:105) (cid:9) basis; as in the previous example,the only purpose of these measurements is to enable theremoval of undesired contributions to the rotation an-gles. The main difference from the previous example isthat a non-unitary gate of the form HR z [ ξ i +1 ] is tele-ported, where ξ i +1 ∈ C . The present goal is therefore tocompensate for the imaginary part of the rotation angle.As discussed in the previous section and Eq. (60), theimaginary part of ξ i +1 is entirely determined by the ra-tio of the singular values of B (2 i +1) , and can be definedas (cid:15) = i ln (cid:0) cot θ (2 i +1) (cid:1) . Consider momentarily the spe-cial case where the (cid:8) B (2 i +1) (cid:9) all have the same singu-lar values. The gate teleported by a measurement ofthe B-transformed qubit 2 i + 1 will then be proportionalto R z [ (cid:15) ( − m i + m i − + ... ], ignoring all rotations aboutreal angles which are entirely determined by the choiceof measurement basis. In short, the sign of the imaginaryangle depends on the outcomes of the previous measure-ments on even-numbered U-transformed qubits.The imaginary component therefore undergoes a ran-dom walk of step-length | (cid:15) | . In particular, the walkertakes its first step to the right when the first measure-ment outcome of an even-numbered qubit is 0, and to theleft if it is 1. Subsequently, a measurement outcome of0 on an even-numbered qubit causes the walker to takeanother step in the same direction as the previous step,while outcome 1 makes the walker take a step in the op-posite direction.The two possible measurement outcomes with oddqubits are always equally likely, but the probabilities witheven qubits depend on the singular values of the B-typeoperators from the (odd) neighboring qubits, and gener-ally speaking the walker is more likely to stray furtherfrom the origin than to step back towards it. For ex-ample, consider measuring the first two qubits of | R (cid:105) inEq. (77) in the {| + (cid:105) , |−(cid:105)} basis. It can easily be shown,using the Schmidt decomposition (38) and the expres-sion (58), that the probabilities of the outcome |±(cid:105) onqubit 1 are equal, and that those on qubit 2 are3 p ± , = 12 (cid:16) ± cos 2 θ (1) cos 2 θ (3) (cid:17) . (78)Here, the random walk effectively begins at positionln (cid:0) tan θ (1) (cid:1) and moves to ln (cid:0) tan θ (1) (cid:1) ± ln (cid:0) tan θ (3) (cid:1) . Fora situation to arise where the walker moves closer to theorigin with probability greater than 1 /
2, one of the twopairs of conditions (cid:12)(cid:12)(cid:12) ln (cid:16) tan θ (1) (cid:17) ± ln (cid:16) tan θ (3) (cid:17)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12) ln (cid:16) tan θ (1) (cid:17)(cid:12)(cid:12)(cid:12) ;(79) ± cos 2 θ (1) cos 2 θ (3) > (cid:8) θ ( i ) (cid:9) are chosen uniformly at random, then the proba-bility of this happening is only about 0 . θ (1) tends to drift awayfrom π/ π/
2, and the range of val-ues of θ (3) for which the walker is likely to turn aroundand walk towards the origin progressively shrinks. Fur-thermore, if (cid:8) θ (1) (cid:9) and (cid:8) θ (3) (cid:9) are equal at any time, thewalker is guaranteed to be more likely to continue in onedirection than to turn around.This procedure constitutes a probabilistic method forimplementing a single-qubit rotation. Unfortunately ifthe walker strays too far from the origin, it becomes ef-fectively impossible to recover and the attempted gateteleportation fails. The entire computation must then berepeated. If the singular values of the B i +1 are chosensuch that the imaginary components of the teleportedangles are all integer multiples of each other, then thebehavior of the random walk is even more deleterious.A judicious two-dimensional arrangement of B − U − Bchains avoids this catastrophe, as discussed in the nextexample.
D. Universal MBQC: Percolated 2D Cluster Statefrom 3D B − U − B state In the previous example using Strategy II, a possiblyinfinite number of steps may be required to teleport anarbitrary single-qubit unitary. But quitting the proto-col results in catastrophic failure: because the computa-tional wire is effectively broken, the entire gate teleporta-tion must be attempted from the beginning. A solutionto these problems is to employ the 3D extension to theprevious resource, corresponding to a cluster state trans-formed by alternating B-type operators and unitaries.This corresponds to a lattice with two interpenetratingcubic sublattices, a B-lattice and a U-lattice.An example of this 3D resource, a cube with side length3, is depicted in Fig. 5. Initially, Z-basis measurementsin the z -direction (as labeled in Fig. 5) are used to carveout a structure in which each B-transformed qubit in the x − y plane, shaded grey, is attached to a long vertical xyz (a) (b)FIG. 5: 3D cluster state with B-type operators and unitariesacting on alternate qubits (a) before carving and (b) aftercarving. Grey qubits are acted upon by B-type operators andwhite qubits by local unitaries. Z-basis measurements aremade in the z direction in (a) to disentangle those verticalchains originating from a white qubit in the x − y plane. Ameasurement protocol along the remaining vertical chains in(b) produces a percolated 2D cluster in the x − y plane. B − U − B chain. Measurements are made on the chainqubits, starting at qubit above the B-transformed qubiton the computational wire and continuing in the verticaldirection until success (defined below) is achieved. Thegoal is to probabilistically produce perfect entanglementin the x − y plane, thereby effectively eliminating theB operators in the horizontal direction. The result is a2D cluster state in this plane with missing entanglementbonds in random locations. As long as the mean densityof broken links exceeds the percolation threshold for atwo-dimensional square lattice, the resource is universalfor MBQC [23].Consider the first vertical chain from the left in Fig. 5.Recall that one can interpret B as a z -rotation by animaginary angle ± i ln λ , as shown in Eq. (59). For sim-plicity, we assume the unitary operators acting on even-numbered qubits are all equal to the identity; were theynot, they could be compensated by a suitable rotation ofthe measurement basis for. The portion of the state cor-responding to the first four qubits of the vertical chain(with the qubit that intersects the horizontal chain la-beled 1), is then (ignoring normalization) | T (cid:105) = B B CZ , CZ , CZ , | + + + + (cid:105) = R z [ i ln λ ] R z [ i ln λ ] | Cl (cid:105) , ignoring normalization factors as usual. First, qubits 2and 3 are measured in the {| + (cid:105) , |−(cid:105)} basis (of course, themeasurement basis for qubit 2 would need to be rotatedif a local unitary U (2) were acting). These are commutingmeasurements, since they are on different qubits and notadaptive. The effect is to teleport the state R z [ i ln λ ] | + (cid:105) from qubit 3 through X m HX m H ≡ X m Z m to qubit 1,yielding the new state | T (cid:48) (cid:105) = R z [ i ln λ ] CZ , X m Z m R z [ i ln λ ] | + + (cid:105) = CZ , R z [ i (( − m + 1)ln λ )] | + + (cid:105) up to overall local unitaries on the final state.4If m = 1, then the imaginary part of the rotationangle is completely canceled. Qubit 4 can then be mea-sured in the computational basis (again, suitably rotatedif necessary) to disentangle the rest of the vertical chainfrom the horizontal chain. The result is a perfect clusteralong the first three qubits in the horizontal direction,and the B-type operator is effectively deleted.If m = 0, then the situation is similar to the original.There is still a B-type operator present in the horizontaldirection, now corresponding to a z-rotation about anangle with imaginary part 2ln λ = ln λ . In other words,the new effective B-type operator in the horizontal chainhas a ratio of singular values that is the square of theoriginal one. In order to remove the effect of the B-typeoperator, the chain qubits must be measured sequentiallyuntil the total number of steps towards the origin exceedsby 1 the total number of steps away.The probabilities p ( k )0 and p ( k )1 of the outcomes 0 and 1on an even qubit in the vertical chain,where k > p ( k )0 = 1 + λ k +2 λ + λ k + λ k +2 ∼ O (1); (81) p ( k )1 = λ + λ k λ + λ k + λ k +2 ∼ O ( λ ) . (82)The total probability p n that the effect of the B will beundone within 2 n measurements is the sum of the prob-abilities of all of the possible trajectories of the walkeron n or fewer steps with initial position 1, final position0 and all intermediate positions strictly positive.The probability p of undoing the B operator after10 attempts (20 measurements) is shown in Fig. 6 as afunction of the ratio of singular values λ . Calculation ofthe exact probability p ∞ is computationally intractable,for two reasons. First, the number of valid trajectoriesfor the walker grows exponentially in the number of stepsallowed. Second, the probability of any particular trajec-tory depends on the full history of the walker, not justthe number of steps. Of course, p ∞ must approach 1 as λ approaches unity (the limit that B becomes a unitarymatrix). In this case, the walk reduces to the simple1D random walk, which is known to sample the originfrequently.If after some predetermined number of measurementsalong a vertical chain one has not yet succeeded in un-doing B, the qubit at the root of the chain (i.e. in thecomputational wire) can be measured in the computa-tional basis and thereby deleted. The result is a brokenlink in the 2D cluster state. The important result shownin Fig. 6 is that there is a critical value of λ , called λ c ,above which the probability of successfully undoing theB rises above the (bond) percolation threshold for a 2Dsquare lattice (approximately 0.593). For this walk, thecritical value obeys λ c (cid:46) . λ c is read off the thick blue curve from Fig. 6. Thus, thisprocedure probabilistically yields a universal resource for Λ FIG. 6: (Color online) Probability of success of the procedurefor deleting a B-type operator in the plane via a random walkin the third dimension, as a function of the ratio λ of thesingular values (thick blue, color online). The probabilities p k of deleting B with exactly k even-qubit measurements arealso shown for k from 1 to 10 (thin, decreasing with increasing k ), and the thick blue line is the sum of these. The red dashedline is the percolation threshold. MBQC provided that λ is sufficiently large. We notethat a similar example was considered in [15], where theresource was a 2D cluster state with identical B-type op-erators acting everywhere and the percolation proceededvia two-element POVMs that either removed the B ordeleted the qubit. There, the critical value of λ was foundto be 0 . V. DISCUSSION AND CONCLUSIONS
Motivated by a desire to identify new resource statesfor measurement-based quantum computing, we haveperformed a (non-exhaustive) search of the equivalenceclass of n -qubit cluster states on a rectangular lattice, un-der the action of GL (2 , C ) ⊗ n . In particular, our aim wasto identify which states within this class could be used asresources for MBQC, by designing explicit protocols forteleporting single-qubit gates and two-qubit entanglinggates, driven by adaptive local projective measurements.We identified a class of one-dimensional states, the so-called N − U − N states, that are deterministically uni-versal for single-qubit rotations, although with a randomnumber of measurements needed to teleport the desiredrotation. We also identified a probabilistically universalresource for single-qubit rotations: the so-called B − U − Bstates. We then described a three-dimensional extensionof B − U − B states that can yield a universal resourcefor deterministic MBQC beyond a percolation threshold,and a 2D N − U − N state that is also universal for de-terministic but random-length MBQC.Several interesting open issues arise as the result of thiswork. First, it is not clear what (if any) relationship ex-ists between the states uncovered in this work and other5known resource states. For example, the probabilisticnature of the protocol with B − U − B states has fea-tures in common with that of other resources for MBQC,such as photonic cluster states prepared via probabilis-tic entangling gates or with unreliable sources [33–36].Likewise, the quasi-deterministic N − U − N states sharevarious characteristics with the AKLT-inspired resourcesof Refs. [5, 10, 15, 20], in particular the exponential spincorrelations and the repeat-until-success measurement-based strategies. One distinction is that only one alter-nating sublattice of the N − U − N states exhibits non-zero correlation functions, whereas in the AKLT chain,every qubit is correlated with every other. Presumably atrue identification of an AKLT-type resource with a N-transformed cluster state will require N-type operatorsto be present on every qubit, a case we have not handledhere.Also, it will be important to better understand therelationships of these states with the universal quantumwires of Ref. [18]. In that work, certain reasonable physi-cal assumptions were imposed on 1D wires at the outset,for example: the possibility of producing a wire via atranslationally invariant nearest-neighbour global entan-gling operation, the asymptotic sharing of an ebit of en-tanglement between the left and right halves of the chain,etc. In our work, it is not clear if a translationally invari-ant scheme exists for producing N − U − N or B − U − Bchains. Evidence from exact calculations on small chainsand the explicit description of the states within the MPSrepresentation reveals that although the left and righthalves of N − U − N chains share an ebit, the halves ofthe B − U − B chains do not. This seems reasonable, asthe N − U − N chain is quasideterministically universalfor single-qubit rotations, while the B − U − B chain isonly probabilistically so.Second, it is conceivable that all states that have beenhitherto identified as universal resources for MBQC arein fact SLOCC-equivalent to the family of cluster states.There is some evidence to support this conjecture. Forexample, the results of Ref. [19] show that many seem-ingly diverse resource states can be reduced to clusterstates via local strategies. Similarly, the proof of the uni-versality of the 2D AKLT state on a honeycomb latticeproceeds via local reduction to a random graph state,which can in turn be reduced to a percolated clusterstate [14]. This reduction is successful despite the fact that the initial resource is defined on qutrits rather thanqubits, and on a non-rectangular lattice. An even moreintriguing possibility (though we believe it to be unlikely)is that all possible states for universal MBQC fall withinthe SLOCC-equivalence class of the cluster states. Atthe very least, the relative size of this class decreases ex-ponentially with the total number of physical qubits, asexpected [16, 17].Third, while we have shown that a certain sub-set of the orbit of the cluster states under SLOCCare useful resources, either probabilistically or quasi-deterministically, it is not clear if the remaining SLOCC-transformed cluster states are also universal resourcesfor MBQC. Generically, single-qubit measurements onthese states teleport gates with byproduct operators thatare rotations about the X axis by complex angles. Onepossibility is that there is a measurement protocol thatcan accommodate all possible byproducts that we sim-ply haven’t found. Perhaps there is another sense, be-sides quasi-deterministic or probabilistic, in which thesestates can be said to be useful for MBQC. Another possi-bility is that there is some map from the full orbit to theparticular subset of states considered in the work. Al-ternatively, the states in the full orbit may not be usefulresources, though we have not attempted to prove this.Finally, it would be useful if the resources presentedin this work could be realized as the ground states of aphysical Hamiltonian. A recent no-go theorem [37] showsthis cannot be the case for frustration-free Hamiltoni-ans on qubits. The question remains open for frustratedHamiltonians, for instance the AKLT Hamiltonian in theHaldane phase [20–22]. Another possibility is that theground states of physical Hamiltonians could be locallyreduced to the resources we have found. Further researchis needed to answer these and related questions.
VI. ACKNOWLEDGEMENTS
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