We present a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. The measurements are used to imprint a quantum logic circuit on the state, thereby destroying its entanglement at the same time. Cluster states are thus one-way quantum computers and the measurements form the program.

We study the entanglement properties of a class of N-qubit quantum states that are generated in arrays of qubits with an Ising-type interaction. These states contain a large amount of entanglement as given by their Schmidt measure. They also have a high persistency of entanglement which means that approximately N/2 qubits have to be measured to disentangle the state. These states can be regarded as an entanglement resource since one can generate a family of other multiparticle entangled states such as the generalized Greenberger-Horne-Zeilinger states of

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.

We present a scheme of fault-tolerant quantum computation for a local architecture in two spatial dimensions. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors.PACS numbers: 03.67. Lx, 03.67.Pp Quantum computation is fragile. Exotic quantum states are created in the process, exhibiting entanglement among large numbers of particles across macroscopic distances. In realistic physical systems, decoherence acts to transform these states into more classical ones, compromising their computational power. Fortunately, the effects of decoherence can be counteracted by quantum error correction [1]. In fact, arbitrarily large quantum computations can be performed with arbitrary accuracy, provided the error level of the elementary components of the quantum computer is below a certain threshold. This is guaranteed by the threshold theorem for quantum computation [2,3,4,5]. Now that the threshold theorem has been established, it is important to devise methods for error correction which yield a high threshold, are robust against variations of the error model, and can be implemented with small operational overhead. An additional desideratum is a simple architecture for the quantum computer, requiring no long-range interaction, for example.Recently, a threshold estimate of 3 × 10 −2 per operation has been obtained for a method using post-selection [6]. An alternative scheme with high threshold combines topological quantum computation with state purification [7]. (See also [8].) In that approach, a subset of the universal gates are assumed to be error-free. Pure topological quantum computation ideally requires no error correction but often picks up a comparable polylogarithmic overhead [9] in the Solovay-Kitaev construction for approximating single-and two-qubit gates (c.f.[10]). fault tolerance is more difficult to achieve in architectures where each qubit can only interact with other qubits in its immediate neighborhood. A fault tolerance threshold for a two-dimensional lattice of qubits with only local and nearest-neighbor gates is 1.9 × 10 −5 [11].In this Letter, we present a scheme for fault-tolerant universal quantum computation on a two-dimensional lattice of qubits, requiring only a nearest-neighbor translation-invariant Ising interaction and single-qubit preparation and measurement. A fault tolerance threshold of 7.5 × 10 −3 for each error source is presented, with moderate resource scaling. This scheme is best suited for implementation with massive qubits where geometric constraints naturally play a role, such as cold atoms in optical lattices [12] or two-dimensional ion traps [13].The presented scheme integrates methods of topological quantum computation, specifically the toric code [14], and magic state distillation [15] into the one-way quantum computer (QC C ) [16] on cluster states. By employing magic state distillation we improve the error threshold significantly beyond [17], with the threshold value and overhead scaling now set by the topologi...

We describe a fault-tolerant version of the one-way quantum computer using a cluster state in three spatial dimensions. Topologically protected quantum gates are realized by choosing appropriate boundary conditions on the cluster. We provide equivalence transformations for these boundary conditions that can be used to simplify fault-tolerant circuits and to derive circuit identities in a topological manner. The spatial dimensionality of the scheme can be reduced to two by converting one spatial axis of the cluster into time. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors. The operational overhead is poly-logarithmic in the circuit size.