The miniaturization of electronic devices has been the principal driving force behind the semiconductor industry, and has brought about major improvements in computational power and energy efficiency. Although advances with silicon-based electronics continue to be made, alternative technologies are being explored. Digital circuits based on transistors fabricated from carbon nanotubes (CNTs) have the potential to outperform silicon by improving the energy-delay product, a metric of energy efficiency, by more than an order of magnitude. Hence, CNTs are an exciting complement to existing semiconductor technologies. Owing to substantial fundamental imperfections inherent in CNTs, however, only very basic circuit blocks have been demonstrated. Here we show how these imperfections can be overcome, and demonstrate the first computer built entirely using CNT-based transistors. The CNT computer runs an operating system that is capable of multitasking: as a demonstration, we perform counting and integer-sorting simultaneously. In addition, we implement 20 different instructions from the commercial MIPS instruction set to demonstrate the generality of our CNT computer. This experimental demonstration is the most complex carbon-based electronic system yet realized. It is a considerable advance because CNTs are prominent among a variety of emerging technologies that are being considered for the next generation of highly energy-efficient electronic systems.

The computing demands of future data-intensive applications will greatly exceed the capabilities of current electronics, and are unlikely to be met by isolated improvements in transistors, data storage technologies or integrated circuit architectures alone. Instead, transformative nanosystems, which use new nanotechnologies to simultaneously realize improved devices and new integrated circuit architectures, are required. Here we present a prototype of such a transformative nanosystem. It consists of more than one million resistive random-access memory cells and more than two million carbon-nanotube field-effect transistors-promising new nanotechnologies for use in energy-efficient digital logic circuits and for dense data storage-fabricated on vertically stacked layers in a single chip. Unlike conventional integrated circuit architectures, the layered fabrication realizes a three-dimensional integrated circuit architecture with fine-grained and dense vertical connectivity between layers of computing, data storage, and input and output (in this instance, sensing). As a result, our nanosystem can capture massive amounts of data every second, store it directly on-chip, perform in situ processing of the captured data, and produce 'highly processed' information. As a working prototype, our nanosystem senses and classifies ambient gases. Furthermore, because the layers are fabricated on top of silicon logic circuitry, our nanosystem is compatible with existing infrastructure for silicon-based technologies. Such complex nano-electronic systems will be essential for future high-performance and highly energy-efficient electronic systems.

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In compressed sensing, one takes n < N samples of an N-dimensional vector x 0 using an n × N matrix A, obtaining undersampled measurements y = Ax 0 . For random matrices with independent standard Gaussian entries, it is known that, when x 0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n,n/N)-phase diagram, convex optimization min || x || 1 subject to y = Ax, x ∈ X N typically finds the sparsest solution, whereas outside that region, it typically fails. It has been shown empirically that the same property-with the same phase transition location-holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical k-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to X N for four different sets X ∈ {[0, 1], R + , R, C}, and the results establish our finding for each of the four associated phase transitions.sparse recovery | universality in random matrix theory equiangular tight frames | restricted isometry property | coherence C ompressed sensing aims to recover a sparse vector x 0 ∈ X N from indirect measurements y = Ax 0 ∈ X n with n < N, and therefore, the system of equations y = Ax 0 is underdetermined. Nevertheless, it has been shown that, under conditions on the sparsity of x 0 , by using a random measurement matrix A with Gaussian i.i.d entries and a nonlinear reconstruction technique based on convex optimization, one can, with high probability, exactly recover x 0 (1, 2). The cleanest expression of this phenomenon is visible in the large n; N asymptotic regime. We suppose that the object x 0 is k-sparse-has, at most, k nonzero entries-and consider the situation where k ∼ ρn and n ∼ δN. Fig. 1A depicts the phase diagram ðρ; δ; Þ ∈ ð0; 1Þ 2 and a curve ρ*ðδÞ separating a success phase from a failure phase. Namely, if ρ < ρ*ðδÞ, then with overwhelming probability for large N, convex optimization will recover x 0 exactly; however, if ρ > ρ*ðδÞ, then with overwhelming probability for large N convex optimization will fail. [Indeed, Fig. 1 depicts four curves ρ*ðδjXÞ of this kind for X ∈ f½0; 1; R + ; R; Cg-one for each of the different types of assumptions that we can make about the entries of x 0 ∈ X N (details below).]How special are Gaussian matrices to the above results? It was shown, first empirically in ref. 3 and recently, theoretically in ref. 4, that a wide range of random matrix ensembles exhibits precisely the same behavior, by which we mean the same phenomenon of separation into success and failure phases with the same phase boundary. Such universality, if exhib...

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