Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.Comment: 13 page

Quantum search by measurement

We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar in spirit to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grover's unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms.Comment: 8 pages, 2 figure

Finding quantum algorithms via convex optimization

In this paper we describe how to use convex optimization to design quantum algorithms for certain computational tasks. In particular, we consider the ordered search problem, where it is desired to find a specific item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve this problem much faster. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find quantum algorithms using 4 log_(605) N ≈ 0.433 log_2 N queries, which improves upon the previously best known exact algorithm

Automotive timing belt life laws and a user design guide

The paper presents a computer-based guide of the effect of layout and loading (tension and torque) on the timing belt life and uses it to show the sensitivity of life to changed conditions in an automotive camshaft drive. The predictions are in line with experience. The guide requires belt property information, such as the tooth and tension member stiffness, the friction coefficient between the belt lands and pulleys and the pitch difference from the pulley, in order to calculate the tooth deflections caused by the belt loadings on the various pulleys in the layout. It also requires information on how the belt life depends on the tooth deflections. Experimental data are presented on the life±deflection relations of a commercial automotive timing belt tested between 100 and 140 8C, although the bulk of the data has been obtained at 120 8C. Four different life laws have been found, depending on whether the failure-initiating deflection occurred on a driver or a driven pulley, and whether at entry to or exit from the pulley. Theoretical analysis of the tooth loading in the partial meshing state shows that, in three cases out of the four, the different life±deflection laws transform to a single relation between the life and the tooth root strain. The exception is failure caused by driven entry conditions; work is continuing to understand better the causes of failure in this circumstanc

Improved quantum algorithms for the ordered search problem via semidefinite programming

One of the most basic computational problems is the task of finding a desired item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find new quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using 4 log_{605} N \approx 0.433 log_2 N queries, which improves upon the previously best known exact algorithm.Comment: 8 pages, 4 figure

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

LIN28 is selectively expressed by primordial and pre-meiotic germ cells in the human fetal ovary

Germ cell development requires timely transition from primordial germ cell (PGC) self-renewal to meiotic differentiation. This is associated with widespread changes in gene expression, including downregulation of stem cell–associated genes, such as OCT4 and KIT, and upregulation of markers of germ cell differentiation and meiosis, such as VASA, STRA8, and SYCP3. The stem cell–expressed RNA-binding protein Lin28 has recently been demonstrated to be essential for PGC specification in mice, and LIN28 is expressed in human germ cell tumors with phenotypic similarities to human fetal germ cells. We have therefore examined the expression of LIN28 during normal germ cell development in the human fetal ovary, from the PGC stage, through meiosis to the initiation of follicle formation. LIN28 transcript levels were highest when the gonad contained only PGCs, and decreased significantly with increasing gestation, coincident with the onset of germ cell differentiation. Immunohistochemistry revealed LIN28 protein expression to be germ cell–specific at all stages examined. All PGCs expressed LIN28, but at later gestations expression was restricted to a subpopulation of germ cells, which we demonstrate to be primordial and premeiotic germ cells based on immunofluorescent colocalization of LIN28 and OCT4, and absence of overlap with the meiosis marker SYCP3. We also demonstrate the expression of the LIN28 target precursor pri-microRNA transcripts pri-LET7a/f/d and pri-LET-7g in the human fetal ovary, and that expression of these is highest at the PGC stage, mirroring that of LIN28. The spatial and temporal restriction of LIN28 expression and coincident peaks of expression of LIN28 and target pri-microRNAs suggest important roles for this protein in the maintenance of the germline stem cell state and the regulation of microRNA activity in the developing human ovary