Number Of Possible Orders Research Articles

Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure

Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.

An inclusive SUSY approach to position dependent mass systems

The supersymmetry (SUSY) formalism for a position dependent mass problem with a more general ordering is yet to be formulated. In this paper, we present an unified SUSY approach for PDM problems of any ordering. Highlighting all non-Hermitian Hamiltonians of PDM problems are of quasi-Hermitian nature, the SUSY operators of these problems are constructed using similarity transformation. The methodology that we propose here is applicable for even more general cases where the kinetic energy term is represented by linear combination of infinite number of possible orderings. We illustrate the method with an example, namely Mathews-Lakshmanan (ML) oscillator. Our results show that the latter system is shape invariant for all possible orderings. We derive eigenvalues and eigenvectors of this nonlinear oscillator for all possible orderings including Hermitian and non-Hermitian ones.

Predicting Babbitt's orderings of time-point classes from array aggregate partitions in None but the Lonely Flute and Around the Horn

Milton Babbitt (1916–2011) is credited with developing several techniques of 12-tone composition that extend beyond pitch. One such technique, the time-point system, figures prominently in his mature rhythmic practice. In many of his works based on the all-partition array, the possible orderings of pitch classes and time-point classes from an aggregate partition are the same. However, the number available is often large and his reasons for choosing one ordering over another are diverse and not clearly understood. In this article, we propose that, when constructing linear orderings of time-point classes from aggregate partitions in an all-partition array, Babbitt attempted to minimize both (1) their dissimilarity to the orderings of pitch classes from the same aggregate partitions and (2) the amount of counter-evidence they provide against a preselected beat. We first review two existing measures, Rothgeb's dissimilarity measure using order inversions and Povel and Essens' clock induction model. We then present a way to determine the exact number of possible orderings of time-point classes (and pitch classes) without repetitions in a particular aggregate partition. Next, we introduce a novel heuristic, based on the aforementioned measures, for predicting from the available orderings of time-point classes those particular orderings chosen by Babbitt. We conclude by evaluating how well this heuristic predicts the orderings of time-point classes found in two of Babbitt's works, None but the Lonely Flute (1991) and Around the Horn (1993).

Optimizing Wait States in the Synthesis of Memory References with Unpredictable Latencies

We consider the problem of synthesizing circuits (from C to Verilog) that are optimized to handle unpredictable latencies of memory operations. Unpredictable memory latencies can occur due to the use of on chip caches, DRAM memory modules, buffers/queues, or multiport memories. Typically, high-level synthesis compilers assume fixed and known memory latencies, and thus are able to schedule the code’s operations efficiently. The operations in the source code are scheduled into states of a state machine whose states will be synthesized to Verilog. The goal is to minimize scheduling length by maximizing the number of operations (and in particular memory operations) that are executed in parallel at the same state. However, with unpredictable latencies, there can be an exponential number of possible orders in which these parallel memory operations can terminate. Thus, in order to minimize the scheduling, we need a different schedule for any such order. This is not practical, and we show a technique of synthesizing a compact state machine that schedules only a small subset of these possible termination orders. Our results show that this compact state machine can improve the execution time compared to a regular scheduling that waits for the termination of all the active memory references in every state.

A bottom-up complement to the logic-based top-down approach to the story arrangement test

Psychometric AI is a type of AI distinguished by the pursuit of intelligent systems able to excel on psychometrically validated human-level tests of cognitive abilities. We seek to build a system that solves a specific sub-test within Psychometric AI: the story arrangment test. Items in this test confront the test-taker with a set of jumbed snapshots (whether diagrammatic or otherwise) which must be ordered to tell a coherent story. We propose a dual-process system that combines bottom-up non- or sub-symbolic processing (e.g. neural network-based modelling) with top-down symbolic processing (e.g. deductive reasoning over declarative information represented as formulae in a logical system) for solving these tests of cognitive ability. The top-down process provides the benefits of a traceable proof, but requires a large amount of pre-existing knowledge. The bottom-up technique sacrifices provability and certainty on some problems for speed, but always yields some level of an answer to a given problem. This demonstrates a natural marriage between the two: the bottom-up approach seems especially powerful when used as a form of pre-processing in conjunction with a logic-based approach, because the latter approach would only need to consider a small number of possible orderings of snapshots.

Dose-finding design for multi-drug combinations

Most of the current designs used for Phase I dose finding trials in oncology will either involve only a single cytotoxic agent or will impose some implicit ordering among the doses. The goal of the studies is to estimate the maximum tolerated dose (MTD), the highest dose that can be administered with an acceptable level of toxicity. A key working assumption of these methods is the monotonicity of the dose-toxicity curve. Here we consider situations in which the monotonicity assumption may fail. These studies are becoming increasingly common in practice, most notably, in phase I trials that involve combinations of agents. Our focus is on studies where there exist pairs of treatment combinations for which the ordering of the probabilities of a dose-limiting toxicity cannot be known a priori. We describe a new dose-finding design which can be used for multiple-drug trials and can be applied to this kind of problem. Our methods proceed by laying out all possible orderings of toxicity probabilities that are consistent with the known orderings among treatment combinations and allowing the continual reassessment method (CRM) to provide efficient estimates of the MTD within these orders. The design can be seen to simplify to the CRM when the full ordering is known. We study the properties of the design via simulations that provide comparisons to the Bayesian approach to partial orders (POCRM) of Wages, Conaway, and O'Quigley. The POCRM was shown to perform well when compared to other suggested methods for partial orders. Therefore, we comapre our approach to it in order to assess the performance of the new design. A limitation concerns the number of possible orders. There are dose-finding studies with combinations of agents that can lead to a large number of possible orders. In this case, it may not be feasible to work with all possible orders. The proposed design demonstrates the ability to effectively estimate MTD combinations in partially ordered dosefinding studies. Because it relaxes the monotonicity assumption, it can be considered a multivariate generalization of the CRM. Hence, it can serve as a link between single and multiple-agent dosefinding trials.

A Novel Method for Estimating Linkage Maps

The goal of linkage mapping is to find the true order of loci from a chromosome. Since the number of possible orders is large even for a modest number of loci, the problem of finding the optimal solution is known as a NP-hard problem or traveling salesman problem (TSP). Although a number of algorithms are available, many either are low in the accuracy of recovering the true order of loci or require tremendous amounts of computational resources, thus making them difficult to use for reconstructing a large-scale map. We developed in this article a novel method called unidirectional growth (UG) to help solve this problem. The UG algorithm sequentially constructs the linkage map on the basis of novel results about additive distance. It not only is fast but also has a very high accuracy in recovering the true order of loci according to our simulation studies. Since the UG method requires n-1 cycles to estimate the ordering of n loci, it is particularly useful for estimating linkage maps consisting of hundreds or even thousands of linked codominant loci on a chromosome.

AntMap: Constructing Genetic Linkage Maps Using an Ant Colony Optimization Algorithm

High-throughput assay of molecular markers enables to utilize large amounts of markers in linkage mapping. To incorporate numerous markers into a linkage map, the development of a highly efficient method for linkage mapping is indispensable. When the number of loci is large, locus ordering is a major difficulty in linkage mapping, since the number of possible orders becomes very large. To address this problem, we developed a new algorithm for locus ordering and used it in a newly developed computer program called AntMap. The algorithm is based on ant colony optimization, which is a set of metaheuristic algorithms inspired by the cooperative behavior of real ants in finding the shortest path from their nest to a food source. Using this algorithm, AntMap seeks the linear order of loci that minimizes the sum of adjacent recombination fractions or that maximizes the log likelihood of locus order. Analyses based on simulated data sets indicated that our algorithm displayed a high efficiency level. The high performance of the algorithm enabled to save time and labor, and also to validate an estimated order by bootstrap tests. Our algorithm and AntMap should enable to construct high-throughput systems for linkage mapping. AntMap is available under a GNU general public license at http://cse.naro.affrc.go.jp/iwatah/antmap/index.html. Source codes and executables of AntMap can be obtained there.

Finite Groups of Matrices Whose Entries Are Integers

Finite groups of matrices appear early as examples in a first course in abstract algebra, and most of the time these examples are given with integral entries. While these groups provide a setting in which to illustrate new concepts and to pose problems, they also have surprising and beautiful properties. For example, Minkowski proved the unexpected result that GL(n, Z), the group of n x n matrices having inverses whose entries are also integers, has only finitely many isomorphism classes of finite subgroups. As a consequence, there are only finitely many possible orders for elements of GL(n, Z); fortunately, the possible orders can be determined using linear algebra. In general, the number of possible orders increases as n increases, but even here we have the surprising result that no new possible orders are obtained when going from GL(2k, Z) to GL(2k + 1, 7). This paper is an exposition of these and other related results and questions. Although finite groups of integral matrices have a long history, most of the beautiful theorems concerning them do not appear in texts and are not known to many algebraists (for example, the first named author). While still an area of active research (see [7], [12], [23], and [24]), major parts are accessible to students; indeed, this paper has its origin in a term paper written by the second author for a beginning modem algebra course that used [9] as a text. One of our goals for this paper is to be a source of problems, readings, and projects for other students; the topic seems ideal since it synthesizes results from number theory, algebra, and linear algebra, gives a natural and somewhat gentle introduction to deeper subjects not yet encountered by a beginning algebra student, has historical features, and could be a subject of continuing student research. With this goal in mind, the authors have included exercises and have tried to organize the paper so that parts can be read independently, or given to students in outline form. Our second goal is to give the reader examples of current research in the area.

Avoiding Cartesian products for multiple joins

Computing the natural join of a set of relations is an important operation in relational database systems. The ordering of joins determines to a large extent the computation time of the join. Since the number of possible orderings could be very large, query optimizers first reduce the search space by using various heuristics and then try to select an optimal ordering of joins. Avoiding Cartesian products is a common heuristic for reducing the search space, but it cannot guarantee optimal ordering in its search space, because the cheapest Cartesian-product-free (CPF for short) ordering could be significantly worse than an optimal non-CPF ordering by a factor of an arbitrarily large number. In this paper, we use programs consisting of joins, semijoins, and projections for computing the join of some relations, and we introduce a novel algorithm that derives programs from CPF orderings of joins. We show that there exists a CPF ordering from which our algorithm derives a program whose cost is within a constant factor of the cost of an optimal ordering. Thus, our result demonstrates the effectiveness of avoiding Cartesian products as a heuristic for restricting the search space of orderings of joins.

The order of points on the second convex hull of a simple polygon

Given a setV of points on the plane, if {q 1,...,q n } is the set of points on the second convex hull ofV, the order in which these points are visited in anyV-gon is characterized. This order must verify two similar conditions to those of Kuratowski's theorem for planar graphs. Moreover, the number of possible orders that verify these conditions is obtained. It isO(5 n ).

Selecting varieties using a series of trials and a combinatorial ordering method

Summary — The interpretation of results from a series of experiments designed to evaluate varietal performance is frequently difficult due to the large interaction observed between varieties and location sites. In fact interactions are only relevant if a great variability in the varietal ranking is observed from one environment to another. However, classical statistical approaches do not take this fact into account, and so we have developed new strategies to establish median orders on the variety set. This raises the need for mathematical tools based on the search for total orders on the whole set of varieties. However, this search requires a large numerical (computer) effort and is impracticable because of the large number of possible orders. Thus, we propose a combinatorial method derived from social choice theory which gives orders minimizing a dissimilarity function between overall orderings and the observed orders in the individual trials. The proposed method is illustrated using an example of 14 rapeseed varieties grown in 60 trials. Its merits and limitations are discussed.