Measurement-based quantum computation on cluster states

The minimum number of required terms to write down a cluster state in explicit form scales exponentially with the number of qubits.

“In fact, to write down a cluster state in its explicit form would be quite space-consuming since the minimum number of required terms scales exponentially with the number of qubits [2]”

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Observables of the form cos(φ)σ_x ± sin(φ)σ_y are measured to realize arbitrary rotations of logical qubits.

“Observables of the form cos(ϕ) σ x ± sin(ϕ) σ y are measured to realize arbitrary rotations of logical qubits.”

A cluster state is completely specified by the eigenvalue equations (1).

“A cluster state is completely specified by the eigenvalue equations (1).”

The required algorithm-specific non-universal quantum resource to run the remainder of a quantum algorithm on the QC_C is a graph state.

“The required algorithm-specific non-universal quantum resource to run the remainder of the quantum algorithm on the QC C is then a graph state [9].”

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The cluster state on a chain of 3 qubits is local unitary equivalent to the GHZ state.

“The state |φ C2 is local unitary equivalent to a Bell state and |φ C3 to the Greenberger-Horne-Zeilinger (GHZ) state.”

Information processing on the QC_C is possible even though the result of every measurement in any direction of the Bloch sphere is completely random.

“It is important to realize here that information processing is possible even though the result of every measurement in any direction of the Bloch sphere is completely random.”

The state |+_C obeys the eigenvalue equations σ_x^(a)|+_C = |+_C for all a ∈ C.

“The state |+ C obviously obeys the eigenvalue equations σ (a) x |+ C = |+ C ∀a ∈ C”

From every algorithm its Clifford part can be removed.

“We show that from every algorithm its Clifford part can be removed.”

All cluster states |φ{κ}_C are equally good for computation.

“All states |φ {κ} C are equally good for computation.”

• Also stated in: § C. Removing the redundant cluster qubits · plain

  “However, any cluster state |φ {κ ′ } CN is equally good for computation as stated in Section II A.”

The overheads in spatial, temporal and operational resources required in QC_C-computations relative to corresponding quantum logic networks are at most polynomial.

“We find that overheads are at most polynomial.”

There do not always need to be overheads in QC_C-computations relative to corresponding quantum logic networks.

“But there do not always need to be overheads.”

Two states that obey equations (1) with the same set {κ}_0 are the same modulo a possible global phase.

“Hence, two states |φ ′ {κ}0 C and |φ {κ}0 C which obey (1) with the same set {κ} 0 are the same modulo a possible global phase.”

The entanglement in the 4-qubit cluster state |φ_C4 cannot be destroyed by a single local operation.

“In particular, the entanglement in |φ C4 cannot be destroyed by a single local operation [2].”

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All QC_C-circuits in the Clifford group have unit logical depth.

“all QC C -circuits in the Clifford group have unit logical depth.”

The reduced density operator for each qubit in the cluster state is one-half times the identity.

“The mathematical expression for the randomness of the measurement results is that the reduced density operator for each qubit in the cluster state is 1 2 1.”

The cluster state on a chain of 4 qubits is not equivalent to a 4-particle GHZ state.

“|φ C4 is not equivalent to a 4-particle GHZ state.”

Measurements of σ_z observables effectively remove the respective lattice qubit from the cluster.

“Measurements of σ z -observables effectively remove the respective lattice qubit from the cluster.”

The QC_C is universal because any unitary quantum logic network can be simulated on it efficiently.

“The QC C is universal since any unitary quantum logic network can be simulated on it efficiently.”

Since the local part of H commutes with the Ising Hamiltonian H_I, the interaction S generated by H is local unitary equivalent to the unitary transformation generated by an Ising Hamiltonian.

“and, since the local part H local of the Hamiltonian commutes with the Ising Hamiltonian H I , the interaction S generated by H is local unitary equivalent to the unitary transformation generated by a Ising Hamiltonian.”

The propagation relations (72)-(74) map Pauli operators onto Pauli operators under propagation, so the byproduct operators remain in the Pauli group when propagated.

“Note that the propagation relations (72) -(74) are such that Pauli operators are mapped onto Pauli operators under propagation and thus the byproduct operators remain in the Pauli group when being propagated.”

The Hadamard and π/2-phase gates are performed only modulo a subsequent byproduct operator determined by the random measurement outcomes.

“As in the cases before, the Hadamard-and the π/2phase gate are performed only modulo a subsequent byproduct operator which is determined by the random measurement outcomes s k”

The values x_i, z_i ∈ {0, 1} are computed from the measurement outcomes s_k.

“The values x i , z i ∈ {0, 1} are computed from the measurement outcomes s k | k ∈ C I (g) ∪ C M (g) .”

The state |φ_C generated from |+_C via the transformation S obeys eigenvalue equations of form (1) with κ_a = 0 for all a ∈ C.

“Thus, the state |φ C generated from |+ C via the transformation S as defined in (5) does indeed obey eigenvalue equations of form (1), with κ a = 0, ∀ a ∈ C.”

The redundant qubits have some remaining influence on computation: after they are measured, the random measurement results enter into the eigenvalues that specify the residual cluster state on C_N.

“After they have been measured, the random measurement results enter into the eigenvalues that specify the residual cluster state |φ {κ ′ } CN on the cluster C N .”

For gates in the Clifford group, U'_g = U_g.

“As discussed above, for gates in the Clifford group we have U ′ g = U g , ∀ g ∈ Clifford group,(77)”

The temporal resources to create the cluster state are constant in the size of the cluster.

“The temporal resources to create the cluster state are constant in the size of the cluster.”

A σ_z-measurement on the state U_Σ|out with results {s} represents the same algorithmic output as a σ_z-measurement of the state |out with results {s'_i}, where s'_i ≡ s_i + x_i mod 2.

“From ( 70) we see that a σ z -measurement on the state U Σ |out with results {s} represents the same algorithmic output as a σ z -measurement of the state |out with the results {s ′ i }, where the sets {s} and {s ′ i } are related by s ′ i ≡ s i + x i mod 2. (71 )”

After measuring redundant qubits in the σ_z eigenbasis, the resulting entangled state on the sub-cluster C_N is again a cluster state obeying equations (1).

“The resulting entangled state |φ {κ ′ } CN on the sub-cluster C N is again a cluster state obeying the set of equations (1).”

Only the byproduct operators of gates g_k earlier than g_i in the network may have an effect on U_{g_i}, i.e. those with k < i.

“From the structure of (68) we see that only the byproduct operators of gates g k earlier than g i in the network may have an effect on U gi , i.e. those with k < i.”

The sequential scheme, in which gates are realized one after another on subclusters, is equivalent to the entangle-once-and-then-only-measure scheme for the whole circuit.

“Thus both realizations, the sequential and the non-sequential, are equivalent.”

The interaction part H_I of the Hamiltonian H generating S is of Ising form.

“We find that the interaction part H I of the Hamiltonian H generating S is of Ising form,”

Cluster states specified by different sets {κ_a} can be obtained by applying Pauli phase flip operators σ_z^(a).

“Cluster states specified by different sets {κ a } can be obtained by applying Pauli phase flip operators σ (a) z .”

Via Procedure 2, the rotation U'_Rot[ξ, η, ζ] = U_Σ,Rot U_Rot[ξ, η, ζ] is realized.

“Via Procedure 2 the rotation U ′ Rot is realized:”

The output state of the sequential realization of two gates g1 and g2 and the output state of the standard procedure applied to the composite circuit are the same for all inputs.

“we find that |Ψ out C(g) , the output state of the sequential realization of the two gates g 1 and g 2 , and |Ψ ′ out C(g) , the output state of the standard procedure applied to the composite circuit, are indeed the same for all inputs |Ψ in C(g) .”

The creation of the cluster state is a two-step process.

“Thus, the creation of the cluster state is a two step process.”

An arbitrary rotation U_Rot ∈ SU(2) can be realized on a chain of 5 qubits.

“An arbitrary rotation U Rot ∈ SU (2) can be realized on a chain of 5 qubits.”

The relation between a rotation U'_{g_i} modified by byproduct operators and the non-modified rotation U_{g_i} is given by equation (78).

“The relation between a rotation U ′ gi modified by the byproduct operators and the non-modified rotation U gi is U ′ gi =   k|k<i U Σ,g k | Oi   U gi   k|k<i U Σ,g k | Oi   † , ∀ U gi ∈ SU (2). (78 )”

From (60), the quantum input and quantum output of the unitary evolution are related by |ψ_out = (product over i of U_{Σ,g_i} U_{g_i}) |ψ_in.

“From (60) it follows that the quantum input |ψ in and the quantum output |ψ out of the unitary evolution are related via |ψ out =   |N | i=1 U Σ,gi U gi   |ψ in .(68)”

Redundant cluster qubits do not have to be removed physically; measuring each of them in the σ_z eigenbasis suffices to make them ineffective to the realized circuit.

“It is sufficient to measure each of them in the σ z -eigenbasis, as will be described in Section II C.”

• Also stated in: § C. Removing the redundant cluster qubits · plain

  “To make them ineffective to the realized circuit, it suffices to measure each of them in the σ z -eigenbasis.”

If it takes the Ising interaction a time T_Ising to create a cluster state from a product state, then the Ising interaction acting for a time 2T_Ising performs the identity operation.

“If it takes the Ising interaction a time T Ising to create a cluster state |φ C from a product state |+ C , then the Ising interaction acting for a time 2T Ising performs the identity operation, S (C) S (C) = 1l (C) .”

For a suitable choice of collision phases φ = π mod 2π, the state resulting from a product state |+_C after interaction is a cluster state obeying eigenvalue equations (1), with {κ_a} specified by the filling pattern of the lattice.

“For a suitable choice of the collision phases ϕ, ϕ = π mod 2π, the state resulting from a product state |+ C after interaction is a cluster state obeying the eigenvalue equations (1), with the set {κ a , a ∈ C} specified by the filling pattern of the lattice.”

The byproduct operators U_{Σ,g_i} can be propagated forward through subsequent gates such that they act on the cluster qubits representing the output register.

“This can be done.To propagate the byproduct operators we use the propagation relations”

Measurement patterns corresponding to the projections P1 and P2 can be patched together to form the measurement pattern on the cluster C(g).

“We find that measurement patterns corresponding to the projections P 1 and P 2 can be patched together to form the measurement pattern on the cluster C(g).”

The measurement pattern for a rotation around the z-axis U_z[β] = exp[-iβ/2 σz] requires 5 qubits for its realization.

“The measurement pattern for a rotation around the z-axis U z [β] = exp[-iβ/2σ z ] is illustrated in Fig. 2. It requires 5 qubits for its realization.”

The gate simulation indeed acts as stated in (83) and (84).

“Thus we find that the gate simulation indeed acts as stated in (83) and (84).”

As (101) shows, |cχ| is indeed equal to unity.

“As (101) shows, |c χ | is indeed equal to unity.”

On the input state |+>, the gate simulation acts as Ũ = e^{iχ} U UΣ.

“Therefore, on the input state |+ the gate simulation acts as Ũ = e iχ U U Σ .(99)”

From (94) it follows that cχ = 1/2^n sum_{z∈{0,1}^n} e^{-iη(z)}.

“from (94) it follows that c χ = 1 2 n z∈{0,1} n e -iη(z) .(102)”

The general CNOT gate can always be realized on a vertical slice of width 6 on the cluster, no matter how far control and target qubit are separated.

“The virtue of this gate is that it can always be realized on a vertical slice of width 6 on the cluster, no matter how far control and target qubit are separated.”

For input |+>, the state |Ψ_out^+> obeys the eigenvalue equations U σ_x[i] U† (CO(g)) |Ψ_out^+> = (-1)^{λx,i+si} |Ψ_out^+>.

“Using the upper line of eigenvalue equations (82), the state |Ψ out (+) C(g) is found to obey the eigenvalue equations U σ [i] x U † (CO(g)) |Ψ out (+) C(g) = (-1) λx,i+si |Ψ out (+) C(g) .”

The byproduct operator for the z-rotation gate is U_{Σ,Uz} = σ^{s2+s4}_x σ^{s1+s3}_z.

“The byproduct operator for this gate is U Σ,Uz = σ s2+s4 x σ s1+s3 z .”

The byproduct operators U_{Σ,g_k}|O_i may modify Euler angles of the one-qubit rotations in the network and consequently change measurement bases.

“There, the byproduct operators U Σ,g k | Oi may modify Euler angles of the one-qubit rotations in the network and consequently change measurement bases.”

Equation (82) is satisfied for U = U_x(-η) and UΣ = σ^{s1}_z σ^{s2}_x.

“and thus we see that equation ( 82) is satisfied for U = U x (-η) and U Σ = σ s1 z σ s2 x .”

The global phase factor e^{iχ} has no effect.

“Therein, the global phase factor e iχ has no effect.”

The measurement pattern M(Ux) consists of a measurement on qubit 2 of the observable r_xy(η)•σ = cosη σx + sinη σy = Uz[η]σx Uz[-η], whose eigenstates lie in the x-y plane at an angle η to the x-axis.

“The measurement pattern M(U x ) consists of a measurement, on qubit 2, of the observable represented by the vector r xy (η) = (cos(η), sin(η), 0), r xy (η) • σ = cos η σ x + sin η σ y = U z [η]σ x U z [-η], (116) whose eigenstates lie in the x-y-plane of the Bloch sphere at an angle of η to the x-axis.”

Theorem 1: If the state |ψ_C(g) = P^{C_M(g)}_{s}(M)|φ_C(g) obeys the 2n eigenvalue equations (82), then the gate g can be realized according to Scheme 1 with measurement directions in C_M(g) described by M and σ_x measurements on C_I(g), with input-output relation |ψ_out = U U_Σ |ψ_in, where U_Σ is given by (84).

“Suppose, the state |ψ C(g) = P (CM (g)) {s} (M) |φ C(g) obeys the 2n eigenvalue equations σ (CI (g),i) x U σ (i) x U † (CO(g)) |ψ C(g) = (-1) λx,i |ψ C(g) σ (CI (g),i) z U σ (i) z U † (CO(g)) |ψ C(g) = (-1) λz,i |ψ C(g) ,(82)”

• Also stated in: § F. Using quantum correlations for quantum computation · plain

  “Then, on the cluster C(g) the gate g acting on an arbitrary quantum input state |ψ in can be realized according to Scheme 1 with the measurement directions in C M (g) described by M (CM (g)) and the measurements of the qubits in C I (g) being σ x -measurements.Thereby, the input-and output state in the simulation of g are related via |ψ out = U U Σ |ψ in ,(83)”

A wire with length one (C_I(H)=1, C_M(H)=∅, C_O(H)=2) implements a Hadamard transformation.

“Also note that a wire with length one (C I (H) = 1, C M (H) = ∅, C O (H) = 2), i.e. half of the above elementary wire, implements a Hadamard transformation.”

With S = h w, one finds S ≤ 24 S_qln^2 T_qln.

“With S = h w one finds that S ≤ 24 S qln 2 T qln .(139)”

n'_O(z) = 0 for all z.

“or, in other words, n ′ O (z) = 0 for all z.”

Since |e^{-iη(z)}| = 1 for all z, the triangle inequality gives |cχ| ≤ 1.

“Thus, with |e -iη(z) | = 1 for all z, it follows from the triangle inequality that |c χ | ≤ 1.”

The realized gate acts as Ũ = U U_Σ D (95).

“We have found in (94) that the realized gate acts as Ũ = U U Σ D (95 )”

The z-rotation gate can be understood as the concatenation U_z[α] = H U_x[α] H.

“The simplest way to understand this gate is regard it as the concatenation U z [α] = H U x [α] H.”

A one-qubit rotation through an angle α about the x-axis U_x[α] = exp[-iα/2 σx] is realized on the same three qubit layout as the identity gate.

“A one-qubit rotation through an angle α about the xaxis U x [α] = exp[-iα/2σ x ] is realized on the same three qubit layout as the identity gate.”

The eigenvalue equations (97) in combination with (96) imply |Ψ_out^+> = e^{iχ} |s_x,CI> ⊗ |m_CM> ⊗ [U UΣ |+>]_CO, with χ a free parameter.

“The eigenvalue equations (97) in combination with (96) imply that |Ψ out (+) C(g) = e iχ |s x,CI (g) ⊗ |m CM (g) ⊗ [U U Σ |+ ] CO (g) ,(98) with χ being a free parameter.”

The full unitary operation realized by the identity gate is Ũ = 1l σ^{s2}_x σ^{s1}_z = σ^{s2}_x σ^{s1}_z 1l.

“The byproduct operator U Σ is obtained from equation (84), and we find that the full unitary operation realized by the gate is Ũ = 1l σ s2 x σ s1 z = σ s2 x σ s1 z 1l.”

Comparing equations (136a)-(136d) to the eigenvalue equations (135a)-(135d), one sees that M does indeed realize a CNOT gate.

“Comparing these equations to the eigenvalue equations (135a) to ( 135d), one sees that M does indeed realize a CNOT gate.”

The arbitrary Euler rotation can be realized by combining the measurement patterns of rotations around x- and z-axes by overlaying input and output qubits of adjacent patterns.

“The arbitrary Euler rotation can be realized by combining the measurement patterns of rotations around xand z-axes by overlaying input and output qubits of adjacent patterns, as described in section II D.”

The modulus of cχ can be unity only if all e^{-iη(z)} are equal.

“The modulus of c χ can be unity only if all e -iη(z) are equal.”

The pattern M(1l) corresponds to a measurement of qubit 2 in the σx basis.

“The pattern M(1l) corresponds to a measurement of qubit 2 in the σ x basis.”

One finds M(Uz) realizes U_z(β) if the basis of the measurement on qubit 3 is chosen to be the eigenbasis of r_xy((-1)^{s2}(-β))•σ.

“One finds M(U z ) realizes the operation U z (β) if the basis of the measurement on qubit 3 is chosen to be the eigenbasis of r xy ((-1) s2 (-β)) • σ, where r xy (η) is defined in equation (116).”

The temporal resources of the QC_C and the network model are related via T ≤ 3 T_qln.

“Thus the temporal resources of the QC C and in the network model are related via T ≤ 3T qln .(138)”

• AssumptionThe authors state this as an assumption: In the network model, a general rotation can be realized in a single step.

Inserting (103) into (94), the gate simulation acts upon every input state in the computational basis, and thus upon every input state, as Ũg = e^{iχ} U UΣ.

“If we now insert (103) into (94) we find that the gate simulation acts upon every input state in the computational basis, and thus upon every input state, as Ũg = e iχ U U Σ .”

The operational resources O are always smaller or equal to the spatial resources S, O ≤ S, since each cluster qubit is measured at most once.

“The operational resources O are always smaller or equal to the spatial resources S, O ≤ S,(137) since each cluster qubit is measured at most once.”

All Hadamard-, π/2-phase- and CNOT-gates of a quantum circuit can be realized simultaneously in the first measurement round, regardless of their location in the network.

“Thus, all the Hadamard-, π/2-phaseand CNOT-gates of a quantum circuit can be realized simultaneously in the first measurement round, regardless of their location in the network.”

The measurement pattern described for the CNOT gate does indeed realize a CNOT gate in the QC_C.

“Labeling the qubits as in Fig. 2, we use the same analysis as above to show that this measurement pattern does indeed realize a CNOT gate in the QC C .”

In larger measurement patterns, whenever pairs of adjacent σx-qubits in a wire are surrounded above and below by either vacant lattice sites or σz-measurements, they can be removed from the pattern without changing the logical operation of the gate.

“In larger measurement patterns, whenever pairs of adjacent σ x -qubits in a wire are surrounded above and below by either vacant lattice sites or σ z -measurements, they can be removed from the pattern without changing the logical operation of the gate.”

To realize a specific rotation U_Rot[ξ,η,ζ] = U_x[ζ]U_z[η]U_x[ξ], the angles α, β, γ specifying the measurement bases of qubits 2, 3, and 4 are dependent on measurement results of other qubits; α = (-1)^{s1}(-ξ), β = (-1)^{s2}(-η), γ = (-1)^{s1+s3}(-ζ).

“One finds that, to realize a specific rotation U Rot [ξ, η, ζ] = U x [ζ]U z [η]U x [ξ], the angles α, β, γ specifying the measurement bases of the qubits 2,3, and 4 are again dependent on the measurement results of other qubits.We see that α = (-1) s1 (-ξ), β = (-1) s2 (-η), γ = (-1) s1+s3 (-ζ).”

The output factor is |ψ_out(z) = e^{iη(z)} U U_Σ |z, where U_Σ is given by (84).

“In (92) only the third factor of the state on the r.h.s. is interesting, and this factor is determined by the eigenvalue equations (89): |ψ out (z) = e iη(z) U U Σ |z ,(93) where U Σ is given by (84).”

Under the assumptions of Theorem 1, P^{C_I(g)}_{s}(X) P^{C_I(g)}_{Z,z} |ψ_C(g) = 0 for all z.

“To make use of the equations (89) we need to prove that P (CI (g)) {s} (X) P (CI (g)) Z,z |ψ C(g) = 0 for all z under the assumptions of theorem 1.”

If qubits 10, 11, 13 and 14 are measured in the σx basis and qubits 2, 3, 4, 5, 6, 8 and 12 in the σy eigenbasis, then the cluster state eigenvalue equations (134a)-(134d) induce eigenvalue equations (135a)-(135d) for the projected state |ψ>.

“If the qubits 10, 11, 13 and 14 are measured in the σ x -and the qubits 2, 3, 4, 5, 6, 8 and 12 are measured in the σ y -eigenbasis, whereby the measurement results s 2 -s 6 , s 8 , s 10 -s 14 are obtained, then the cluster state eigenvalue equations (134a) -(134d) induce the following eigenvalue equations for the projected state |ψ”

Quantum circuits which consist only of such gates, i.e. circuits in the Clifford group, can be realized in a single time step.

“In particular, quantum circuits which consist only of such gates, i.e. circuits in the Clifford group, can be realized in a single time step.”

After the sigma_x measurements of the qubits in C_M, the projected state |psi_C(swap) satisfies the eigenvalue equations (142).

“After the σ x -measurements of the qubits in C M we obtain for the projected state equations”

Realizing the measurement pattern M leads to correlations between the ith input qubit and the n+1-i output qubit.

“It can be verified using the methods introduced above that realizing M leads to correlations between the ith input qubit and the n + 1ith output qubit.”

The parameters lambda_k,x and lambda_k,z in {0,1} depend linearly on the measurement outcomes {s(i,j)}.

“Therein, the parameters λ k,x , λ k,z ∈ {0, 1} depend linearly on the measurement outcomes {s (i,j) }.”

With the set of equations (155), the assumptions (82) of theorem 1 are fulfilled.

“With the set of equations (155) the assumptions (82) of theorem 1 are fulfilled.”

The entire computation amounts to just measurements of the cluster qubits in a certain order and basis.

“The entire computation amounts to just measurements of the cluster qubits in a certain order and basis.”

The measurement outcomes from all the cluster qubits contribute to the result of the computation.

“The measurement outcomes from all the cluster qubits contribute to the result of the computation.”

After the sigma_x measurements of the qubits in C_M, there remains a strict sigma_x correlation for the state |psi_C(swap).

“It is now easy to see that after the σ x measurements of the qubits in C M there remains a strict σ (I,1) x σ (O,4) x correlation for the state |ψ C(swap) .”

A multi-qubit swap gate can be realized in a simple way on the QC_C as shown in Fig. 9a.

“This can be realized in a simple way on the QC C , as shown in Fig. 9a.”

The contribution U_Sigma,Q of the cluster qubits measured in tilted bases to the byproduct operator U_Sigma in (84) contains only a z-part but no x-part.

“The contribution U Σ,Q ր of the cluster qubits measured in tilted bases to the byproduct operator U Σ in (84) contains only a z-part but no x-part.”

In the second measurement round, one measures on qubit (3,4) the operator given in (147), where lambda_M = 1 + lambda + sum lambda_z,i mod 2.

“That is, in the second measurement round we measure on the qubit (3, 4) the operator given in (147), where λ M = 1 + λ + 4 i=1 λ z,i mod 2. (161”

The rotation U_zz[phi/4] is realized via the measurement of the cluster qubit at lattice site (3,1) in the appropriate basis.

“An initial cluster-state correlations which is used for the realization of a generalized rotation is shown in Fig. 18: the rotation U (c1,c2) zz [φ/4] is realized via the measurement of the cluster qubit at the lattice site (3, 1) in the appropriate basis.”

The sign factors of the angles that specify the measurement bases depend on the outcome of sigma_x-measurements only.

“The sign factors of the angles that specify the measurement bases depend on the outcome of σ x -measurements only.”

In this way we obtain a measurement pattern in which there are adjacent cluster qubits in wires that are measured in the sigma_x-eigenbasis.

“In this way we obtain a measurement pattern in which there are adjacent cluster qubits in “wires” that are measured in the σ x -eigenbasis.”

By removing adjacent pairs of sigma_x-measured cluster qubits, the sigma_y-measurements of the Hadamard transformations are moved into the subsequent conditional phase gates, i.e. a cluster qubit which was not from a wire is removed.

“Note, that by removing adja- cent pairs of σ x -measured cluster qubits we have moved the σ y -measurements of the Hadamard transformations “into” the subsequent conditional phase gates, i.e. we removed a cluster qubit which was not from a wire.”

The {lambda_x,i}, {lambda_z,i} and lambda depend linearly on the measurement outcomes {s(i,j)} obtained in the first measurement round.

“The {λ x,i }, {λ z,i } and λ depend linearly on the measurement outcomes {s (i.j) } obtained in the first measurement round.”

To realize U_d of the cluster qubit (6,7), the observable U_z[±pi/4] sigma_x U_z[∓pi/4] = ±sigma_y is measured.

“To realize U d of the cluster qubit (6, 7) the observable U z ± π 4 σ x U z ∓ π 4 = ±σ y (178”

The n-qubit QC C adder requires a cluster of height 8n-5.

“the adder requires a cluster of height 8n-5 and of constant width 38”

For the n-qubit QC C adder, spatial and operational resources are, to leading order, S = O = 304 n.

“Thus the spatial and operational resources are, to leading order, S = O = 304 n.(181)”

In the Toffoli gate and the three-qubit controlled gate, the ancilla qubits are located on cluster qubits which have only one next neighbor.

“As can be seen from the Toffoli gate and the three-qubit controlled gate displayed in Figs. 17 and 19, the ancilla qubits are located on cluster qubits which have only one next neighbor.”

The n-qubit QC C adder requires a cluster of constant width 38.

“the adder requires a cluster of height 8n-5 and of constant width 38”

The temporal complexity T of an n-qubit QC C adder is T = n, plus one step of σx-, σy- and σz-measurements for the entire circuit.

“The temporal complexity T of an n-qubit QC C -adder is T = n,(182) plus one step of σ x -,σ y -and σ z -measurements for the entire circuit.”

For the quantum Fourier transform and the addition circuit, the authors ultimately imitated network composition.

“For the quantum Fourier transform and the addition circuit we have, though playing with some tricks, ultimately imitated network composition.”

In the Hamiltonian simulation circuit of Section IV B, it is easy to realize generalized rotations exp(iφ σ(J)) where σ(J) is a composite Pauli operator.

“In the Hamiltonian simulation circuit of Section IV B we found that it is easy to realize generalized rotations exp(iϕ σ (J) ) where σ (J) is a composite Pauli operator, σ (J) = a∈J σ (a) ka , k a = x, y, z.”

Any method that creates a state obeying equations (1) with a specific set {κ_a} creates the same state.

“Consequently, any method that creates a state obeying equations (1) with a specific set {κ a |a ∈ C} creates the same state.”

The QC_C can be explained as a simulator of quantum logic networks.

“The QC C can thus be explained as a simulator of quantum logic networks.”

All that remains of the Clifford part is a mathematical graph specifying the graph state.

“All that remains of the Clifford part is a mathematical graph specifying this graph state.”

Using a modified scheme of repeated (re-)entangling via the Ising interaction alternating with rounds of one-qubit measurements, it is possible to split long computations such that they fit piecewise on a small cluster.

“Using this modified scheme it is possible to split long computations such that they fit piecewise on a small cluster.”

The computational model that emerges for the QC_C makes no reference to the concept of unitary evolution.

“However, the computational model that emerges for the QC C [7] makes no reference to the concept of unitary evolution”

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The network model does not provide the most suitable description for the QC_C.

“it shall be pointed out from the beginning that the network model does not provide the most suitable description for the QC C .”

The one-way quantum computer (QC_C) is a scheme of quantum computation that consists entirely of one-qubit measurements on cluster states.

“We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states.”

• Also stated in: § I. INTRODUCTION · plain

  “This scheme uses a given entangled state, the so-called cluster state [2], as its central physical resource.”

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The random byproduct operator U_Σ,Rot can be corrected for at the end of the computation.

“It can be corrected for at the end of the computation, as will be explained in Section II E.”

The described measurement-based scheme of quantum computation can be decomposed into the three procedures for gate simulation.

“As will be shown in Section II D, the three procedures stated are precisely of such a form that the described measurement-based scheme of quantum computation can be decomposed into them.”

Starting with a product state at t = 0 evolving under permanent Ising interaction, stroboscopic measurements may be performed at times (2k + 1)T_Ising.

“Therefore, starting with a product state |+ C at time t = 0 evolving under permanent Ising interaction, stroboscopic measurements may be performed at times (2k + 1) T Ising , k ∈ N.”

A temporal ordering among the measurements is inferred from the requirement to keep the computation on the QC_C deterministic despite the randomness introduced by the measurements.

“A temporal ordering among the measurements is inferred from the requirement to keep the computation on the QC C deterministic in spite of of the randomness introduced by the measurements.”

• AssumptionRests on an assumption: Individual measurement outcomes in the QC_C scheme are random and uncontrollable.

One can cope with the randomness of the measurement results provided the byproduct operators U_{Σ,g_i} in (68) can be propagated forward through subsequent gates such that they act on the cluster qubits representing the output register.

“Thus we find that one can cope with the randomness of the measurement results provided the byproduct operators U Σ,gi in (68) can be propagated forward through the subsequent gates such that they act on the cluster qubits representing the output register.”

• AssumptionRests on an assumption: Individual measurement outcomes in the QC_C scheme are random and uncontrollable.

A permanent Ising interaction instead of a globally tunable one is sufficient for creating cluster states if the measurement process is much faster than the characteristic time scale for the Ising interaction, i.e. if the measurements are stroboscopic.

“Even this condition can be relaxed.A permanent Ising interaction instead of a globally tunable one is sufficient, if the measurement process is much faster than the characteristic time scale for the Ising interaction, i.e. if the measurements are stroboscopic.”

Once the gates g_k for all k < i have been realized, U_{g_i} can be deduced from U'_{g_i} for each gate g_i because the byproduct operators U_{Σ,k} are then known for all k < i.

“Once the gates g k for all k < i have been realized, this can be done for each gate g i since the byproduct operators U Σ,k are then known for all k < i.”

The scheme works with unit efficiency despite the randomness of the individual measurement results.

“The scheme works with unit efficiency despite the randomness of the individual measurement results, as will be discussed in Section II E.”

The scheme of quantum computation with the QC_C consists of a single entangling operation which creates the resource cluster state and, subsequently, a series of one-qubit measurements on that state.

“The scheme of quantum computation with the QC C consists of a single entangling operation which creates the resource cluster state and, subsequently, of a series of one-qubit measurements on that state.”

In QC_C, temporal complexity is determined by the temporal ordering of measurements, unlike in quantum logic networks where it depends on the ordering of gates.

“First note that in case of the QC C the basic primitive are measurements.Thus, the temporal complexity will be determined by the temporal ordering of these measurements, unlike in quantum logic networks, where it depends on the ordering of gates.”

The operations in the sequential scheme may be reordered so that all entangling operations act on the initial state first and afterwards all measurements are performed.

“Thus, the operations may be reordered in such a way that in a first step all entangling operations S (C(gi)) act on the initial state and afterwards all the measurements are performed.”

To prove the functioning of a gate g realized via Scheme 1, it is sufficient to demonstrate that a cluster state on C(g) exhibits certain quantum correlations.

“To prove the functioning of a gate g realized via Scheme 1 it is sufficient to demonstrate that a cluster state on C(g) exhibits certain quantum correlations.”

A cluster qubit q can only be measured once all cluster qubits p_i for which q ∈ fc(p_i) have been measured earlier.

“The rule is that a cluster qubit q can only be measured once all cluster qubits p i for which q ∈ fc(p i ) have been measured earlier.”

Qubit 2 must be measured prior to qubit 3.

“Qubit 2 must thus be measured prior to qubit 3.”

With adaptive measurement bases the effect of the randomness introduced by the measurements can be counteracted.

“With adaptive measurement bases the effect of the randomness introduced by the measurements can be counteracted.”

The fact that (94) and (98) hold simultaneously imposes stringent conditions on the phases η(z).

“The fact that (94) and (98) hold simultaneously imposes stringent conditions on the phases η(z).”

Gates in the Clifford group do not contribute to the temporal complexity of a QC_C algorithm.

“Gates in the Clifford group do not contribute to the temporal complexity of a QC Calgorithm, see Section II I.”

The scaling of spatial and temporal resources is at worst polynomial as compared to the network model.

“The purpose of this section was to demonstrate that the scaling of spatial and temporal resources is at worst polynomial as compared to the network model.”

Instead of describing a quantum algorithm as a network of gates applied to a fiducial input state, a quantum algorithm may be characterized by a graph specifying the quantum resource and the structure for processing the measurement outcomes.

“In conclusion, instead of describing a quantum algorithm as a network of gates applied to some fiducial input state, a quantum algorithm may (arguably more effectively) be characterized by a graph specifying the quantum resource and the structure [7] for the processing of the measurement outcomes.”

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The eigenvalue equations (142) can be derived from corresponding eigenvalue equations for the cluster state |phi_C(swap).

“The eigenvalue equations (142) can be derived from corresponding eigenvalue equations for the cluster state |φ C(swap) on the cluster C(swap).”

The QC_C-construction is not affected by the presence of non-Clifford operations.

“The QC C -construction, on the other hand, is not affected by this.”

With theorem 1 it follows that the measurement pattern displayed in Fig. 10 realizes a unitary transformation U_sim = U_4[-(-1)^lambda theta] U_swap U_Sigma.

“With theorem 1 it follows that the measurement pattern displayed in Fig. 10 realizes a unitary transformation U sim = U 4 [-(-1) λ θ]U swap U Σ ,(156)”

With the eigenvalue equations (142), the assumptions of theorem 1 are fulfilled and thus via the described measurement pattern a unitary operation U = SWAP is realized modulo a byproduct operator as specified in (84).

“With the eigenvalue equations (142) the assumptions of theorem 1 are fulfilled and thus via the described measurement pattern a unitary operation U = SW AP is realized modulo a byproduct operator as specified in (84).”

The price is that the universal quantum resource, the cluster state, is changed into a non-universal, algorithm-specific resource, a graph state on fewer qubits.

“The price is that the universal quantum resource, the cluster state, is changed into a non-universal, algorithm-specific resource -a graph state (21)-on fewer qubits.”

For the QC_C the notions of quantum input and quantum output have no genuine meaning if we restrict ourselves to the situation where the input state is known.

“For the QC C the notions of “quantum input” and “quantum output” have no genuine meaning if we restrict ourselves to the situation where the input state is known.”

The measurement of the qubits in Q0 realizes the Clifford part of a quantum circuit.

“Now note that the measurement of the qubits in Q 0 realize the Clifford part of a quantum circuit.”

The measurement pattern introduces a large amount of redundancy into the description of a QC_C-algorithm.

“Thus, the measurement pattern introduces a large amount of redundancy into the description of a QC C -algorithm.”

Only the non-Clifford gates require quantum resources.

“Only the non-Clifford gates require quantum resources.”

Not only qubits measured in the sigma_z eigenbasis may be removed, but all qubits of which one of the Pauli operators sigma_x, sigma_y, or sigma_z is measured, i.e. all qubits in Q_0.

“Not only the qubits measured in the σ z -eigenbasis may be removed from the cluster but instead all those qubits of which one of the Pauli operators σ x , σ y or σ z is measured, i.e. all the qubits which form the set Q 0 .”

For an n-qubit network algorithm, the classical data that the QC_C has to keep track of is entirely contained in a 2n-component binary-valued information flow vector I(t).

“If the network algorithm runs on n qubits then the classical data that the QC C has to keep track of is entirely contained in a 2ncomponent binary valued vector, which we have called the information flow vector I(t) [7].”

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As shown in [7], there is no difference in the way measurements of cluster qubits in I, O or M contribute to the final result of the computation.

“As shown in [7], it turns out that there is none.”

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Each quantum network algorithm in question may be reduced by its Clifford part.

“Each quantum network algorithm in question may be reduced by its Clifford part.”

The classical processing of measurement outcomes in a QC_C-computation increases the total time of computation only marginally.

“Concerning the resources for the classical processing of the measurement outcomes in a QC C -computation, we point out that this processing increases the total time of computation only marginally [7].”

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The fact that the quantum resource can be reduced by these qubits means that the Clifford part can be removed from each quantum algorithm.

“The fact that we can reduce the quantum resource by these qubits means that we can remove from each quantum algorithm its Clifford part.”

The QC_C reduction of the Clifford part represents an extension to the Knill-Gottesman theorem.

“This represents, in a way, an extension to the Knill-Gottesman-Theorem [19]”

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For the QC_C, the Clifford part of each algorithm can be removed.

“For the QC C , the Clifford part of each algorithm can be removed.”

Allowing an algorithm-specific graph state as the quantum resource of a QC_C-computation reduces the redundancy of both the description and the realization of a quantum algorithm.

“To allow for an algorithm-specific graph state as the quantum resource of a QC C -computation reduces the redundancy of both the description and the realization of a quantum algorithm.”

In a QC_C-computation with a special graph state as the quantum resource and the first measurement round omitted, the processing of classical information is the same as in a QC_C-computation with a universal resource and the first measurement round performed.

“Thus, in a QC C -computation with a special graph state as the quantum resource and the first measurement round omitted, the way of processing the classical information is the same as in a QC C -computation with a universal resource and the first measurement round performed.”

This redundancy may be reduced to a large extent by allowing for non-universal, algorithm-specific quantum resources.

“This redundancy may be reduced to a large extent by allowing for non-universal, algorithm-specific quantum resources.”

The state of the unmeasured qubits after measurement of the cluster qubits in Q_0 is again local equivalent to a graph state.

“The state of the unmeasured qubits that emerges after the measurement of the cluster qubits in Q 0 is again (local equivalent to) a graph state.”

Any stabilizer code over the alphabet A = F_p^m is local unitary equivalent to a graph code.

“The cited theorem states that any stabilizer code over the alphabet A = F p m is [local unitary] equivalent to a graph code.”

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With only a single non-Clifford operation in the circuit, such as a one-qubit rotation about most axes and angles, the efficient classical formalism upon which the Gottesman-Knill theorem rests can no longer be applied.

“With only a single non-Clifford operation in the circuit, such as a one-qubit rotation about most axes and angles, the efficient classical formalism upon which the Gottesman-Knill theorem rests can no longer be applied.”

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The graphs associated with states (141) are in general not unique.

“The graphs associated with states (141) are in general not unique [20].”

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The Clifford part of the network algorithm specifies the corresponding graph.

“The Clifford part of the network algorithm specifies the corresponding graph.”

The controlled phase gate U_CPG(theta) can be realized without decomposing it into CNOTs and rotations.

“Here, we give an example of another two-qubit gate which can be realized without decomposing it into CNOTs and rotations, the controlled phase gate U CPG (θ).”

The quantum logic circuit realized by the whole measurement pattern uses these sub-patterns to swap the positions of adjacent qubits.

“The quantum logic circuit realized by the whole measurement pattern, illustrated on the left-hand side of Fig. 13 uses these sub-patterns to swap the positions of adjacent qubits.”

To realize the controlled phase gate (162) together with a swap-gate, the observables (168) measured in the second round have to be chosen with angles alpha_(2,1)=(-1)^{1+s(2,2)+s(1,3)} theta/2, alpha_(1,2)=(-1)^{s(1,1)+s(2,2)+s(1,3)} theta/2, and alpha_(2,3)=(-1)^{s(1,1)+s(2,2)+1} theta/2.

“Using (170) one finds the following result: To realize the controlled phase gate (162) together with a swap-gate, the observables (168) measured in the second round have to be chosen with the angles α (2,1) = (-1) 1+s (2,2) +s (1,3) θ/2, α (1,2) = (-1) s (1,1) +s (2,2) +s (1,3) θ/2 and α (2,3) = (-1) s (1,1) +s (2,2) +1 θ/2.”

This brings non-neighboring qubits together so that a CNOT operation may be performed on them.

“This brings non-neighboring qubits together so that a CNOT operation may be performed on them.”

To realize U_4[phi] with phi specified in (148), one must choose theta = (-1)^{1+lambda+sum lambda_z,i} phi.

“Thus, in order to realize U 4 [ϕ] with ϕ specified in (148) we must choose θ = (-1) 1+λ+ 4 i=1 λz,i ϕ.(160)”

The realization of the gate U_CARRY requires two measurement rounds.

“The realization of the gate requires two measurement rounds.”

The measurements in tilted bases may all be performed simultaneously in the second round.

“The reason why the measurements in the tilted bases may all be performed simultaneously in the second round can be seen as follows.”

Each of the generalized rotations that make up the Toffoli phase gate is directly associated with one of the measurements made in the eigenbasis of U_z[±phi/4] sigma_x U_z[±phi/4]^dagger.

“Each of the generalized rotations that make up the gate is directly associated with of one of the measurements made in the eigenbasis of U z [±φ/4]σ x U z [±φ/4] † .”

The measurement sub-pattern in Fig. 12a also realizes a controlled pi-phase gate, also known as a controlled sigma_z gate.

“However, as well as swapping the qubits, this pattern also realizes the a controlled π-phase gate, also known as a controlled σ z gate, illustrated in Fig. 12b.”

Such pairs of cluster qubits may be removed from the measurement pattern.

“As described in Section II G 2, such pairs of cluster qubits may be removed from the measurement pattern.”

Because U_Sigma,Q has only a z-part, it commutes with U_a-g.

“Because U Σ,Q ր has only a z-part it commutes with U a-g .”

The sub-circuit described simulates the unitary evolution according to a particular four-particle Hamiltonian in a two-step process of measurements.

“The sub-circuit we have described in this section simulates the unitary evolution according to a particular fourparticle Hamiltonian in a two-step process of measurements.”

For each generalized rotation there is one cluster qubit in C_M(U_a-g) whose measurement basis specifies the respective rotation angle.

“For each generalized rotation there is one cluster qubit in C M (U a-g ) whose measurement basis specifies the respective rotation angle.”

The measurement sub-pattern illustrated in Fig. 12a crosses the wires for the logical qubits between the target and control qubit.

“The “wires” for the logical qubits in between the target-and the control qubit are crossed using the measurement sub-pattern illustrated in Fig. 12a.”

Therefore, the results of measurements in a tilted basis do not mutually affect the choice of their measurement bases.

“Therefore, the results of measurements in a tilted basis do not mutually affect the choice of their measurement bases.”

The Toffoli phase gate can be represented as a product of multi-qubit rotations (174).

“Like the controlled phase gate it can be represented as a product of multi-qubit rotations, U (c1,c2,t) Toffoli [φ] = U (c1,c2,t) zzz φ 4 U (c1,c2) zz -φ 4 U (c1,t) zz -φ 4 U (c2,t) zz -φ 4 U (c1) z φ 4 U (c2) z φ 4 U (t) z φ 4 .”

The time for which the simulated Hamiltonian acts is encoded in the basis of the measurement in the second round.

“The time for which the simulated Hamiltonian acts is encoded in the basis of the measurement in the second round.”

The whole gate U_CARRY can be performed on the QC_C in two measurement rounds.

“The first is to note that the whole gate U CARRY can be performed on the QC C in two measurement rounds.”

The target qubit may travel through the gate backwards.

“Thus, the target qubit may travel through the gate backwards.”

The randomness of the measurement outcome in preparing an ancilla qubit in the state |+ does not jeopardize the deterministic character of the circuit.

“(the randomness of the measurement outcome does not jeopardize the deterministic character of the circuit)”

This property also holds for the Toffoli phase gate.

“This property also holds for the Toffoli phase gate.”

The resource overheads in one realization as compared to the other are only constant.

“Therefore, the resource overheads in one realization as compared to the other one are only constant.”

Equation (139) yields an upper bound on S which is ∼ n^3.

“Equation (139) yields an upper bound on S which is ∼ n 3 and (140) gives bounds on O and S which are ∼ n 2 .”

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Measuring a qubit of a cluster state which only has one next neighbor in the σx eigenbasis projects the neighboring cluster qubit into an eigenstate of σz.

“to measure a qubit of a cluster state which only has one next neighbor in the eigenbasis of σ x also has the effect of projecting this neighboring cluster qubit into an eigenstate of σ z”

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For U_CARRY, the target-input and target-output can be interchanged.

“Second, note that the for U CARRY the target-input and the target-output can be interchanged, see Fig. 22.”

The more compact realization discussed below requires only a linear overhead.

“However, the more compact realization discussed below requires only a linear overhead.”

The QC_C-circuit may be formed in such a way that the horizontal size of the required cluster is constant such that the cluster size increases only linearly with the number n of logical qubits.

“The QC C -circuit may be formed in such a way that the horizontal size of the required cluster is constant such that the cluster size increases only linearly with the number n of logical qubits.”

The horizontal dimension of the cluster to realize the adder circuit does not grow linearly with the number of logical qubits n.

“At first sight it appears as if the horizontal dimension of the cluster to realize the adder circuit would grow linearly with the number of logical qubits n.This is, however, not the case.”

For both the network and the QC C realization of the quantum adder, the spatial, temporal and operational resources scale linearly with n.

“Thus we find for both the network and the QC Crealization of the quantum adder that the spatial, temporal and operational resources scale linearly with n.”

This holds because the conditional phase-flip on the target qubit is its own inverse.

“This holds because the (conditional) phase-flip on the target qubit is its own inverse.”

With the same argument, the cluster qubits carrying the ancillas in their output state, and their next neighbors may also be removed.

“With the same argument, the cluster qubits carrying the ancillas in their output state, and their next neighbors may also be removed.”

If on the QC_C a quantum logic network is simulated, the temporal axis is converted into an additional spatial axis.

“If on the QC C a quantum logic network is simulated, the temporal axis is converted into an additional spatial axis.”

For the QC C, the resource scaling is much better than indicated by bounds (138), (139) and (140), in particular for spatial and operational resources.

“For the QC C this is much better than what is indicated by the bounds (138), ( 139) and (140), in particular for the spatial and operational resources.”

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The method of removing pairs of adjacent cluster qubits measured in the σx eigenbasis still works even when one qubit in each pair has more than two neighbors.

“But the method still works as can be easily verified.”

Cluster qubits projected into an eigenstate of σz may be removed from the cluster.

“Such cluster qubits may be removed from the cluster as explained in Section II C.”

To choose the right measurement bases, neither of the measurements in a tilted basis that realizes one of the rotations U_a, .., U_i needs to wait for another measurement in a tilted basis.

“Thus, to choose the right measurement bases neither of the measurements in a tilted basis that realizes one of the rotations U a , .. , U i needs to wait for another measurement in a tilted basis.”

The quantum adder is an example for which these bounds are very loose.

“Thus, the quantum adder is an example for which these bounds are very loose.”

A disadvantage is that the Clifford part of the circuit may no longer be performed in a single time step.

“A disadvantage is that the Clifford part of the circuit may no longer be performed in a single time step.”

The composition principle for these elements will be different from gate composition.

“As a consequence, the composition principle for these elements will be different from gate composition.”

For the QC C, it is not the one-qubit and two-qubit operations that are particularly simple.

“First, for the QC C it is not the one-qubit and two-qubit operations that are particularly simple.”

The elementary constituents of the QC C are not gate simulations.

“The elementary constituents of the QC C are not gate simulations.”

This might, for any error rate, limit the duration of a computation.

“This might, for any error rate, limit the duration of a computation.”

The above construction fulfills a basic requirement to make the QC C fault-tolerant, namely decoherence can be controlled.

“Perhaps the most important advantage of the above construction is that in this way a basic requirement to make the QC C fault-tolerant can be fulfilled.Namely, decoherence can be controlled.”

The cluster states on subclusters C(g1) and C(g2) are mutually entangled.

“The reason for this is that the cluster states on the subclusters C(g 1 ) and C(g 2 ) are mutually entangled.”

It is possible to divide the circuit into sub-circuits each of which fits onto the cluster.

“It is thus possible to divide the circuit into sub-circuits each of which fits onto the cluster.”

The two central conditions (49) and (50) and constraint (51) are fulfilled if the subgraphs G(C(g_i), E(g_i)) are chosen in accordance with assignment (A2) and constraint (A3).

“To summarize, the two central conditions (49) and (50) for the decomposition of the edges E CN of the initial graph and the constraint (51) are fulfilled if the subgraphs G(C(g i ), E(g i )) are chosen in accordance with the assignment (A2) and the constraint (A3).”

If the cluster provided by a device is too small for a measurement pattern, the QC C algorithm can still be run on the device by splitting the computation into parts that fit on the cluster.

“If the cluster that may be provided by a specific device is too small for a certain measurement pattern it does not mean that the respective QC C -algorithm cannot be run on this device.Instead, the QC C -computation may be split into several parts such that each of those parts fits on the cluster.”

The QC C has no quantum register.

“However, as we have explained in [7] and also briefly sketched in Section III A, the QC C has no quantum register.”

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An advantage of this modified procedure is that one gets by with smaller clusters.

“An advantage of this modified procedure is that one gets by with smaller clusters.”

The QC C can be regarded as a simulator of quantum logic networks.

“We have shown that the QC C can be regarded as a simulator of quantum logic networks.”

This mutual entanglement is caused by precisely those interactions S_ab that correspond to edges in E_CN which could not be included in either subgraph.

“This is caused by precisely those interactions S ab that correspond to the edges in E CN which could not be included in either of the subgraphs.”

The circuit displayed in Fig. 27a cannot be understood from its components on the subclusters C(g1) and C(g2).

“It is necessary to exclude a decomposition as in Fig 27a as the circuit displayed there cannot be understood from its components on the sub-clusters C(g 1 ) and C(g 2 ).”

A decomposition of a cluster into subclusters C(g1) and C(g2) as in Fig. 27a is not allowed.

“Therefore, condition (49) cannot be satisfied and consequently the decomposition of C N into C(g 1 ) and C(g 2 ) is not allowed.”