Symmetry protected topological order as a requirement for measurement-based quantum gate teleportation

All known resource states for measurement-based quantum teleportation in correlation space possess symmetry protected topological order, but is this a sufficient or even necessary condition? This work considers two families of one-dimensional qubit states to answer this question in the negative. The first is a family of matrix-product states with bond dimension two that includes the cluster state as a special case, protected by a global non-onsite symmetry, which is characterized by a finite correlation length and a degenerate entanglement spectrum in the thermodynamic limit but which is unable to deterministically teleport a universal set of single-qubit gates. The second are states with bond dimension four that are a resource for deterministic universal teleportation of finite single-qubit gates, but which possess no symmetry.


I. INTRODUCTION
The measurement-based model of quantum computation (MBQC) [1,2] is wholly equivalent to the quantum circuit model in its ability to effect arbitrary quantum gates [3], but is advantageous for practical implementations where the application of local entangling gates on demand is challenging.In MBQC, the entanglement is present at the outset, in the form of a specific resource state, and quantum gates are teleported by means of adaptive single-qubit measurements.A long-standing open problem has been to identify the essential characteristics required of a resource state, and accordingly much attention has been focused on one-dimensional systems which are able to perform measurement-based gate teleportation (MBQT) of arbitrary single-qubit gates.To date, all resource states for MBQT that have been identified possess symmetry protected topological (SPT) order [4][5][6][7][8], which passively protects the quantum information from certain kinds of errors [9]; these include the cluster states of the original one-way quantum computation model [3,10], and Haldane-phase states [11] such as the ground states of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state [12,13], its generalizations to two dimensions and higher spin [14][15][16][17][18], and a two-dimensional state with genuine SPT order [19].
For all resource states with SPT order, MBQT is performed in correlation space in the matrix product state (MPS) representation [20][21][22][23].The group cohomology [4] ensures that the teleported gate in correlation space can be expressed as a rotation operator in a tensor product with an unimportant 'junk' matrix [6,7].Unfortunately, the teleported gates throughout the universal SPT phase are not strictly in the protected 'wire basis,' which restricts the target teleported gates to infinitesimal rotations in correlation space [8,24] except for the cluster state itself.Another key signature of SPT is the degeneracy of the entanglement spectrum (ES) [7,25,26].
While SPT order has been a powerful approach to classifying resource states for MBQT, a plethora of key questions remain, even for the simplest case of one-dimensional qubits.Are resource states with SPT order required for MBQT?If not, what other kinds of resource states are possible?Can any resources other than the cluster state effect the teleportation of universal singlequbit gates based on finite, rather than infinitesimal, unitary rotations?What is the relationship between the ability of a state to be a resource for MBQT and the structure of the teleported gates?
This work partially addresses these questions by considering two specific examples.The first is a family of SPT states with bond dimension D = 2 that includes the cluster state as a special case, which is protected by a global Z 2 × Z 2 symmetry that is generally neither unitary nor onsite.States within this family have finite correlation length and exhibit a degenerate ES for arbitrary boundary conditions in the thermodynamic limit, but (except for the cluster state itself) are unable to deterministically teleport a universal set of protected singlequbit gates in correlation space.The second example is an extension of the cluster state to a family of non-SPT states with bond dimension D = 4, which are a resource for the deterministic teleportation of single-qubit gates, based on finite rotations.The results demonstrate that SPT order is neither sufficient nor necessary for a state to be an MBQT resource.

II. TECHNICAL BACKGROUND
A one-dimensional state for n qubits can be written in the MPS representation as The , where i n = {0, 1}, are rectangular matrices and their product (indexed by the strings i 1 • • • i n ) therefore constitute the amplitudes of |ψ in the computational basis.Given that A [n] [i n ] (A [1] [i 1 ]) is a row (column) vector, it is conventional to introduce matrices B [k] [i k ] and boundary vectors |L and |R such that A [1] [i 1 ] = B [1] [i 1 ]|L , and In this work, the MPS matrices B [k] [0] and B [k] [1] are arbitrary complex matrices with fixed 'bond' dimension D, and are normalized in (left) site canonical form, where I is the identity matrix.
A general single-qubit measurement can be effected by first performing a unitary gate Ũ on the qubit, and then measuring it in the computational basis by projecting the result onto |m m|, m = 0, 1.This is equivalent to applying the operator |m m| Ũ = |m φ m |, where |φ m = Ũ † |m constitute a basis for the unitary: Consider the action of this operator on the first qubit of |ψ in Eq. (2).Ignoring normalization, one obtains where the left boundary state in correlation space is transformed into |L ′ = B [1] [φ m ] |L by the operator B [1] [φ m ] = i1 φ m |i 1 B [1] [i 1 ]; in general, one obtains for measurements of 0 and 1, respectively.Successive measurements therefore apply a sequence of gates to |L .For MBQT, however, B [k] [φ m ] must correspond to a unitary operator for all m, a severe restriction on possible resource states which are defined by the matrices One-dimensional cluster states of qubits with open boundary conditions provide a convenient reference for the work presented here.It is straightforward to verify that these are (non-uniquely) described by an MPS representation with matrices for k = 1, . . ., n, where |± = (|0 ± |1 ) / √ 2, R| = 0|, and |L = √ 2|+ .Because the matrices are independent of site, the MPS representation is said to be translationally invariant, even though the state itself is defined with open boundary conditions.One obtains which is unitary if where c is a constant related to the (re)normalization of |ψ after the measurement.These two conditions require , and it is straightforward to verify that they together imply B [k] [φ m ] = X m HR Z (θ) and c = 1/2 ignoring overall phase factors, where and R Z (θ) = exp(iZθ).

III. SPT STATES UNABLE TO EFFECT MBQT
A. MPS matrices and the state The (unitary) teleported gate ( 5) is chosen to have the form yU , where U = e iφ1 cos θ e iφ2 sin θ e −iφ2 sin θ −e −iφ1 cos θ (11) with all parameters assumed to be real, and y is a proportionality factor to account for the renormalization of the state after measurement; this case corresponds to an MPS with bond dimension D = 2. Consider first the simplest case of a translationally invariant system.The derivation is straightforward but unwieldy, and is relegated to Appendix A. Choosing the MPS matrices to be in column form as in the cluster-state case, Eq. ( 7), and ensuring that they do not depend on the measurement angles, restricts both the measurement basis and the teleported gates; one choice corresponds to ϕ 2 = −ϕ 1 := −ϕ, ϑ = (2k + 1)π/4, k ∈ Z, and y = 1/ √ 2. These yield the measurement basis Ũ † = HR Z (ϕ), exactly as in clusterstate teleportation.One obtains the (non-unique) expressions for the MPS matrices, Eq. (A10): which in turn yield the measurement-dependent teleported unitary gates, Eq. (A11): (14) where Y = iXZ.
Assuming a translationally invariant MPS, consistently measuring |0 would yield successive rotations about Z and R Y (−θ)R Z (ϕ)R Y (θ), a measurementdependent rotation around Z conjugated by a fixed rotation around Y .These are non-parallel axes, which allows for the teleportation of any single-qubit unitary; for the cluster state, θ = θ c := (2k + 1)π/4, k ∈ Z, and the latter rotation axis is X.However, the byproduct operator when measuring |1 is not easily compensated for.Consider the teleported gates on two successive measurements: If 0 is a rotation around Z conjugated by a rotation around Y in the opposite direction than would be the case for m 1 = 0.One strategy would be to choose ϕ 2 = 0 on the next measurement, but if one instead obtains m 2 = 1 one is left with a second unwanted rotation R Y (−2θ) that would somehow need to be compensated for on the third measurement.The MBQT protocol would therefore become non-deterministic.Another strategy might be to restrict the ϕ i to infinitesimal angles, but in this case the error induced by the byproduct is Because the error accumulates, the teleported state would be indistinguishable from noise after several iterations.The inability of a state defined by the MPS (12) to effect deterministic MBQT of either finite or infinitesimal gates is in marked contrast from the 'oblivious wire' protocol that ensures that SPT states have uniform computational power to effect MBQT [24,27].In that case the gates are infinitesimally displaced from the symmetryprotected wire basis, in order to compensate for the fact that the junk matrices are generally measurementdependent.But this is not possible for the simple D = 2 case under consideration here.Over all MPS matrices (12), only the cluster state can effect deterministic teleportation.
The (unnormalized) state |ψ can be constructed either directly from Eq. ( 2) or using the machinery in Ref. [21].With left and right boundary states and site-dependent values of θ, the state takes an especially simple form after some straightforward algebra: acts on qubits j and j + 1.With θ j = π/4 for j > 1, so that , the state coincides with the one-dimensional cluster state with rotated left and right physical qubits.As C j,j+1 θj is only a unitary operator for θ j = θ c ∀j, Eq. ( 17) should be considered as an expression of the state rather than as a procedure for generating it.

B. SPT order
The real-space representation of the state, Eq. ( 17), allows for the explicit construction of the symmetry operators.As shown in Appendix B, the state possesses an exact Z 2 × Z 2 symmetry O(g 1 , g 2 )|ψ = |ψ , where O(g 1 , g 2 ) = X g1 odd X g2 even and g 1 , g 2 ∈ {0, 1}.The operators X odd and X even are the analogs of the X symmetry operators that act on odd-labeled and even-labeled sites of the cluster state, respectively, and for an even number of sites are given by Eq. (B25): where Alternatively, these can be written as X odd = S 1,2 j S 2j+1,2j+2 and X even = j S 2j,2j+1 S n−1,n , where S j,j+1 = P j−1,j θj P j,j+1 θj+1 Z j−1 X j Z j+1 are (nonlocal) stabilizer generators for the state, Eq. (17).While these symmetry operators square to the identity and commute with one another, as shown in Appendix B, they are neither unitary nor onsite.
Consider the left boundary qubit.The X and Z gates are transformed by the C θ operators into effective Pauli gates: Then one may determine the effective operators X ′ 1 and Z ′ 1 corresponding to X 1 and Z 1 conjugated by O(g 1 , g 2 ), respectively.Straightforward algebra presented in Appendix B reveals Z ′ 1 = (−1) g1 Z 1 and X ′ 1 = (−1) g2 X.The transformations on Z and X by the Z 2 × Z 2 operators are therefore equivalent to conjugation under an effective operator O eff (g 1 , g 2 ) = X g1 Z g2 , which is the same as for the regular cluster state.A similar result holds for the right boundary.Thus, the state belongs to the same maximally non-commutative phase as the cluster state [6,8].In the cluster-state limit θ j = θ c ∀j, the symmetry operators (20) reduce to X odd = j=1 X 2j−1 and X even = j=1 X 2j , as expected.The onsite symmetry even , which acts in parallel on adjacent two-site blocks so that U (g) ⊗n/2 |ψ = |ψ , is shared by the MPS matrices themselves via [21,34,35] where i and j are bitstrings of length 2, and 2 and φ g = 0.For any choices of θ j = θ c , however, O(g) = O(g 1 , g 2 ) is non-onsite, and there is no analog of Eq. ( 22) that can be expressed in block-injective form for any length smaller than n.Rather, where i, and j are now bitstrings of length n and g = (1010 for any V (g): the only non-zero term on the left is j = i ⊕ g, and |O(g) i,i⊕g | = 1.Thus, the symmetry of the real-space state is no longer shared by the (product of) MPS matrices for a global non-onsite symmetry.

C. Entanglement Spectrum
Consistent with the SPT order of the state defined either by the MPS matrices (7) or the state (17), the ES is asymptotically degenerate in the thermodynamic limit for all choices of boundary conditions.The ES corresponds to the eigenvalues of the reduced density matrix associated with a partition of the one-dimensional state with ℓ qubits on the left and n − ℓ qubits on the right.It can be obtained by diagonalizing the reduced density matrix, but more efficiently from the MPS matrices.Following Prosen [36], one may express the amplitudes of the state (2) as here, and |j are computational basis states.The elements of covariance matrices V L n and V R n are obtained via where the sum is over all internal indices.The ES coincides with the eigenvalues of V R ℓ V L ℓ .The calculations for the solution (12) with boundary conditions specified in Eq. ( 16) are given in Appendix C. For a bulk bipartition where 2 < ℓ < n − 1, one obtains Eq. (C5: where k=1 cos(2θ k ), with x 1 , x 2 defined below Eq.( 17).Note that the matrix elements depend explicitly on the boundary states.If ℓ and V L ℓ are proportional to the identity and the ES is degenerate.This condition is automatically satisfied for the cluster state, θ k = θ c , ∀k.If θ k = θ c , however, both matrices are strictly diagonal only if α and β do not depend on the choice of ℓ, corresponding to which includes the state that is fully invariant under O(g 1 , g 2 ).
In general, the (unnormalized) eigenvalues of V R ℓ V L ℓ are given by The ES becomes asymptotically degenerate in the thermodynamic limit, for any boundary conditions.For 0 < θ k < θ c , one has 0 < cos(2θ k ) < 1 so that α, β → 0 as n → ∞ for any bulk bipartition, ℓ ∼ n/2; in that case, V L ℓ , V R ℓ → I/2.Alternatively, in the translationally invariant case θ k = θ one may write cos(2θ) = e −1/ξ , where ξ is the correlation length.This yields α ξ , so that α, β → 0 exponentially quickly on finite chains as long as ℓ, n − ℓ ≫ ξ.In this aspect, the system behaves much like the AKLT chain [37].
To summarize the results of this section: SPT order on qubits is not a sufficient condition for the state to be a resource for deterministic MBQT with finite or infinitesimal gates.

IV. NON-SPT STATES THAT EFFECT MBQT
Consider next the case where the teleported gate in the D = 4 correlation space is a direct sum U ⊕ J of a 2×2 unitary U , given again by Eq. (11), and an arbitrary junk matrix with all parameters real.The U at each measurement step acts on the {|00 , |01 } computational subspace of the virtual two-qubit state, which can be considered as encoding a single qubit, while J acts on the complementary subspace.Assuming a direct sum is notationally convenient in what follows, but choosing any other subset of registers yields an equivalent description.For example, if U and J act on the odd-parity and evenparity subspaces {|01 , |10 } and {|00 , |11 } respectively, the output has the characteristic structure of a matchgate [38,39], and indeed would correspond exactly to a matchgate if det(U ) = det(J).
The procedure follows closely the strategy above.Setting ϕ 2 = −ϕ 1 := −ϕ, φ 1 = φ p = φ r = ϕ, and φ 2 = φ q = φ s = −ϕ, one obtains Enforcing the canonical normalization conditions requires sec 2 ϑ = csc 2 ϑ, which is satisfied again by setting ϑ = (2k + 1)π/4, k ∈ Z, in which case the teleported single-qubit unitaries coincide with Eqs. ( 13) and ( 14) according to the measurement outcome.The normalization conditions also require y = 1/ √ 2 and p 2 + r 2 = q 2 + s 2 = 1.These last conditions can be conveniently incorporated by setting p = cos γ, r = sin γ, q = sin δ, and s = cos δ, in which case the junk matrices become J[0] = e iϕ cos γ e −iϕ sin δ e iϕ sin γ e −iϕ cos δ A major motivation is to explore the possibility that the direct-sum format can yield new resource states for deterministic MBQT with finite rotations, so the primary focus is on the θ = π/4 case, in which case U [m] = X m HR Z (ϕ).Assuming site-dependent junk matrices, and general boundary states where and C j,j+1 γj ,δj = √ 2diag(cos γ j , sin δ j , sin γ j , cos δ j ) j,j+1 .(35) The state in Eq. ( 32) allows for the teleportation of deterministic single-qubit unitaries (with feed-forward) for all choices of junk-matrix angles γ j and δ j , because the computational subspace acts like a cluster state and is orthogonal to (and therefore remains independent of) the junk subspace.
The matrices (30) are in block-diagonal form, so that the MPS is not injective [21].This further implies that the state (32) cannot be the unique ground state of a local frustration-free parent Hamiltonian, but rather that the ground-state degeneracy of such a parent Hamiltonian is two, corresponding to the number of blocks; this however doesn't preclude the possibility of preparing the state directly via a quantum circuit.In principle, the noninjectivity could affect readout of the final state [8].In practice, the state can be chosen such that C n−1,n γn−1,δn−1 = I via γ n−1 = δ n−1 = π/4 so that no entanglement is generated between the last two qubits in the junk sector.This prevents any information reaching the junk output state c L |0 + d L |1 , which can be defined in any convenient way, and therefore the quantum information encoded in the cluster sector remains uncontaminated.
Similar to state (17), the C j,j+1 γj ,δj in Eq. ( 32) are not generally unitary and are (potentially) site-dependent.Because ( 32) is described by a superposition of states, each defined by a different set of generalized stabilizers, it no longer possesses SPT order.Thus, neither SPT order nor injectivity are necessary conditions for a state to be a resource for MBQT.

V. CONCLUSIONS AND DISCUSSION
The results presented in this work demonstrate that the presence of symmetry-protected topological order is neither a sufficient nor necessary condition for a quantum state to be a resource for deterministic measurementbased quantum gate teleportation.On the one hand, a family of states of one-dimensional qubits with a nononsite SPT symmetry is unable to deterministically teleport universal one-qubit gates in correlation space, while on the other a family of states with no SPT order is able to do so.All identified states can be considered to be analogs of cluster states, but where the C Z entangling gates in their description are generally replaced by diagonal non-unitary operators.
The family of states with non-onsite symmetries identified here belong to the same SPT phase as the cluster state, and therefore can be prepared from the cluster state via a constant-depth quantum circuit comprised of non-overlapping k-local unitaries [40].The fact that such a unitary transformation maps a resource state for deterministic MBQT to a non-resource state suggests that a large number of states in a given SPT phase may not be resources for MBQT.Rather, perhaps only the subset of transformations that preserve the onsite nature of the symmetry would ensure that the state remains within the same computational phase.

Appendix B: SPT order
In this Appendix we show that the states defined by Eq. ( 17) have non-trivial symmetry-protected topological order, by explicitly constructing the symmetry operators in real space.

Review of SPT order in cluster states
It is useful to review the basics of SPT order in cluster states, and the following analysis follows expands on Ref. 41.The Z 2 ×Z 2 symmetry of the cluster state with an even number of sites n and periodic boundary conditions is explicitly generated by the operators X even = j X 2j and X odd = j X 2j+1 .Because (X ⊗ I)C Z = C Z (X ⊗ Z), applying the X j operator on so that all Z factors cancel on the application of either X odd or X even .The symmetry operators leave the cluster state invariant because all qubits are originally set to |+ which is an eigenstate of X.
For a cluster state with open boundary conditions, the symmetry operators need to be slightly modified.Again assume n is even.The additional Z operators resulting from the action of X on the first and last sites, will be cancelled by other Z gates arising from the adjacent odd or even sites, respectively.But the action of X on site 2 and n − 1, operators on the first and last sites that aren't cancelled by other X gates in X even or X odd .The symmetry operators therefore become X even = Z 1 j X 2j and X odd = j X 2j+1 Z n .
The C Z gates are diagonal and therefore commute with Z operators, so the cluster state is the unique +n eigenstate of the n-fold sum of stabilizer generators in the bulk S j = Z j−1 X j Z j+1 (2 ≤ j ≤ n − 1) and at the boundaries S 1 = X 1 Z 2 and S n = Z n−1 X n .Pauli gates at the boundaries are transformed by the C Z operators into effective Pauli gates odd X g2 even , where g 1 , g 2 ∈ {0, 1}, the effective Pauli operators on the left site are transformed as U (g 1 , g 2 )X 1 U (g 1 , g 2 ) † = (−1) g2 X 1 and U (g 1 , g 2 )Z 1 U (g 1 , g 2 ) † = (−1) g1 Z 1 , which is equivalent to an effective transformation U eff (g 1 , g 2 ) = X g1 Z g2 .A similar result holds for the right edge.