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.

We describe a fault-tolerant one-way quantum computer on cluster states in three dimensions. The presented scheme uses methods of topological error correction resulting from a link between cluster states and surface codes. The error threshold is 1.4% for local depolarizing error and 0.11% for each source in an error model with preparation-, gate-, storage- and measurement errors.Comment: 26 page

We study the ±J random-plaquette Z2 gauge model (RPGM) in three spatial dimensions, a three-dimensional analog of the two-dimensional ±J random-bond Ising model (RBIM). The model is a pure Z2 gauge theory in which randomly chosen plaquettes (occuring with concentration p) have couplings with the "wrong sign" so that magnetic flux is energetically favored on these plaquettes. Excitations of the model are one-dimensional "flux tubes" that terminate at "magnetic monopoles" located inside lattice cubes that contain an odd number of wrong-sign plaquettes. Electric confinement can be driven by thermal fluctuations of the flux tubes, by the quenched background of magnetic monopoles, or by a combination of the two. Like the RBIM, the RPGM has enhanced symmetry along a "Nishimori line" in the p-T plane (where T is the temperature). The critical concentration pc of wrong-sign plaquettes at the confinement-Higgs phase transition along the Nishimori line can be identified with the accuracy threshold for robust storage of quantum information using topological error-correcting codes: if qubit phase errors, qubit bit-flip errors, and errors in the measurement of local check operators all occur at rates below pc, then encoded quantum information can be protected perfectly from damage in the limit of a large code block. Through Monte-Carlo simulations, we measure pc0, the critical concentration along the T = 0 axis (a lower bound on pc), finding pc0 = .0293 ± .0002. We also measure the critical concentration of antiferromagnetic bonds in the two-dimensional RBIM on the T = 0 axis, finding pc0 = .1031 ± .0001. Our value of pc0 is incompatible with the value of pc = .1093 ± .0002 found in earlier numerical studies of the RBIM, in disagreement with the conjecture that the phase boundary of the RBIM is vertical (parallel to the T axis) below the Nishimori line. The model can be generalized to a rank-r antisymmetric tensor field in d dimensions, in the presence of quenched disorder.