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.

We describe a phase transition for long-range entanglement in a three-dimensional cluster state affected by noise. The partially decohered state is modeled by the thermal state of a suitable Hamiltonian. We find that the temperature at which the entanglement length changes from infinite to finite is nonzero. We give an upper and lower bound to this transition temperature.Comment: 7 page

The theoretical existence of photon-number-splitting attacks creates a security loophole for most quantum key distribution (QKD) demonstrations that use a highly attenuated laser source. Using ultra-low-noise, highefficiency transition-edge sensor photo-detectors, we have implemented the first version of a decoy state protocol that incorporates finite statistics without the use of Gaussian approximations in a one-way QKD system, enabling the creation of secure keys immune to photon-number-splitting attacks and highly resistant to Trojan horse attacks over 107 km of optical fiber.PACS numbers: 03.67. Dd, 03.67.Hk, 85.25.Oj Quantum key distribution (QKD), which enables users to create a shared key with secrecy guaranteed by the laws of physics [1], is arguably the most advanced application in the growing field of quantum information science. Since the first demonstration in 1992 [2], the field has advanced sufficiently that commercial systems are now available. Most current QKD implementations use "prepare and measure" protocols that involve the sender (Alice) preparing a single photon in a quantum state and sending it to the receiver (Bob), who then measures the photon. Attempts by an eavesdropper (Eve) to obtain information about the state of the single photon will introduce an error rate in the transmission, which alerts the users to Eve's presence.For example, to implement the Bennett-Brassard 1984 (BB84) protocol [3], Alice randomly encodes a single photon with either a 0 or a 1 in one of two conjugate bases and sends the photon to Bob. Bob performs a measurement in one of the two bases, and communicates the time slots for which he obtained detection events. Alice and Bob then create a sifted key by only retaining events where they used the same basis. Ideally, Alice's sifted bits should be perfectly correlated with Bob's if Eve did not attack the transmission, but any real system has error rates due to experimental imperfections. Error correction [4] removes these errors, leaving Alice and Bob with a perfectly correlated key. However, this key is not yet completely secret because, in principle, the errors may have arisen from Eve attacking the system. Therefore, a final step of privacy amplification [5] is used to obtain a shorter, secret key about which Eve has negligible information.The lack of readily available single-photon sources, especially at telecom wavelengths where most fiber-based QKD systems operate, modifies the simple picture outlined above considerably. If the source emits more than one photon, Eve could remove one of the photons and store it until Bob announces his basis choice, at which time she would measure the photon in the correct basis and learn the bit value without introducing any errors. Therefore, in addition to assuming that all errors arise from Eve's interaction with single photons, it is also necessary to assume that Eve can gain full information about any sifted bits that arose from multi-photon events.To determine the number of sifted bits that were encoded in single photo...

We show that qubits traveling along closed timelike curves are a resource that a party can exploit to distinguish perfectly any set of quantum states. As a result, an adversary with access to closed timelike curves can break any prepare-and-measure quantum key distribution protocol. Our result also implies that a party with access to closed timelike curves can violate the Holevo bound.PACS numbers: 03.65. Wj, 03.67.Dd, 03.67.Hk, 04.20.Gz Introduction-The theory of general relativity points to the possible existence of closed timelike curves (CTCs) [1,2]. The grandfather paradox is one criticism raised to their existence, but Deutsch resolved this paradox by presenting a method for finding self-consistent solutions of CTC interactions [3].Recently, several quantum information researchers have assumed that CTCs exist and have examined the consequences of this assumption for computation [4,5,6]. Brun showed that a classical treatment (assuming a lack of contradictions) allows NP-hard problems to be computed with a polynomial number of gates [4]. Bacon followed with a purely quantum treatment that demonstrates the same reduction of NP-hard problems to P, along with a sketch of how to perform this reduction in a fault-tolerant manner [5]. Aaronson and Watrous have recently established that either classical or quantum computers interacting with closed timelike curves can compute any function in PSPACE in polynomial time [6].In this Letter, we show how a party with access to CTCs, or a "CTC-assisted" party, can perfectly distinguish among a set of non-orthogonal quantum states. The result has implications for fundamental protocols in quantum communication because a simple corollary is that a CTC-assisted party can break any prepareand-measure quantum key distribution protocol [7,8,9]. (The security of such a scheme relies on the informationdisturbance tradeoff for identifying quantum states.) Furthermore, the capacity for quantum systems to carry classical information becomes unbounded.Our work here raises fundamental questions concerning the nature of a physical world in which closed timelike curves exist because it challenges the postulate of quantum mechanics that non-orthogonal states cannot be perfectly distinguished. A full theory of quantum gravity would have to resolve this apparent contradiction between the implication of CTCs and the laws of quantum mechanics. Note that any alternative source of nonlinearity would raise similar questions.We structure this Letter as follows. First, we give some background on Deutsch's formalism regarding CTCs in

We study the properties of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional Hilbert space. The stabilizer group of such a code is associated with a symplectically integral lattice in the phase space of 2N canonical variables. From the existence of symplectically integral lattices with suitable properties, we infer a lower bound on the quantum capacity of the Gaussian quantum channel that matches the one-shot coherent information optimized over Gaussian input states.