Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering, and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of d
iscrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms -- (i) an adaption of quantum-walk based fast forwarding algorithm (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.
We present the chemical compositions of four K giants CS 22877-1, CS 22166-16, CS22169-35 and BS 16085 - 0050 that have [Fe/H] in the range -2.4 to -3.1. Metal-poor stars with [Fe/H] < -2.5 are known to exhibit considerable star - to - star variation
s of many elements. This quartet confirms this conclusion. CS 22877-1 and CS 22166-16 are carbon-rich. There is significant spread for [$alpha$/Fe] within our sample where [$alpha$/Fe] is computed from the mean of the [Mg/Fe], and [Ca/Fe] ratios. BS 16085 - 0050 is remarkably $alpha$ enriched with a mean [$alpha$/Fe] of $+$0.7 but CS 22169-35 is $alpha$-poor. The aluminium abundance also shows a significant variation over the sample. A parallel and unsuccessful search among high-velocity late-type stars for metal-poor stars is described.
We provide detailed abundance analyses of 8 candidate super-metal-rich stars. Five of them are confirmed to have [Fe/H] > 0.2 dex, the generally-accepted limit for super-metal-richness. Furthermore, we derive abundances of several elements and find t
hat the stars follow trends seen in previous studies of metal-rich stars. Ages are estimated from isochrones and velocities calculated. We find that there do exist very metal-rich stars that are older than 10 Gyr. This is contrary to what is found in several recent studies of the galactic age-metallicity relation. This is tentative evidence that there might not exist a one-to-one relation between age and metallicity for all stars. This is not surprising considering the current models of the independent evolution of the different galactic components. We also find that one star, HD 182572, could with ~ 75 % chance be a thick disk star with, for the thick disk, an extremely high metallicity at 0.34 dex. This star is, intriguingly, also somewhat enhanced in the alpha-elements.
