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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
In the paper
“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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“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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“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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“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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“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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“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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“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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“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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“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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“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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“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 the paper
“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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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
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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.
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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
“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.
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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.
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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.
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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.
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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.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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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.
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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.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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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.
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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.
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It may state two separate claims as one.
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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.
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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.
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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 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.
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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.
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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.
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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.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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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
“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.
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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.
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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
“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.
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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
“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.
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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
“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.
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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
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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
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