Fault-tolerant quantum computation with cluster states

The one-way quantum computing model introduced by Raussendorf and Briegel [Phys. Rev. Lett. 86, 5188 (2001)] shows that it is possible to quantum compute using only a fixed entangled resource known as a cluster state, and adaptive single-qubit measurements. This model is the basis for several practical proposals for quantum computation, including a promising proposal for optical quantum computation based on cluster states [M. A. Nielsen, Phys. Rev. Lett. (to be published), quant-ph/0402005]. A significant open question is whether such proposals are scalable in the presence of physically realistic noise. In this paper we prove two threshold theorems which show that scalable fault-tolerant quantum computation may be achieved in implementations based on cluster states, provided the noise in the implementations is below some constant threshold value. Our first threshold theorem applies to a class of implementations in which entangling gates are applied deterministically, but with a small amount of noise. We expect this threshold to be applicable in a wide variety of physical systems. Our second threshold theorem is specifically adapted to proposals such as the optical cluster-state proposal, in which nondeterministic entangling gates are used. A critical technical component of our proofs is two powerful theorems which relate the properties of noisy unitary operations restricted to act on a subspace of state space to extensions of those operations acting on the entire state space. We expect these theorems to have a variety of applications in other areas of quantum-information science.

A 2D numerical model for simulating the physics of fault systems

Simulations provide a powerful means to help gain the understanding of crustal fault system physics required to progress towards the goal of earthquake forecasting. Cellular Automata are efficient enough to probe system dynamics but their simplifications render interpretations questionable. In contrast, sophisticated elasto-dynamic models yield more convincing results but are too computationally demanding to explore phase space. To help bridge this gap, we develop a simple 2D elastodynamic model of parallel fault systems. The model is discretised onto a triangular lattice and faults are specified as split nodes along horizontal rows in the lattice. A simple numerical approach is presented for calculating the forces at medium and split nodes such that general nonlinear frictional constitutive relations can be modeled along faults. Single and multi-fault simulation examples are presented using a nonlinear frictional relation that is slip and slip-rate dependent in order to illustrate the model.

Apparent strain localization and shear wave dispersion in elastic fault gouge with microrotations

Shear deformation of fault gouge or other particulate materials often results in observed strain localization, or more precisely, the localization of measured deformation gradients. In conventional elastic materials the strain localization cannot take place therefore this phenomenon is attributed to special types of non-elastic constitutive behaviour. For particulate materials however the Cosserat continuum which takes care of microrotations independent of displacements is a more appropriate model. In elastic Cosserat continuum the localization in displacement gradients is possible under some combinations of the generalized Cosserat elastic moduli. The same combinations of parameters also correspond to a considerable dispersion in shear wave propagation which can be used for independent experimental verification of the proposed mechanism of apparent strain localization in fault gouge.

Convective instability of 3-D fluid-saturated geological fault zones heated from below

We conduct a theoretical analysis to investigate the convective instability of 3-D fluid-saturated geological fault zones when they are heated uniformly from below. In particular, we have derived exact analytical solutions for the critical Rayleigh numbers of different convective flow structures. Using these critical Rayleigh numbers, three interesting convective flow structures have been identified in a geological fault zone system. It has been recognized that the critical Rayleigh numbers of the system have a minimum value only for the fault zone of infinite length, in which the corresponding convective flow structure is a 2-D slender-circle flow. However, if the length of the fault zone is finite, the convective flow in the system must be 3-D. Even if the length of the fault zone is infinite, since the minimum critical Rayleigh number for the 2-D slender-circle flow structure is so close to that for the 3-D convective flow structure, the system may have almost the same chance to pick up the 3-D convective flow structures. Also, because the convection modes are so close for the 3-D convective flow structures, the convective flow may evolve into the 3-D finger-like structures, especially for the case of the fault thickness to height ratio approaching zero. This understanding demonstrates the beautiful aspects of the present analytical solution for the convective instability of 3-D geological fault zones, because the present analytical solution is valid for any value of the ratio of the fault height to thickness. Using the present analytical solution, the conditions, under which different convective flow structures may take place, can be easily determined.

Double diffusion-driven convective instability of three-dimensional fluid-saturated geological fault zones heated from below

We conduct a theoretical analysis to investigate the double diffusion-driven convective instability of three-dimensional fluid-saturated geological fault zones when they are heated uniformly from below. The fault zone is assumed to be more permeable than its surrounding rocks. In particular, we have derived exact analytical solutions to the total critical Rayleigh numbers of the double diffusion-driven convective flow. Using the corresponding total critical Rayleigh numbers, the double diffusion-driven convective instability of a fluid-saturated three-dimensional geological fault zone system has been investigated. The related theoretical analysis demonstrates that: (1) The relative higher concentration of the chemical species at the top of the three-dimensional geological fault zone system can destabilize the convective flow of the system, while the relative lower concentration of the chemical species at the top of the three-dimensional geological fault zone system can stabilize the convective flow of the system. (2) The double diffusion-driven convective flow modes of the three-dimensional geological fault zone system are very close each other and therefore, the system may have the similar chance to pick up different double diffusion-driven convective flow modes, especially in the case of the fault thickness to height ratio approaching 0. (3) The significant influence of the chemical species diffusion on the convective instability of the three-dimensional geological fault zone system implies that the seawater intrusion into the surface of the Earth is a potential mechanism to trigger the convective flow in the shallow three-dimensional geological fault zone system.

Numerical simulation of double-diffusion driven convective flow and rock alteration in three-dimensional fluid-saturated geological fault zones

Numerical methods are used to simulate the double-diffusion driven convective pore-fluid flow and rock alteration in three-dimensional fluid-saturated geological fault zones. The double diffusion is caused by a combination of both the positive upward temperature gradient and the positive downward salinity concentration gradient within a three-dimensional fluid-saturated geological fault zone, which is assumed to be more permeable than its surrounding rocks. In order to ensure the physical meaningfulness of the obtained numerical solutions, the numerical method used in this study is validated by a benchmark problem, for which the analytical solution to the critical Rayleigh number of the system is available. The theoretical value of the critical Rayleigh number of a three-dimensional fluid-saturated geological fault zone system can be used to judge whether or not the double-diffusion driven convective pore-fluid flow can take place within the system. After the possibility of triggering the double-diffusion driven convective pore-fluid flow is theoretically validated for the numerical model of a three-dimensional fluid-saturated geological fault zone system, the corresponding numerical solutions for the convective flow and temperature are directly coupled with a geochemical system. Through the numerical simulation of the coupled system between the convective fluid flow, heat transfer, mass transport and chemical reactions, we have investigated the effect of the double-diffusion driven convective pore-fluid flow on the rock alteration, which is the direct consequence of mineral redistribution due to its dissolution, transportation and precipitation, within the three-dimensional fluid-saturated geological fault zone system. (c) 2005 Elsevier B.V. All rights reserved.

A sensitive vorticity gauge using rotated porphyroblasts, and its application to rocks adjacent to the Alpine Fault, New Zealand

Variable aspect ratio porphyroblasts deformed in non-coaxial flow. and internally containing rotated relicts of an external foliation, can be used to characterise plane strain flow regimes. The distribution obtained by plotting the orientation of the long axis of such grains, classified by aspect ratio, against the orientation of the internal foliation is potentially a sensitive gauge of both the bulk shear strain (as previously suggested) and kinematic vorticity number. We illustrate the method using rotated biotite porphyroblasts in the Alpine Schist: a sequence of mid-crustal rocks that have been ramped to the surface along the Alpine Fault. a major transpressional plate boundary. Results indicate that, at distances greater than or equal to similar to1 km from the fault, the rocks have undergone a combination of irrotational fattening and dextral-oblique, normal-sense shear, with a bulk shear strain of similar to0.6 and kinematic vorticity number of similar to0.2. The vorticity analysis is compatible with estimates of strongly oblate bulk strain of similar to 75% maximum shortening. Dextral-reverse transpressional flow characterises higher strain S-tectonite mylonite within similar to1 km of the Alpine Fault. These relationships provide insight into the kinematics of flow and distribution of strain in the hangingwall of the Alpine Fault and place constraints on numerical mechanical models for the exhumation of these mid-crustal rocks. (C) 2001 Elsevier Science Ltd. All rights reserved.

Ductile fabrics in the zone of active oblique convergence near the Alpine Fault, New Zealand: identifying the neotectonic overprint

The mid-crustal Alpine Schist in central Southern Alps, New Zealand has been exhumed during the past similar to3 m.y. on the hanging wall of the oblique-slip Alpine Fault. These rocks underwent ductile deformation during their passage through the similar to 150-km-wide Pacific-Australia plate boundary zone. Likely to be Cretaceous in age, peak metamorphism predates the largely Pliocene and younger oblique convergence that continues to uplift the Southern Alps today. Late Cenozoic ductile deformation constructively reinforced a pre-existing fabric that was well oriented to accommodate a dextral-transpressive overprint. Quartz microstructures below a recently exhumed brittle-ductile transition zone reflect a late Cenozoic increment of ductile strain that was distributed across deeper levels of the Pacific Plate. Deformation was transpressive, including a dextral-normal shear component that bends and rotates a delaminated panel of Pacific Plate crust onto the oblique footwall ramp of the Alpine Fault. Progressive ductile shear in mylonites at the base of the Pacific Plate overprints earlier fabrics in a dextral-reverse sense, a deformation that accompanies translation of the schists up the Alpine Fault. Ductile shear along that structure affects not only the 12-km-thick section of Alpine mylonites, but is distributed across several kilometres of overlying nonmylonitic rocks. (C) 2001 Elsevier Science Ltd. All rights reserved.

Simulation of the influence of rate- and state-dependent friction on the macroscopic behavior of complex fault zones with the lattice solid model

In order to understand the earthquake nucleation process, we need to understand the effective frictional behavior of faults with complex geometry and fault gouge zones. One important aspect of this is the interaction between the friction law governing the behavior of the fault on the microscopic level and the resulting macroscopic behavior of the fault zone. Numerical simulations offer a possibility to investigate the behavior of faults on many different scales and thus provide a means to gain insight into fault zone dynamics on scales which are not accessible to laboratory experiments. Numerical experiments have been performed to investigate the influence of the geometric configuration of faults with a rate- and state-dependent friction at the particle contacts on the effective frictional behavior of these faults. The numerical experiments are designed to be similar to laboratory experiments by DIETERICH and KILGORE (1994) in which a slide-hold-slide cycle was performed between two blocks of material and the resulting peak friction was plotted vs. holding time. Simulations with a flat fault without a fault gouge have been performed to verify the implementation. These have shown close agreement with comparable laboratory experiments. The simulations performed with a fault containing fault gouge have demonstrated a strong dependence of the critical slip distance D-c on the roughness of the fault surfaces and are in qualitative agreement with laboratory experiments.

Parallel simulation system for earthquake generation: Fault analysis modules and parallel coupling analysis

Solid earth simulations have recently been developed to address issues such as natural disasters, global environmental destruction and the conservation of natural resources. The simulation of solid earth phenomena involves the analysis of complex structures including strata, faults, and heterogeneous material properties. Simulation of the generation and cycle of earthquakes is particularly important, but such simulations require the analysis of complex fault dynamics. GeoFEM is a parallel finite-element analysis system intended for solid earth field phenomena problems. This paper describes recent development in the GeoFEM project for the simulation of earthquake generation and cycles.