Let F (z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural ε-clusters of roots of F (z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is twofold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schönhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

Through introducing controlled alternate quantum walks, we present controlled alternate quantum walks (CAQW) based quantum hash function. CAQW based quantum hash function have excellent security, outstanding statistical performance and splendid expansibility. Furthermore, due to the structure of alternate quantum walks, implementing CAQW based quantum hash function significantly reduces the resources necessary for its feasible experimental realization than implementing other quantum hash functions.

Learning low dimensional representation is a crucial issue for many machine learning tasks such as pattern recognition and image retrieval. In this article, we present a quantum algorithm and a quantum circuit to efficiently perform A-Optimal Projection for dimensionality reduction. Compared with the best-know classical algorithms, the quantum A-Optimal Projection (QAOP) algorithm shows an exponential speedup in both the original feature space dimension n and the reduced feature space dimension k. We show that the space and time complexity of the QAOP circuit are O [log 2 (nk/ )] and O[log 2 (nk)poly log 2 −1 ] respectively, with fidelity at least 1 − . Firstly, a reformation of the original QAOP algorithm is proposed to help omit the quantum-classical interactions during the QAOP algorithm. Then the quantum algorithm and quantum circuit with performance guarantees are proposed. Specifically, the quantum circuit modules for preparing the initial quantum state and implementing the controlled rotation can be also used for other quantum machine learning algorithms. *

he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.

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