General framework for quantum search algorithms

Grover's quantum search algorithm drives a quantum computer from a prepared initial state to a desired final state by using selective transformations of these states. Here, we analyze a framework when one of the selective trasformations is replaced by a more general unitary transformation. Our framework encapsulates several previous generalizations of the Grover's algorithm. We show that the general quantum search algorithm can be improved by controlling the transformations through an ancilla qubit. As a special case of this improvement, we get a faster quantum algorithm for the two-dimensional spatial search.Comment: revised versio

Semi Quantitative Expression Analysis of Potential Cancer Biomarkers

A cancer biomarker refers to a substance or process that is indicative of the presence of cancer it may be a molecule secreted by a tumor or a specific response of the body to the presence of cancer. Background: The aim of the present study is to investigate the feasibility and potential of the molecules Cytokeratin 20, Cytokeratin 17, Cytokeratin 10 and Anillin as tumor specific markers to rapidly detect cancer. Materials and Methods: The expression of biomarkers was analyzed by preparing RNA and synthesized cDNA. Specific primers were synthesized for the biomarkers Cytokeratin 10, Cytokeratin 20 Cytokeratin 17 and Anillin and semi quantitative gene expression was carried out using mMLuv reverse transcriptase. Results: All the three biomarkers showed an enhanced expression compared with normal fibroblasts while the expression of cytokeratin 10 did not show much expression in HCT116 while Cytokeratin 20 and Anillin showed better expression in both cell lines. Conclusion: All the biomarkers showed comparatively good expression indicating the use of these genes as potential makers for detection of cancer

SYMBOL LEVEL DECODING FOR DUO-BINARY TURBO CODES

This paper investigates the performance of three different symbol level decoding algorithms for Duo-Binary Turbo codes. Explicit details of the computations involved in the three decoding techniques, and a computational complexity analysis are given. Simulation results with different couple lengths, code-rates, and QPSK modulation reveal that the symbol level decoding with bit-level information outperforms the symbol level decoding by 0.1 dB on average in the error floor region. Moreover, a complexity analysis reveals that symbol level decoding with bit-level information reduces the decoding complexity by 19.6 % in terms of the total number of computations required for each half-iteration as compared to symbol level decoding

Quantum walks can find a marked element on any graph

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are known.Comment: 50 page

On the relationship between continuous- and discrete-time quantum walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio

New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

Investigating splicing variants uncovered by next-generation sequencing the Alzheimer’s disease candidate genes, CLU, PICALM, CR1, ABCA7, BIN1, the MS4A locus, CD2AP, EPHA1 and CD33

Late onset Alzheimer’s disease (LOAD), the most common cause of late onset dementia, has a strong genetic component. To date, 21 disease-risk loci have been identified through genome wide association studies (GWAS). However, the causative functional variant(s) within these loci are yet to be discovered. This study aimed to identify potential functional splicing mutations in the nine original GWAS-risk genes: CLU, PICALM, CR1, ABCA7, BIN1, the MS4A locus, CD2AP, EPHA1 and CD33. Target enriched next generation sequencing (NGS) was used to resequence the entire genetic region for each of these GWAS-risk loci in 96 LOAD patients and in silico databases were used to annotate the variants for functionality. Predicted splicing variants were further functionally characterised using splicing prediction software and minigene splicing assays. Following in silico annotation, 21 variants were predicted to influence splicing and, upon further annotation, four of these were examined utilising the in vitro minigene assay. Two variants, rs881768 A>G in ABCA7 and a novel variant 11: 60179827 T>G in MS4A6A were shown, in these cell assays, to affect the splicing of these genes. The method employed in the paper successfully identified potential splicing variants in GWAS-risk genes. Further investigation will be needed to understand the full effect of these variants on LOAD risk. However, these results suggest a possible pipeline in order to identify putative functional variants as a result of NGS in disease-associated loci although improvements are needed within the current prediction programme in order to reduce the number of false positives

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa