Qubit Complexity of Continuous Problems

The number of qubits used by a quantum algorithm will be a crucial computational resource for the foreseeable future. We show how to obtain the classical query complexity for continuous problems. We then establish a simple formula for a lower bound on the qubit complexity in terms of the classical query complexityComment: 6 pages, 2 figure

Non-Computability and Intractability: Does It Matter to Physics?

Should the impossibility results of theoretical computer science be of concern to physics

Associated Polynomials and Uniform Methods for the Solution of Linear Problems

To every polynomial P of degree n we associate a sequence of n-1 polynomials of increasing degree which we call the associated polynomials of P. The associated polynomials depend in a particularly simple way on the coefficients of P. These polynomials have appeared in many guises in the literature, usually related to some particular application and most often going unrecognized. They have been called Horner polynomials and Laguerre polynomials. Often what occurs is not an associated polynomial itself but a number which is an associated polynomial evaluated at a zero of P. The properties of associated polynomials have never been investigated in themselves. We shall try to demonstrate that associated polynomials provide a useful unifying concept. Although many of the results of this paper are new, we shall also present known results in our framework

Construction of Globally Convergent Iteration Functions for the Solution of Polynomial Equations

Iteration functions for the approximation of zeros of a polynomial P are usually given as explicit functions of P and its derivatives. We introduce a class of iteration functions which are themselves constructed according to a certain algorithm given below. The construction of the iteration functions requires only simple polynomial manipulation which may be performed on a computer

Variational Calculations of the 2³S State of Helium

With a 12-parameter Hylleraas-type wave function containing only positive powers, a new calculation has been carried out for the 2³S state of helium by the Ritz variational principle. The energy was minimized by a descent process. A nonrelativistic energy of -1.0876088 Hylleraas units was reached as compared with the best previously published value of -1.0876015 Hylleraas units from a 6-parameter function. When masspolarization and a2Ry corrections are included, the 12-parameter function gives an ionization potential 2.52 cm-1 less than the experimental value of 38 454.64 cm-1. The electron density at the nucleus is also calculated and compared with the experimental hyperfine-spectrum value. All numerical work was carried out on an I.B.M. 650 computer