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Measurement-based quantum computation on cluster states
Open accessAbstract
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. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation, and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size.
Overview
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TL;DR
The paper gives a detailed account of the one-way quantum computer (QC_C), in which computation is performed solely by one-qubit measurements on a cluster state, and proves its universality by efficient simulation of quantum logic networks. It also relates algorithms to graphs, analyzes resource scaling, and provides example circuits including quantum Fourier transformation and an n-qubit adder.
Key findings
- The QC_C is universal: any unitary quantum logic network can be simulated efficiently. A CNOT gate is realized on a cluster of 15 qubits, and an arbitrary one-qubit rotation on a chain of 5 qubits; all Clifford-group circuits have unit logical depth and can be realized in a single time step.
- Resource bounds relative to the network model are polynomial: \(T \le 3T_{\text{qln}}\), \(S \le 24 S_{\text{qln}}^2 T_{\text{qln}}\), and \(S \le 24 O_{\text{qln}} S_{\text{qln}}\), \(O \le 24 O_{\text{qln}} S_{\text{qln}}\). For quantum Fourier transformation, \(T=n\), \(S,O=2n^2\), versus network \(T_{\text{qln}}=2n\), \(S_{\text{qln}}=n\), \(O_{\text{qln}}=n^2/2\). For the n-qubit adder, \(S=O=304n\), \(T=n\), versus network \(S_{\text{qln}}=3n\), \(O_{\text{qln}}=T_{\text{qln}}=8n\); the QC_C uses about 100 times more spatial and 38 times more operational resources while being 8 times faster.
- The Clifford part of any quantum algorithm can be removed; the remaining algorithm-specific non-universal quantum resource is a graph state, and only non-Clifford gates require quantum resources.
- Computation on finite clusters is possible by repeated re-entangling and measurement rounds, which can split long computations and bound the decoherence exposure time of each cluster qubit.
Methods
- Theoretical and constructive analysis: cluster states are defined by eigenvalue equations and related to graph states; they can be created via an Ising-type interaction.
- Universality is proved by showing that gate simulations can be concatenated, using Scheme 1/Scheme 2 and Theorem 1, a criterion based on \(2n\) eigenvalue equations, byproduct operators, and adaptive measurement bases.
- The paper analyzes resource scaling and gives explicit measurement patterns for gates and subcircuits, including CNOT, arbitrary rotations, controlled phase, Toffoli, CARRY, swap, Hamiltonian simulation, quantum Fourier transformation, and addition.
Limitations
- The paper states that the resource upper bounds “should not be taken for estimates,” that a spatial overhead always exists, and that in the finite-cluster variant the Clifford part may no longer be performed in a single time step. It also states that a proposed new composition principle based on generalized rotations is “Not yet” clear.
- Evident from the text: the work is theoretical and does not report an experimental implementation or a fault-tolerance threshold analysis; decoherence is discussed qualitatively as a motivation for splitting computations, not quantified.
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140 No basis given · 132 Own interpretation · 27 Not checked yet · 8 Own result
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“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]”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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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.”
§ Measurement-based quantum computation on cluster states > quantum gate information flow
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“A cluster state is completely specified by the eigenvalue equations (1).”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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].”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“The state |φ C2 is local unitary equivalent to a Bell state and |φ C3 to the Greenberger-Horne-Zeilinger (GHZ) state.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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.”
§ Measurement-based quantum computation on cluster states > II. UNIVERSALITY OF QUANTUM COMPUTATION VIA ONE-QUBIT-MEASUREMENTS
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“The state |+ C obviously obeys the eigenvalue equations σ (a) x |+ C = |+ C ∀a ∈ C”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“We show that from every algorithm its Clifford part can be removed.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“We find that overheads are at most polynomial.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“But there do not always need to be overheads.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“In particular, the entanglement in |φ C4 cannot be destroyed by a single local operation [2].”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“all QC C -circuits in the Clifford group have unit logical depth.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“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.”
§ Measurement-based quantum computation on cluster states > II. UNIVERSALITY OF QUANTUM COMPUTATION VIA ONE-QUBIT-MEASUREMENTS
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“|φ C4 is not equivalent to a 4-particle GHZ state.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“Measurements of σ z -observables effectively remove the respective lattice qubit from the cluster.”
§ Measurement-based quantum computation on cluster states > quantum gate information flow
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“The QC C is universal since any unitary quantum logic network can be simulated on it efficiently.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“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”
§ Measurement-based quantum computation on cluster states > 3. Measure qubits 1 -4.
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“The values x i , z i ∈ {0, 1} are computed from the measurement outcomes s k | k ∈ C I (g) ∪ C M (g) .”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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 .”
§ Measurement-based quantum computation on cluster states > C. Removing the redundant cluster qubits
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“As discussed above, for gates in the Clifford group we have U ′ g = U g , ∀ g ∈ Clifford group,(77)”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“The temporal resources to create the cluster state are constant in the size of the cluster.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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 )”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“The resulting entangled state |φ {κ ′ } CN on the sub-cluster C N is again a cluster state obeying the set of equations (1).”
§ Measurement-based quantum computation on cluster states > C. Removing the redundant cluster qubits
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“Thus both realizations, the sequential and the non-sequential, are equivalent.”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“We find that the interaction part H I of the Hamiltonian H generating S is of Ising form,”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“Cluster states specified by different sets {κ a } can be obtained by applying Pauli phase flip operators σ (a) z .”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“Via Procedure 2 the rotation U ′ Rot is realized:”
§ Measurement-based quantum computation on cluster states > 3.
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“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) .”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“Thus, the creation of the cluster state is a two step process.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“An arbitrary rotation U Rot ∈ SU (2) can be realized on a chain of 5 qubits.”
§ Measurement-based quantum computation on cluster states > B. A universal set of quantum gates
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“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 )”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“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)”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“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.”
§ Measurement-based quantum computation on cluster states > 3. Measure qubits 1 -4.
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“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) .”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“This can be done.To propagate the byproduct operators we use the propagation relations”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“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).”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“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.”
§ Measurement-based quantum computation on cluster states > 4. Rotation around z-axis
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“Thus we find that the gate simulation indeed acts as stated in (83) and (84).”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“As (101) shows, |c χ | is indeed equal to unity.”
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“Therefore, on the input state |+ the gate simulation acts as Ũ = e iχ U U Σ .(99)”
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“from (94) it follows that c χ = 1 2 n z∈{0,1} n e -iη(z) .(102)”
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“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.”
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“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) .”
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“The byproduct operator for this gate is U Σ,Uz = σ s2+s4 x σ s1+s3 z .”
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“and thus we see that equation ( 82) is satisfied for U = U x (-η) and U Σ = σ s1 z σ s2 x .”
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“Therein, the global phase factor e iχ has no effect.”
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“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.”
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“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)”
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“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.”
§ Measurement-based quantum computation on cluster states > 1. Identity gate
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“With S = h w one finds that S ≤ 24 S qln 2 T qln .(139)”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“or, in other words, n ′ O (z) = 0 for all z.”
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“Thus, with |e -iη(z) | = 1 for all z, it follows from the triangle inequality that |c χ | ≤ 1.”
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“We have found in (94) that the realized gate acts as Ũ = U U Σ D (95 )”
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“The simplest way to understand this gate is regard it as the concatenation U z [α] = H U x [α] H.”
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“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.”
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“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.”
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“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.”
§ Measurement-based quantum computation on cluster states > 1. Identity gate
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“Comparing these equations to the eigenvalue equations (135a) to ( 135d), one sees that M does indeed realize a CNOT gate.”
§ Measurement-based quantum computation on cluster states > 7. The CNOT gate
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“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.”
§ Measurement-based quantum computation on cluster states > 5. Arbitrary Rotation
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“The modulus of c χ can be unity only if all e -iη(z) are equal.”
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“The pattern M(1l) corresponds to a measurement of qubit 2 in the σ x basis.”
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“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).”
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“Thus the temporal resources of the QC C and in the network model are related via T ≤ 3T qln .(138)”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“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 .”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“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.”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“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.”
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“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 .”
§ Measurement-based quantum computation on cluster states > 7. The CNOT gate
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“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.”
§ Measurement-based quantum computation on cluster states > 2. Removing unnecessary measurements
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“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).”
§ Measurement-based quantum computation on cluster states > 5. Arbitrary Rotation
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“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).”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“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.”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“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”
§ Measurement-based quantum computation on cluster states > 7. The CNOT gate
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“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.”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“After the σ x -measurements of the qubits in C M we obtain for the projected state equations”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“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.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“Therein, the parameters λ k,x , λ k,z ∈ {0, 1} depend linearly on the measurement outcomes {s (i,j) }.”
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“With the set of equations (155) the assumptions (82) of theorem 1 are fulfilled.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“The entire computation amounts to just measurements of the cluster qubits in a certain order and basis.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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“The measurement outcomes from all the cluster qubits contribute to the result of the computation.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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“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) .”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“This can be realized in a simple way on the QC C , as shown in Fig. 9a.”
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“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.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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“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”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“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.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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“The sign factors of the angles that specify the measurement bases depend on the outcome of σ x -measurements only.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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“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.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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“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.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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“The {λ x,i }, {λ z,i } and λ depend linearly on the measurement outcomes {s (i.j) } obtained in the first measurement round.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“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”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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“the adder requires a cluster of height 8n-5 and of constant width 38”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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“Thus the spatial and operational resources are, to leading order, S = O = 304 n.(181)”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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“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.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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“the adder requires a cluster of height 8n-5 and of constant width 38”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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“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.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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“For the quantum Fourier transform and the addition circuit we have, though playing with some tricks, ultimately imitated network composition.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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“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.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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“Consequently, any method that creates a state obeying equations (1) with a specific set {κ a |a ∈ C} creates the same state.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“The QC C can thus be explained as a simulator of quantum logic networks.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“All that remains of the Clifford part is a mathematical graph specifying this graph state.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“Using this modified scheme it is possible to split long computations such that they fit piecewise on a small cluster.”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“However, the computational model that emerges for the QC C [7] makes no reference to the concept of unitary evolution”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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- Not checked yetcites [7] source
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“it shall be pointed out from the beginning that the network model does not provide the most suitable description for the QC C .”
§ Measurement-based quantum computation on cluster states > I. INTRODUCTION
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“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.”
§ Measurement-based quantum computation on cluster states > Abstract
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- Also stated in: § I. INTRODUCTION · plain
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“It can be corrected for at the end of the computation, as will be explained in Section II E.”
§ Measurement-based quantum computation on cluster states > 3.
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“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.”
§ Measurement-based quantum computation on cluster states > 3. Measure qubits 1 -4.
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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- Rests on an assumption: Individual measurement outcomes in the QC_C scheme are random and uncontrollable.
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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- Rests on an assumption: Individual measurement outcomes in the QC_C scheme are random and uncontrollable.
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“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.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“The scheme works with unit efficiency despite the randomness of the individual measurement results, as will be discussed in Section II E.”
§ Measurement-based quantum computation on cluster states > 3. Measure qubits 1 -4.
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“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.”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“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.”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“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.”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“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.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“Qubit 2 must thus be measured prior to qubit 3.”
§ Measurement-based quantum computation on cluster states > 4. Rotation around z-axis
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“With adaptive measurement bases the effect of the randomness introduced by the measurements can be counteracted.”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“The fact that (94) and (98) hold simultaneously imposes stringent conditions on the phases η(z).”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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“Gates in the Clifford group do not contribute to the temporal complexity of a QC Calgorithm, see Section II I.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“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.”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“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.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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- Not checked yetcites [7] source
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“The eigenvalue equations (142) can be derived from corresponding eigenvalue equations for the cluster state |φ C(swap) on the cluster C(swap).”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“The QC C -construction, on the other hand, is not affected by this.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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“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)”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“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).”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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“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.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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“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.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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“Now note that the measurement of the qubits in Q 0 realize the Clifford part of a quantum circuit.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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“Thus, the measurement pattern introduces a large amount of redundancy into the description of a QC C -algorithm.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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“Only the non-Clifford gates require quantum resources.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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“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 .”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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“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].”
§ Measurement-based quantum computation on cluster states > A. Processing of information
What the sources say
- Not checked yetcites [7] source
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In the paper
“As shown in [7], it turns out that there is none.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
What the sources say
- Not checked yetcites [7] source
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In the paper
“Each quantum network algorithm in question may be reduced by its Clifford part.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“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].”
§ Measurement-based quantum computation on cluster states > A. Processing of information
What the sources say
- Not checked yetcites [7] source
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In the paper
“The fact that we can reduce the quantum resource by these qubits means that we can remove from each quantum algorithm its Clifford part.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“This represents, in a way, an extension to the Knill-Gottesman-Theorem [19]”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
What the sources say
- Not checked yetcites [19]
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In the paper
“For the QC C , the Clifford part of each algorithm can be removed.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“This redundancy may be reduced to a large extent by allowing for non-universal, algorithm-specific quantum resources.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“The cited theorem states that any stabilizer code over the alphabet A = F p m is [local unitary] equivalent to a graph code.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
What the sources say
- Not checked yetcites [20]
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
What the sources say
- Not checked yetcites [19]
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In the paper
“The graphs associated with states (141) are in general not unique [20].”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
What the sources say
- Not checked yetcites [20]
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In the paper
“The Clifford part of the network algorithm specifies the corresponding graph.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“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).”
§ Measurement-based quantum computation on cluster states > D. Controlled Phase Gate
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > C. CNOT between non-neighbouring qubits
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > D. Controlled Phase Gate
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In the paper
“This brings non-neighboring qubits together so that a CNOT operation may be performed on them.”
§ Measurement-based quantum computation on cluster states > C. CNOT between non-neighbouring qubits
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In the paper
“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)”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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In the paper
“The realization of the gate requires two measurement rounds.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“The reason why the measurements in the tilted bases may all be performed simultaneously in the second round can be seen as follows.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“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] † .”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > C. CNOT between non-neighbouring qubits
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In the paper
“As described in Section II G 2, such pairs of cluster qubits may be removed from the measurement pattern.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“Because U Σ,Q ր has only a z-part it commutes with U a-g .”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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In the paper
“For each generalized rotation there is one cluster qubit in C M (U a-g ) whose measurement basis specifies the respective rotation angle.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > C. CNOT between non-neighbouring qubits
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In the paper
“Therefore, the results of measurements in a tilted basis do not mutually affect the choice of their measurement bases.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“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 .”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“The time for which the simulated Hamiltonian acts is encoded in the basis of the measurement in the second round.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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In the paper
“The first is to note that the whole gate U CARRY can be performed on the QC C in two measurement rounds.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“Thus, the target qubit may travel through the gate backwards.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“(the randomness of the measurement outcome does not jeopardize the deterministic character of the circuit)”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“This property also holds for the Toffoli phase gate.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“Therefore, the resource overheads in one realization as compared to the other one are only constant.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“Equation (139) yields an upper bound on S which is ∼ n 3 and (140) gives bounds on O and S which are ∼ n 2 .”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
What the sources say
- Not checked yetcites [139]
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In the paper
“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”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
What the sources say
- Not checked yetcites [1] source
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In the paper
“Second, note that the for U CARRY the target-input and the target-output can be interchanged, see Fig. 22.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“However, the more compact realization discussed below requires only a linear overhead.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“This holds because the (conditional) phase-flip on the target qubit is its own inverse.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“With the same argument, the cluster qubits carrying the ancillas in their output state, and their next neighbors may also be removed.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“If on the QC C a quantum logic network is simulated, the temporal axis is converted into an additional spatial axis.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
What the sources say
- Not checked yetcites [138]
- Not checked yetcites [139]
- Not checked yetcites [140]
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“But the method still works as can be easily verified.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“Such cluster qubits may be removed from the cluster as explained in Section II C.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“Thus, the quantum adder is an example for which these bounds are very loose.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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“A disadvantage is that the Clifford part of the circuit may no longer be performed in a single time step.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
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“As a consequence, the composition principle for these elements will be different from gate composition.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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“First, for the QC C it is not the one-qubit and two-qubit operations that are particularly simple.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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“The elementary constituents of the QC C are not gate simulations.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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In the paper
“This might, for any error rate, limit the duration of a computation.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
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“The reason for this is that the cluster states on the subclusters C(g 1 ) and C(g 2 ) are mutually entangled.”
§ Measurement-based quantum computation on cluster states > (A1)
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“It is thus possible to divide the circuit into sub-circuits each of which fits onto the cluster.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
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In the paper
“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).”
§ Measurement-based quantum computation on cluster states > Additional information
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“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.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
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In the paper
“However, as we have explained in [7] and also briefly sketched in Section III A, the QC C has no quantum register.”
§ Measurement-based quantum computation on cluster states > H. Remarks
What the sources say
- Not checked yetcites [7] source
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“An advantage of this modified procedure is that one gets by with smaller clusters.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
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“We have shown that the QC C can be regarded as a simulator of quantum logic networks.”
§ Measurement-based quantum computation on cluster states > VI. CONCLUSION
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In the paper
“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.”
§ Measurement-based quantum computation on cluster states > (A1)
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In the paper
“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 ).”
§ Measurement-based quantum computation on cluster states > (A1)
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In the paper
“Therefore, condition (49) cannot be satisfied and consequently the decomposition of C N into C(g 1 ) and C(g 2 ) is not allowed.”
§ Measurement-based quantum computation on cluster states > (A1)
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- Probably not a substantial claim. 46
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In the paper
“The state |φ C2 is local unitary equivalent to a Bell state and |φ C3 to the Greenberger-Horne-Zeilinger (GHZ) state.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“Measurements in the σ x -(and σ y -) eigenbasis are used for”
§ Measurement-based quantum computation on cluster states > quantum gate information flow
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“For these cluster qubits, the basis in which each of them is measured depends on the results of preceding measurements.This introduces a temporal order in which the measurements have to be performed.”
§ Measurement-based quantum computation on cluster states > quantum gate information flow
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“So what remains is the first factor, and we find that the unitary evolution of the input state |ψ in that is realized is composed of the modified gates U ′ gi .The gates we will realize are thus the U ′ gi , not U gi .”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“Now using the above propagation relations, (68) can be rewritten in the following way”
- Also stated in: § E. Randomness of the measurement results · plain
“Therein, U Σ,gi | Ω are forward propagated byproduct operators resulting from the byproduct operators U Σ,gi of the gates g i .They accumulate to the total byproduct operator U Σ whose effect on the result of the computation is contained in (71),”
- Also stated in: § E. Randomness of the measurement results · plain
“Further, the U ′ gi are the gates modified under the propagation of the byproduct operators.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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In the paper
“After the 5 qubits are entangled by the unitary transformation S, the state |ψ in can be rotated by measuring qubits 1 to 4. At the same time, the state is also swapped to site 5.”
§ Measurement-based quantum computation on cluster states > B. A universal set of quantum gates
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“Gates which are not in the Clifford group are modified by byproduct operator propagation.Specifically, the general rotations (25) are conjugated as can be seen from (73).”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“The graph G(C, E C ) which describes a cluster state is that of a square lattice in 2D and that of a simple cubic lattice in 3D”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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In the paper
“A CNOT-gate can be realized on a cluster state of 15 qubits, as shown in Fig. 2. All measurements can be performed simultaneously.”
§ Measurement-based quantum computation on cluster states > B. A universal set of quantum gates
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“For CNOT-, Hadamard-and π/2-phase gates the byproduct operator changes under propagation while the gate remains unchanged.This holds for all gates in the Clifford group, because the propagation relations for Clifford gates are of the form 72) and (74), i.e. the byproduct operator U Σ is conjugated under the gate, and the Clifford group by its definition as the normalizer of the Pauli group maps Pa”
- Also stated in: § E. Randomness of the measurement results · hedged
“For gates which are not in the Clifford group this would in general not work and therefore, for rotations which are not in the Clifford group, the propagation relations are different.There, the gate is conjugated under the byproduct operator; and thus the byproduct operator remains unchanged in propagation while the gate is modified.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“The entangling operation S 1 on C(g 1 ) and S 2 on C(g 2 ) combined give the entangling operation S (C(g)) on C(g), S 2 S 1 = S (C(g)) ,(66) because of the central rule (50).”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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In the paper
“Note that cluster states (1) are a particular case of graph states (21).”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“The entanglement operations at the various stages of the sequential scheme commute with all the measurements carried out earlier.This holds because both operations, entangling operation and earlier measurement, act on different particles.”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“A cluster state on a two-dimensional cluster of rectangular shape, say, is a resource that allows for any computation that fits on the cluster.”
§ Measurement-based quantum computation on cluster states > C. Removing the redundant cluster qubits
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In the paper
“Among these rotations are, for example, the Hadamard gate and the π/2-phase gate.These gates can be realized on a chain of 5 qubits in the following way:”
§ Measurement-based quantum computation on cluster states > 3.
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“Initial individual preparation of the cluster qubits in |+ a∈C can also be done in parallel.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“Note that all operations S ab in S mutually commute and that they can therefore be carried out at the same time.”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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“In both cases, the forward propagation leaves the byproduct operators in the Pauli group.In particular, their tensor product structure is maintained.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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“The extra rotations U Σ,g are always in the Pauli group, i.e.”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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“The difference with respect to Procedure 2 for general rotations is that in Procedure 3 no measurement bases need to be adjusted according to previous measurement results and therefore the measurements can all be performed at the same time.”
§ Measurement-based quantum computation on cluster states > 3. Measure qubits 1 -4.
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“To realize a general CNOT on the QC C takes one step of measurements, to realize a general rotation takes at most three.”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“Thus, the two measurements have mapped a cluster state onto a cluster state and thus do not contribute to the logical operation realized by M, which, in the case where both s c and s d equal 0, is completely equivalent to the reduced measurement pattern M ′ , from which these adjacent σ x measurements have been removed.”
§ Measurement-based quantum computation on cluster states > 2. Removing unnecessary measurements
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“When σ z is propagated through U x (η) the sign of the angle is reversed, so we find that the gate operation realized by this M(U x ) in the QC C is U g = U x [(-1) s1 (-η)] .(123)”
§ Measurement-based quantum computation on cluster states > 3. One-qubit rotation around x-axis
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“Thus, we also have S ≤ 24 O qln S qln , O ≤ 24 O qln S qln .”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
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“In the subsequent measurement rounds only operators of the form cos ϕ σ x ± sin ϕ σ y are measured where |ϕ| < π/2, ϕ = 0.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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In the paper
“Finally, the result of the computation is calculated from the measurement outcomes obtained in all the measurement rounds.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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“The measurement outcomes from the qubits in Q 0 determine the measurement bases for the qubits in Q 1 , which are measured in the second round, those from Q 0 and Q 1 together determine the bases for the measurements of the qubits in Q 2 which are measured in the third round, and so on.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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“The cluster qubits within each set Q t can be measured simultaneously and the sets are measured one after another.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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“The measurement bases are adaptive in these rounds, i.e. they are adapted to measurement results obtained in previous rounds.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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In the paper
“Furthermore, quantum gates are not constitutive elements of the QC C ; these are instead one-qubit measurements performed in a certain temporal order and in a spatial pattern of adaptive measurement bases.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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In the paper
“Note that U swap and U 4 commute.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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In the paper
“The temporal spatial and operational resources T, S and O are, to leading order T = n, S, O = 2n 2 . (172”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“Note that the rotation angle of U d is twice as big as for the other rotations.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“The resultant state of the remaining qubits |ψ ′ , after this sub-set of the measurements has been carried out, is defined by the following set of eigenvalue equations.”
§ Measurement-based quantum computation on cluster states > D. Controlled Phase Gate
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In the paper
“The corresponding network resources are T qln = 2n, S qln = n and O qln = n 2 /2.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“If the pre-factors are compared, one finds that for the realization of a quantum adder the QC C requires about 100 times more spatial and 38 times more operational resources, while it is 8 times faster.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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In the paper
“The corresponding network resources are to leading order S qln = 3n and O qln = T qln = 8n.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
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- The authors state this as an assumption: For counting the operational and temporal network resources, the three-qubit controlled spinflip gate used in the addition circuit is composed of two Toffoli gates and one CNOT-gate, and the CNOT and Toffoli gates are regarded as elementary.
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In the paper
“For the particular set of gate simulations used in the QC C universality proof in Section II, the CNOT-gate and arbitrary one-qubit rotations, there is only a single instance of where one of these gates has been used as part of a more complicated gate in all examples of this section.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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In the paper
“All the examples for QC C -gate simulations displayed in this paper are of this type.”
§ Measurement-based quantum computation on cluster states > Additional information
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In the paper
“Instead, more compact measurement patterns have been found.”
§ Measurement-based quantum computation on cluster states > VI. CONCLUSION
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In the paper
“Furthermore, in the subsequent examples of the multi-qubit gates in Sections IV D and IV F we have decomposed the gates into such generalized rotations rather than into known standard gates on fewer qubits.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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In the paper
“Note that the simulations of the universal gates required in the universality proof are hardly used.”
§ Measurement-based quantum computation on cluster states > VI. CONCLUSION
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In the paper
“Namely, the next-neighbor CNOT-gate has been used as part of the long-distance CNOT described in Section IV C.”
§ Measurement-based quantum computation on cluster states > H. Remarks
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In the paper
“The individual measurement results are random but correlated, and these correlations enable quantum computation on the QC C .”
§ Measurement-based quantum computation on cluster states > II. UNIVERSALITY OF QUANTUM COMPUTATION VIA ONE-QUBIT-MEASUREMENTS
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In the paper
“Cluster states can be created efficiently in any system with a quantum Ising-type interaction (at very low temperatures) between two-state particles in a lattice configuration.”
§ Measurement-based quantum computation on cluster states > II. UNIVERSALITY OF QUANTUM COMPUTATION VIA ONE-QUBIT-MEASUREMENTS
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In the paper
“Cold atoms representing the qubits can be arranged on a two-or three dimensional lattice and state-dependent interaction phases may be acquired via cold collisions between neighboring atoms [12] or via tunneling [13].”
§ Measurement-based quantum computation on cluster states > A. Cluster states and their quantum correlations
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In the paper
“Please note that it is a sufficient criterion for the realization of the gate g i that all gates g k with k < i must have been realized before, but not a necessary one.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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In the paper
“This composition can be iterated so that the entire circuit can be realized via the standard procedure of Scheme 1.The measurement pattern of the circuit is thereby obtained by patching together the measurement patterns of the gates the circuit is composed of.”
§ Measurement-based quantum computation on cluster states > D. Concatenation of gate simulations
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In the paper
“First note that a byproduct operator U Σ that acts after the final unitary gate U g |N | does not jeopardize the scheme.Its only effect is that the results of the readout measurements have to be reinterpreted.”
§ Measurement-based quantum computation on cluster states > E. Randomness of the measurement results
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In the paper
“What remains is a random byproduct operator which does not affect the deterministic character of a QC C -computation and which is accounted for in the post-processing of the measurement results.”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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In the paper
“Second, Theorem 1 is a tool to demonstrate the functioning of QC C -circuits without having to repeat the whole universality proof for each particular circuit under consideration.”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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In the paper
“To realize a specific rotation deterministically, qubit 2 must thus be measured before qubits 3 and 4, and qubit 3 before qubit 4, in the bases specified in Section II B.”
§ Measurement-based quantum computation on cluster states > 5. Arbitrary Rotation
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In the paper
“As we will see, from these two cases it can be concluded that the gate simulation works for all input states of the computational basis.This is sufficient because of the linearity of the applied operations; if the gate simulations work for states of the computational basis then they work for superpositions of such inputs as well.”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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In the paper
“In [7] it has been shown, as stated in Section III A, that the required classical processing increases the computation time only marginally (logarithmically in the number n of logical qubits) and thus there is no exponential overhead in either classical or quantum resources.”
§ Measurement-based quantum computation on cluster states > H. Upper bounds on resource consumption
What the sources say
- Not checked yetcites [7] source
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“In order to realize U x [α] deterministically, the angle of the measurement, η, on qubit 2 must be η = (-1) s1 (-α), thus this measurement can only be realized after the measurement of qubit 1.”
§ Measurement-based quantum computation on cluster states > 3. One-qubit rotation around x-axis
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In the paper
“The significance of the above theorem is that it provides a comparatively simple criterion for the functioning of gate simulations on the QC C .”
§ Measurement-based quantum computation on cluster states > F. Using quantum correlations for quantum computation
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In the paper
“These updates and the final identification of the computational result from I(t max ) are all elementary.”
§ Measurement-based quantum computation on cluster states > A. Processing of information
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In the paper
“stating that a quantum computation that consist only of quantum input state preparation in the computational basis, unitary gates in the Clifford group, measurement of observables in the Pauli group, and gates in the Clifford group conditioned on the outcomes of such measurements, may be simulated efficiently classically and thus requires no quantum resources at all.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
What the sources say
- Not checked yetcites [19]
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In the paper
“Every algorithm may be run with a graph state as the quantum resource since the cluster state is a particular graph state.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“The remaining state on the sub-cluster C N is again a cluster state.Hence it is also a graph state.”
§ Measurement-based quantum computation on cluster states > B. Algorithms and graphs
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In the paper
“The generalization of the simulation of the 4-particle Hamiltonian H 4 , shown in Fig. 10, to an arbitrary number n of qubits, i.e. the simulation of the Hamiltonian H n = n i=1 σ [i] z , is straightforward.”
§ Measurement-based quantum computation on cluster states > IV. EXAMPLES OF PRACTICAL INTEREST A. Multi-qubit swap gate
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In the paper
“Thus, the scaling of the QC C spatial resources is worse than in the network model, but the temporal and operational resources scale in the same way as the corresponding resources for the network.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“As can be seen from Fig. 15, the quantum Fourier transform consists of Hadamard gates and combined gates which perform a conditional phase shift and a swap.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
What the sources say
- Not checked yetcites [22]
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In the paper
“While this could be achieved using a combination of the CNOT gate, introduced above, and the swap gate, the width of the measurement pattern needed to realize this would grow linearly with the separation of the two logical qubits.”
§ Measurement-based quantum computation on cluster states > C. CNOT between non-neighbouring qubits
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In the paper
“The induced eigenvalue equations for the state |ψ are of the form of equation (82), and the unitary operation realized by the gate can be read off from them using theorem 1.”
§ Measurement-based quantum computation on cluster states > D. Controlled Phase Gate
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In the paper
“The networks on the left and on the right of Fig. 13 act identically, and thus the measurement pattern displayed in Fig. 11 realizes a distant CNOT-gate.”
§ Measurement-based quantum computation on cluster states > C. CNOT between non-neighbouring qubits
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In the paper
“The measurement pattern realizes the generalized rotations directly and is not derived from a quantum logic network.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“It can be easily verified that this is an allowed extension of the method described in Section II G 2.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“This realizes the gate U Σ U CP G [θ], where the byproduct operator U Σ generated by the measurements may be read off from equation (170).”
§ Measurement-based quantum computation on cluster states > D. Controlled Phase Gate
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In the paper
“Thus, after all σ x -measurements have been performed, the measurement bases for the remaining qubits can be deduced and the Toffoli phase-gate is realized in a single further time-step.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“The QC C -simulation of the network displayed in Fig. 15 requires half as many time steps and four times as many operations, albeit only one-qubit operations.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“All the byproduct operators created with the measurements in tilted bases have only a zbut no x-part.Therefore they all commute with U CARRY .”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“Thus, the realization of U d belongs to the first round of measurements.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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In the paper
“The QC C -circuit, shown in Fig. 16 for the case of four qubits, is straightforwardly generalized to an arbitrary number n of logical qubits.”
§ Measurement-based quantum computation on cluster states > E. Quantum Fourier transformation
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In the paper
“Strictly speaking, this measurement round does not belong to the gate but to the circuit as a whole since all standard measurements are performed simultaneously.”
§ Measurement-based quantum computation on cluster states > F. Multi-qubit controlled gates
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The temporal axis in a QC Ccomputation emerges anew.It has no counterpart in the network model.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If we modify a quantum logic network in such a way that qubits travel from right to left, as done in Fig. 24, it does not mean that we propose to use particles that travel backwards in time because we do not need to respect the temporal axis implied by the network model.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, since we compare different objects these ratios do not mean much apart from the fact that they are constant.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
plainSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This arrangement of gates makes the circuit more compact.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“In general they should not be mistaken as estimates.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
What the sources say
- Not checked yetcites [138]
- Not checked yetcites [139]
- Not checked yetcites [140]
plainSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The QC C -version of the quantum adder corresponds to the quantum logic network as given in [23], see Fig. 23.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
What the sources say
- Not checked yetcites [23]
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“Equation (139) yields an upper bound on S which is ∼ n 3 and (140) gives bounds on O and S which are ∼ n 2 .”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
What the sources say
- Not checked yetcites [140]
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A straightforward simulation of this network on the QC C would result in a quadratic scaling of spatial resources.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The twoqubit controlled gate in this network is the Tofolli phase gate as discussed in Section IV F.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“With these neighboring qubits removed the cluster qubits on which the initial ancilla qubits were located become disconnected from the remaining cluster and may thus be removed as well.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Concerning the operational and temporal resources, the QC C only uses one-qubit measurements while the corresponding network uses two-and threequbit gates as elementary operations.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Nevertheless, it appears that the generalized rotations should be reflected in what may emerge as elementary constituents and composition principle for the QC C .”
§ Measurement-based quantum computation on cluster states > H. Remarks
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In such a case the assignment of the edges to the sub-clusters is not as simple as (A2).Instead, each edge had to be assigned to only one subcluster while respecting condition (51).”
§ Measurement-based quantum computation on cluster states > G(C(g
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The QC C may simulate, for example, general one-qubit rotations and Toffoli gates alike.”
§ Measurement-based quantum computation on cluster states > H. Remarks
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“So, the generalized rotations and their concatenation at least have to be reformulated to fit the description of the QC C .”
§ Measurement-based quantum computation on cluster states > H. Remarks
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It relates unitary transformations to quantum correlations exhibited by certain pure states.”
§ Measurement-based quantum computation on cluster states > VI. CONCLUSION
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In particular, they have to be made compatible with the graph states identified in Section III B as characteristic quantum resource to represent algorithms.”
§ Measurement-based quantum computation on cluster states > H. Remarks
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It may be argued that in case of the QC C spatial resources are not as precious as they usually are, for to create cluster states one needs a system with non-selective uniform interaction only while for quantum logic networks one generally requires a system with selective interactions among the qubits.”
§ Measurement-based quantum computation on cluster states > G. Circuit for addition
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“On the contrary, if the computation is split then the size of the sub-circuits may be adjusted such that each of them can be performed within a fixed time T and in this way, each cluster qubit is, before being measured, exposed to a bounded amount of decoherence specified by T .”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The assignment convention (A2) together with the condition (A3) exclude decompositions of a cluster like the one displayed in Fig 27b.”
§ Measurement-based quantum computation on cluster states > G(C(g
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In principle, however, such a decomposition is possible.”
§ Measurement-based quantum computation on cluster states > G(C(g
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“They correspond to the usual gate composition rules.”
§ Measurement-based quantum computation on cluster states > G(C(g
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If a single large cluster is used the computation might reach certain cluster qubits only after a long time such that the cluster would have already decohered significantly and it is not clear how error-correction could help in such a situation.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Scheme 1 is applicable to any gate or sub-circuit.”
§ Measurement-based quantum computation on cluster states > V. COMPUTATION WITH LIMITED SPATIAL RESOURCES AND IN THE PRESENCE OF DECOHERENCE
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
The authors state this as an assumption.
The readout measurements are performed in the σ_z eigenbasis.
“Without loss of generality, we assume in this paper that the readout measurements are performed in the σ z -eigenbasis.”
The authors state this as an assumption.
For the sub-cluster C_N, κ'_a = 0 for all a ∈ C_N.
“We assume that κ ′ a = 0 for all a ∈ C N .”
The paper relies on this without support.
Individual measurement outcomes in the QC_C scheme are random and uncontrollable.
“Thus one can perform measurements specified by a spatial and temporal pattern of measurement bases but one cannot control into which of the two eigenstates the qubits are projected.”
The authors state this as an assumption.
In the network model, a general rotation can be realized in a single step.
“Rather we assume that in the network model a rotation can be realized in a single step.”
The authors state this as an assumption.
For counting the operational and temporal network resources, the three-qubit controlled spinflip gate used in the addition circuit is composed of two Toffoli gates and one CNOT-gate, and the CNOT and Toffoli gates are regarded as elementary.
“For the counting of the operational and temporal network resources we have assumed that the three-qubit controlled spinflip gate used in the addition circuit is composed of two Toffoli gates and one CNOT-gate as described in [23], and that the CNOT-and the Toffoli-gate are regarded as elementary.”
The paper relies on this without support.
Any unitary transformation may be decomposed into a unitary transformation in the Clifford group followed by generalized rotations.
“Any unitary transformation may be decomposed into a unitary transformation in the Clifford group followed by generalized rotations.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“Another significant step is the discovery of”
From Multiphoton entanglement and interferometry (2012) · cites this paper as Raussendorf et al., 2003
The cited source says what the claim attributes to it.
In the citing paper
“Cluster states can be created by a controllable Ising-type interaction (Briegel and Raussendorf, 2001;Raussendorf and Briegel, 2001;Raussendorf et al., 2003).”
From Multiphoton entanglement and interferometry (2012) · cites this paper as Raussendorf et al., 2003
In the citing paper
“BQC uses the concept of one-way quantum computing [33-37], a measurement-based model of computation [38, 39] which represents a paradigm shift in the understanding of complex data processing by clearly separating the classical and quantum parts of a computation.”
From Demonstration of Blind Quantum Computing (2012) · cites this paper as [34]
In the citing paper
“in the one-way quantum computer-which will be the focus of this article-universal quantum computation can be achieved with single-qubit measurements only”
From Measurement-based quantum computation (2009) · cites this paper as [12]
In the citing paper
“The existence of a non-zero threshold was first proven by reduction to the circuit model 65-67 .”
From Measurement-based quantum computation (2009) · cites this paper as [65]
In the citing paper
“A Z measurement removes a qubit from a cluster and severs all the bonds that it had with the cluster (Raussendorf et al. 2003;Hein et al. 2004).An X measurement on a qubit in a cluster removes that qubit from the cluster, and it will transfer all the bonds of the original qubit to a neighbour.All the other neighbours become single connected qubits to the neighbour that inherited the bonds (Rauss”
From Linear optical quantum computing with photonic qubits (2007) · cites this paper as Raussendorf et al. 2003
Citations
Cites
Quantum Computation and Quantum Information
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895
A fast quantum mechanical algorithm for database search
Grover, L. K. (1996) · Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96 · doi:10.1145/237814.237866 · Cited by 8,830
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
Shor, P. W. (1997) · SIAM Journal on Computing · doi:10.1137/s0097539795293172 · Cited by 8,472
Fault-tolerant quantum computation by anyons
Kitaev, A. Y. (2003) · Annals of Physics · doi:10.1016/s0003-4916(02)00018-0 · Cited by 7,238
Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms
Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W., & Bloch, I. (2002) · Nature · doi:10.1038/415039a · Cited by 5,935
A scheme for efficient quantum computation with linear optics
Knill, E., Laflamme, R., & Milburn, G. J. (2001) · Nature · doi:10.1038/35051009 · Cited by 5,880
Quantum theory, the Church–Turing principle and the universal quantum computer
Deutsch, D. (1985) · Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences · doi:10.1098/rspa.1985.0070 · Cited by 4,597
Raussendorf, R., & Briegel, H. J. (2001) · Physical Review Letters · doi:10.1103/physrevlett.86.5188 · Cited by 4,477
Self-citationElementary gates for quantum computation
Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., & Weinfurter, H. (1995) · Physical Review A · doi:10.1103/physreva.52.3457 · Cited by 4,311
Quantum Mechanics Helps in Searching for a Needle in a Haystack
Grover, L. K. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.325 · Cited by 4,306
Research works printed but not linked: 1
- R. Raussendorf, 2002, Quantum Inf. Comput., 6, 443
Cited by
Sorted by citations
Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009) · Reviews of Modern Physics · doi:10.1103/revmodphys.81.865 · Cited by 10,118
Entanglement in many-body systems
Amico, L., Fazio, R., Osterloh, A., & Vedral, V. (2008) · Reviews of Modern Physics · doi:10.1103/revmodphys.80.517 · Cited by 3,745
Surface codes: Towards practical large-scale quantum computation
Fowler, A. G., Mariantoni, M., Martinis, J. M., & Cleland, A. N. (2012) · Physical Review A · doi:10.1103/physreva.86.032324 · Cited by 3,059
Georgescu, I. M., Ashhab, S., & Nori, F. (2014) · Reviews of Modern Physics · doi:10.1103/revmodphys.86.153 · Cited by 2,932
“…have a positive impact on the development of other fields such as adiabatic quantum computation (Farhi et al., 2001), measurement-based quantum computation (Raussendorf et al., 2003), topological quantum computation (Kitaev, 2003) and the theory of quantum computation (Vollbrecht and Cirac, 2008).”
BackgroundLinear optical quantum computing with photonic qubits
Kok, P., Munro, W. J., Nemoto, K., Ralph, T. C., Dowling, J. P., & Milburn, G. J. (2007) · Reviews of Modern Physics · doi:10.1103/revmodphys.79.135 · Cited by 2,904
Metadata
- Authors
- Raussendorf, R., Browne, D. E., & Briegel, H. J.
- Affiliations
- Ludwig-Maximilians-Universität München (DE)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2003-08-25
- Volume
- 68
- Issue
- 2
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2103368228
- Semantic Scholar
- adb89e8fcb7226a67a97929bfe85843c9243acaf
- Cited by
- 1,655
- References
- 45
- Citations by year
- 2026: 71 · 2025: 121 · 2024: 132 · 2023: 106 · 2022: 83 · 2021: 84 · 2020: 82 · 2019: 72 · 2018: 65 · 2017: 73 · 2016: 63 · 2015: 42 · 2014: 69 · 2013: 59 · 2012: 71
- FWCI
- 45.44
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Elektronische Hochschulschriften der LMU München (Ludwig-Maximilians-Universität München) (submittedVersion)
- UCL Discovery (University College London) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutsche Forschungsgemeinschaft
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- measurement-based quantum computation, cluster states, universality, entangled states, quantum networks, quantum algorithms, graph theory, resource scaling, quantum Fourier transform, quantum adder, finite clusters
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-09, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-09 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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