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Register a new userAlmost Exact Exchange At Almost No Cost
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A recently developed semiclassical approximation to exchange in one dimension is shown to be almost exact, with essentially no computational cost. The variational stability of this approximation is tested, and its far greater accuracy relative to local density functional calculations demonstrated. Even a fully orbital-free potential-functional calculation (no orbitals of any kind) yields little error relative to exact exchange, for more than one orbital.
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To identify which principles characterize quantum correlations, it is essential to understand in which sense this set of correlations differs from that of almost quantum correlations. We solve this problem by invoking the so-called no-restriction hypothesis, an explicit and natural axiom in many reconstructions of quantum theory stating that the set of possible measurements is the dual of the set of states. We prove that, contrary to quantum correlations, no generalised probabilistic theory satisfying the no-restriction hypothesis is able to reproduce the set of almost quantum correlations. Therefore, any theory whose correlations are exactly, or very close to, the almost quantum correlations necessarily requires a rule limiting the possible measurements. Our results suggest that the no-restriction hypothesis may play a fundamental role in singling out the set of quantum correlations among other non-signalling ones.
We present a new method for sampling the Levy area for a two-dimensional Wiener process conditioned on its endpoints. An efficient sampler for the Levy area is required to implement a strong Milstein numerical scheme to approximate the solution of a stochastic differential equation driven by a two-dimensional Wiener process whose diffusion vector fields do not commute. Our method is simple and complementary to those of Gaines-Lyons and Wiktorsson, and amenable to quasi-Monte--Carlo implementation. It is based on representing the Levy area by an infinite weighted sum of independent Logistic random variables. We use Chebychev polynomials to approximate the inverse distribution function of sums of independent Logistic random variables in three characteristic regimes. The error is controlled by the degree of the polynomials, we set the error to be uniformly 10^(-12). We thus establish a strong almost-exact Levy area sampling method. The complexity of our method is square logarithmic. We indicate how our method can contribute to efficient sampling in higher dimensions.
This work presents an algorithm to evaluate Coulomb and exchange matrices in Fock operator using range separation techniques at various aspects. This algorithm is particularly favorable for the scenario of (1) all-electron calculations or (2) computing exchange matrix for a large number of $mathbf{k}$-point samples. An all electron Hartree-Fock calculation with 110k basis functions is demonstrated in this work.
We provide a new non-asymptotic analysis of distributed TD(0) with linear function approximation. Our approach relies on one-shot averaging, where $N$ agents run local copies of TD(0) and average the outcomes only once at the very end. We consider two models: one in which the agents interact with an environment they can observe and whose transitions depends on all of their actions (which we call the global state model), and one in which each agent can run a local copy of an identical Markov Decision Process, which we call the local state model. In the global state model, we show that the convergence rate of our distributed one-shot averaging method matches the known convergence rate of TD(0). By contrast, the best convergence rate in the previous literature showed a rate which, in the worst case, underperformed the non-distributed version by $O(N^3)$ in terms of the number of agents $N$. In the local state model, we demonstrate a version of the linear time speedup phenomenon, where the convergence time of the distributed process is a factor of $N$ faster than the convergence time of TD(0). As far as we are aware, this is the first result rigorously showing benefits from parallelism for temporal difference methods.
The recently developed semistochastic heat-bath configuration interaction (SHCI) method is a systematically improvable selected configuration interaction plus perturbation theory method capable of giving essentially exact energies for larger systems than is possible with other such methods. We compute SHCI atomization energies for 55 molecules which have been used as a test set in prior studies because their atomization energies are known from experiment. Basis sets from cc-pVDZ to cc-pV5Z are used, totaling up to 500 orbitals and a Hilbert space of $10^{32}$ Slater determinants for the largest molecules. For each basis, an extrapolated energy well within chemical accuracy (1 kcal/mol or 1.6 mHa/mol) of the exact energy for that basis is computed using only a tiny fraction of the entire Hilbert space. We also use our almost exact energies to benchmark coupled-cluster [CCSD(T)] energies. The energies are extrapolated to the complete basis set limit and compared to the experimental atomization energies. The extrapolations are done both without and with a basis-set correction based on density-functional theory. The mean absolute deviations from experiment for these extrapolations are 0.46 kcal/mol and 0.51 kcal/mol, respectively. Orbital optimization methods used to obtain improved convergence of the SHCI energies are also discussed.
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