SEARCH · Search NASA

Results for “RECURSION FORMULA”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

4 records

All sources NASA NTRS DOE OSTI

DOE OSTI2024

Generalized Quantum Signal Processing

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that | P | ≤ 1 , a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 10 7 . We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ -approximate fractional query problem that requires O ( ( 1 / δ ) + log ( 1 / ϵ ) ) queries to perform where O ( 1 / δ ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O ( d log N + log 2 N ) 1 and 2-qubit gates for a filter of lengths d . Published by the American Physical Society 2024

Motlagh, Danial↗

DOE OSTI2024

On the computation of moments in the Super-Transition-Arrays model for radiative opacity calculations

In the Super-Transition-Array statistical method for the computation of radiative opacity of hot dense matter, the moments of the absorption or emission features involve partition functions with reduced degeneracies, occurring through the calculation of averages of products of subshell populations. Here, in the present work, we discuss several aspects of the computation of such peculiar partition functions, insisting on the precautions that must be taken in order to avoid numerical difficulties. In a previous work, we derived a formula for supershell partition functions, which takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients for which we obtained a recursion relation and an explicit formula. We show that such an expansion can be combined with the recurrence relation for shifted partition functions. We also propose, neglecting the effect of fine structure as a first step, a positive-definite formula for the Super-Transition-Array moments of any order, providing an insight into the asymmetry and sharpness of the latter. The corresponding formulas are free of alternating sums. Several ways to speed up the calculations are also presented.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2025

Smooth splitting and zeros from on-shell recursion

We describe a new approach to understanding the origins of recently discovered “hidden zeros” and “smooth splitting” of tree-level amplitudes in Tr ϕ 3 , Non-Linear Sigma Model (NLSM), Yang-Mill-Scalar (YMS) and the special Galileon. Introducing a new type of linear shift in kinematic space we demonstrate that the mysterious splitting formulae follow from a simple contour integration argument in the style of on-shell recursion. The argument makes use of only standard notions of tree-level factorization on propagators, but assumes improved UV behavior in the form of the absence of a residue at infinity. In the case of Tr ϕ 3 and NLSM this is proven by identifying our shift as a special case of a more general construction called a g-vector shift; in the case of YMS it remains an unproven conjecture. This recursive perspective leads to numerous new results: we derive generalizations of the splitting formulae on more relaxed near-zero kinematics, including interesting new kinematic limits in which the amplitude splits into a triple-product; we also demonstrate that the uncolored special Galileon model has improved UV scaling and hence also splits. We also investigate the possible realization of hidden zeros in four dimensions. The conditions under which the dimensionality constraints are compatible with zero kinematics is investigated in detail for Tr ϕ 3 and YMS; for the latter we find they can be realized only with certain restrictions on external helicity states. The realizable 4d zeros are proven by a similar recursive argument based on BCFW and is found to generalize to a new class of intrinsically 4d “helicity zeros” present in all sectors of YM and also gravity.

effective field theories↗

DOE OSTI2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node’s importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its “global” character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability P ( K ) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank- 1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗