2,055 research outputs found
The Bravyi-Kitaev transformation for quantum computation of electronic structure
Quantum simulation is an important application of future quantum computers
with applications in quantum chemistry, condensed matter, and beyond. Quantum
simulation of fermionic systems presents a specific challenge. The
Jordan-Wigner transformation allows for representation of a fermionic operator
by O(n) qubit operations. Here we develop an alternative method of simulating
fermions with qubits, first proposed by Bravyi and Kitaev [S. B. Bravyi, A.Yu.
Kitaev, Annals of Physics 298, 210-226 (2002)], that reduces the simulation
cost to O(log n) qubit operations for one fermionic operation. We apply this
new Bravyi-Kitaev transformation to the task of simulating quantum chemical
Hamiltonians, and give a detailed example for the simplest possible case of
molecular hydrogen in a minimal basis. We show that the quantum circuit for
simulating a single Trotter time-step of the Bravyi-Kitaev derived Hamiltonian
for H2 requires fewer gate applications than the equivalent circuit derived
from the Jordan-Wigner transformation. Since the scaling of the Bravyi-Kitaev
method is asymptotically better than the Jordan-Wigner method, this result for
molecular hydrogen in a minimal basis demonstrates the superior efficiency of
the Bravyi-Kitaev method for all quantum computations of electronic structure
A Sparse Reformulation of the Green's Function Formalism Allows Efficient Simulations of Morphological Neuron Models
We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs and the inputs are restricted to a spatially discrete, well chosen set of points, the Green's function (GF) formalism can be rewritten to scale as O (n) with the number n of inputs locations, contrary to the previously reported O (n(2)) scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected are discussed.Peer reviewedFinal Accepted Versio
Substructured formulations of nonlinear structure problems - influence of the interface condition
We investigate the use of non-overlapping domain decomposition (DD) methods
for nonlinear structure problems. The classic techniques would combine a global
Newton solver with a linear DD solver for the tangent systems. We propose a
framework where we can swap Newton and DD, so that we solve independent
nonlinear problems for each substructure and linear condensed interface
problems. The objective is to decrease the number of communications between
subdomains and to improve parallelism. Depending on the interface condition, we
derive several formulations which are not equivalent, contrarily to the linear
case. Primal, dual and mixed variants are described and assessed on a simple
plasticity problem.Comment: in International Journal for Numerical Methods in Engineering, Wiley,
201
A Continuous/Discontinuous FE Method for the 3D Incompressible Flow Equations
A projection scheme for the numerical solution of the incompressible Navier-Strokes equation is presented. Finite element discontinuous Galerkin (dG) discretization for the velocity in the momentum equations is employed. The incompressibility constraint is enforced by numerically solving the Poisson equation for pressure using a continuous Galerkin (cG) discretization. The main advantage of the method is that is does not require the velocity and pressure approximation spaces to satisfy the usual inf-sup condition, thus equal order finite element approximations for both velocity and pressure can be used. Furthermore, by using cG discretization for the Poisson equation, no auxiliary equations are needed as it is required for dG approximations of second order derivatives. In order to enable large time steps for time marching to steady-state and time evolving problems, implicit scheme is used in connection with high order implicit RK methods. Numerical tests demonstrate that the overall scheme is accurate and computationally efficient
Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones
We show that a wide variety of non-linear cellular automata (CAs) can be
decomposed into a quasidirect product of linear ones. These CAs can be
predicted by parallel circuits of depth O(log^2 t) using gates with binary
inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of
inputs are allowed. Thus these CAs can be predicted by (idealized) parallel
computers much faster than by explicit simulation, even though they are
non-linear.
This class includes any CA whose rule, when written as an algebra, is a
solvable group. We also show that CAs based on nilpotent groups can be
predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p''
gates respectively.
We use these techniques to give an efficient algorithm for a CA rule which,
like elementary CA rule 18, has diffusing defects that annihilate in pairs.
This can be used to predict the motion of defects in rule 18 in O(log^2 t)
parallel time
An assessment of methods of moments for the simulation of population dynamics in large-scale bioreactors
A predictive modelling for the simulation of bioreactors must account for both the biological and hydrodynamics complexities. Population balance models (PBM) are the best approach to conjointly describe these complexities, by accounting for the adaptation of inner metabolism for microorganisms that travel in a large-scale heterogeneous bioreactor. While being accurate for solving the PBM, the Class and Monte-Carlo methods are expensive in terms of calculation and memory use. Here, we apply Methods of Moments to solve a population balance equation describing the dynamic adaptation of a biological population to its environment. The use of quadrature methods (Maximum Entropy, QMOM or EQMOM) is required for a good integration of the metabolic behavior over the population. We then compare the accuracy provided by these methods against the class method which serves as a reference. We found that the use of 5 moments to describe a distribution of growth-rate over the population gives satisfactory accuracy against a simulation with a hundred classes. Thus, all methods of moments allow a significant decrease of memory usage in simulations. In terms of stability, QMOM and EQMOM performed far better than the Maximum Entropy method. The much lower memory impact of the methods of moments offers promising perspectives for the coupling of biological models with a fine hydrodynamics depiction
A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems
This paper presents a method for the numerical treatment of
reaction-convection-diffusion problems with parameter-dependent coefficients
that are arbitrary rough and possibly varying at a very fine scale. The
presented technique combines the reduced basis (RB) framework with the recently
proposed super-localized orthogonal decomposition (SLOD). More specifically,
the RB is used for accelerating the typically costly SLOD basis computation,
while the SLOD is employed for an efficient compression of the problem's
solution operator requiring coarse solves only. The combined advantages of both
methods allow one to tackle the challenges arising from parametric
heterogeneous coefficients. Given a value of the parameter vector, the method
outputs a corresponding compressed solution operator which can be used to
efficiently treat multiple, possibly non-affine, right-hand sides at the same
time, requiring only one coarse solve per right-hand side.Comment: 27 pages, 6 figure
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