107,096 research outputs found
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
Renormalization and Computation II: Time Cut-off and the Halting Problem
This is the second installment to the project initiated in [Ma3]. In the
first Part, I argued that both philosophy and technique of the perturbative
renormalization in quantum field theory could be meaningfully transplanted to
the theory of computation, and sketched several contexts supporting this view.
In this second part, I address some of the issues raised in [Ma3] and provide
their development in three contexts: a categorification of the algorithmic
computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra
renormalization of the Halting Problem.Comment: 28 page
PennyLane: Automatic differentiation of hybrid quantum-classical computations
PennyLane is a Python 3 software framework for optimization and machine
learning of quantum and hybrid quantum-classical computations. The library
provides a unified architecture for near-term quantum computing devices,
supporting both qubit and continuous-variable paradigms. PennyLane's core
feature is the ability to compute gradients of variational quantum circuits in
a way that is compatible with classical techniques such as backpropagation.
PennyLane thus extends the automatic differentiation algorithms common in
optimization and machine learning to include quantum and hybrid computations. A
plugin system makes the framework compatible with any gate-based quantum
simulator or hardware. We provide plugins for Strawberry Fields, Rigetti
Forest, Qiskit, Cirq, and ProjectQ, allowing PennyLane optimizations to be run
on publicly accessible quantum devices provided by Rigetti and IBM Q. On the
classical front, PennyLane interfaces with accelerated machine learning
libraries such as TensorFlow, PyTorch, and autograd. PennyLane can be used for
the optimization of variational quantum eigensolvers, quantum approximate
optimization, quantum machine learning models, and many other applications.Comment: Code available at https://github.com/XanaduAI/pennylane/ .
Significant contributions to the code (new features, new plugins, etc.) will
be recognized by the opportunity to be a co-author on this pape
A renormalization fixed point for Lorenz maps
A Lorenz map is a Poincar\'e map for a three-dimensional Lorenz flow. We
describe the theory of renormalization for Lorenz maps with a critical point
and prove that a restriction of the renormalization operator acting on such
maps has a hyperbolic fixed point. The proof is computer assisted and we
include a detailed exposition on how to make rigorous estimates using a
computer as well as the implementation of the estimates.Comment: 29 pages, 2 figure
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