Fast Conversion Algorithms for Orthogonal Polynomials

We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation

Efficient Quantum Algorithms for State Measurement and Linear Algebra Applications

We present an algorithm for measurement of $k$-local operators in a quantum state, which scales logarithmically both in the system size and the output accuracy. The key ingredients of the algorithm are a digital representation of the quantum state, and a decomposition of the measurement operator in a basis of operators with known discrete spectra. We then show how this algorithm can be combined with (a) Hamiltonian evolution to make quantum simulations efficient, (b) the Newton-Raphson method based solution of matrix inverse to efficiently solve linear simultaneous equations, and (c) Chebyshev expansion of matrix exponentials to efficiently evaluate thermal expectation values. The general strategy may be useful in solving many other linear algebra problems efficiently.Comment: 17 pages, 3 figures (v2) Sections reorganised, several clarifications added, results unchange

Gauss–Lobatto to Bernstein polynomials transformation

AbstractThe aim of this paper is to transform a polynomial expressed as a weighted sum of discrete orthogonal polynomials on Gauss–Lobatto nodes into Bernstein form and vice versa. Explicit formulas and recursion expressions are derived. Moreover, an efficient algorithm for the transformation from Gauss–Lobatto to Bernstein is proposed. Finally, in order to show the robustness of the proposed algorithm, experimental results are reported