We prove that
Massive gully land consolidation projects, launched in China’s Loess Plateau, aim to restore 2667
- NSF-PAR ID:
- 10197459
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 10
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -depth local random quantum circuits with two qudit nearest-neighbor gates on a$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ D -dimensional lattice withn qudits are approximatet -designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$${{\,\textrm{poly}\,}}(t)\cdot n$$ . We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($$D=1$$ ) is infinite and that certain counting problems are$${{\,\mathrm{\textsf{PH}}\,}}$$ -hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$\#{\textsf{P}}$$ depth suffices for anti-concentration. The proof is based on a previous construction of$$O(\sqrt{n})$$ t -designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size corresponding to depth$$O(n\ln ^2 n)$$ . We also show a lower bound of$$O(\ln ^3 n)$$ for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$\Omega (n \ln n)$$ (size$$O(\ln n \ln \ln n)$$ ) using a different model.$$O(n \ln n \ln \ln n)$$ -
Abstract We present a proof of concept for a spectrally selective thermal mid-IR source based on nanopatterned graphene (NPG) with a typical mobility of CVD-grown graphene (up to 3000
), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of an$$\hbox {cm}^2\,\hbox {V}^{-1}\,\hbox {s}^{-1}$$ in-plane electric field. The localized surface plasmons (LSPs) on the NPG sheet, partially hybridized with graphene phonons and surface phonons of the neighboring materials, allow for the control and tuning of the thermal emission spectrum in the wavelength regime from to 12$$\lambda =3$$ m by adjusting the size of and distance between the circular holes in a hexagonal or square lattice structure. Most importantly, the LSPs along with an optical cavity increase the emittance of graphene from about 2.3% for pristine graphene to 80% for NPG, thereby outperforming state-of-the-art pristine graphene light sources operating in the near-infrared by at least a factor of 100. According to our COMSOL calculations, a maximum emission power per area of$$\upmu$$ W/$$11\times 10^3$$ at$$\hbox {m}^2$$ K for a bias voltage of$$T=2000$$ V is achieved by controlling the temperature of the hot electrons through the Joule heating. By generalizing Planck’s theory to any grey body and deriving the completely general nonlocal fluctuation-dissipation theorem with nonlocal response of surface plasmons in the random phase approximation, we show that the coherence length of the graphene plasmons and the thermally emitted photons can be as large as 13$$V=23$$ m and 150$$\upmu$$ m, respectively, providing the opportunity to create phased arrays made of nanoantennas represented by the holes in NPG. The spatial phase variation of the coherence allows for beamsteering of the thermal emission in the range between$$\upmu$$ and$$12^\circ$$ by tuning the Fermi energy between$$80^\circ$$ eV and$$E_F=1.0$$ eV through the gate voltage. Our analysis of the nonlocal hydrodynamic response leads to the conjecture that the diffusion length and viscosity in graphene are frequency-dependent. Using finite-difference time domain calculations, coupled mode theory, and RPA, we develop the model of a mid-IR light source based on NPG, which will pave the way to graphene-based optical mid-IR communication, mid-IR color displays, mid-IR spectroscopy, and virus detection.$$E_F=0.25$$ -
Abstract Let
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Abstract The double differential cross sections of the Drell–Yan lepton pair (
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