Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

In this paper, the stability of a waveboard, a human propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels, is analysed. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a Differential-Algebraic system of equations (DAE system), are linearized along the steady forward motion resorting to a recently validated linearization procedure, which allows the maximum possible reduction of the linearized equations of motion of constrained multibody systems. The approach enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of spurious null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

Correction: Lehtonen et al. The Potentials of Tangible Technologies for Learning Linear Equations. Multimodal Technol. Interact. 2020, 4, 77

In the original publication, there was a mistake in Table 3 as published [...]

Exploring students’ procedural flexibility in three countries

Background In this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies. Results Our results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees. Conclusions Although it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning.

Pembelajaran Kooperatif Tipe STAD untuk Meningkatkan Aktivitas dan Hasil Belajar Matematika dengan Alat Peraga Gelas Variabel bagi Siswa SMPN 3 Kayangan

This study aims to improve student activity and learning outcomes on the subject matter of linear equations with one variable with the help of variable glass props through STAD Type Cooperative learning for students of SMPN 3 Kayangan, North Lombok Regency. This research method uses classroom action research with qualitative and quantitative approaches, which are carried out in 2 cycles. Each cycle consists of stages of implementation, giving action, observation, and reflection. The subjects of this study were students of class VII.2 SMPN 3 Kayangan, totaling 33 students. Data collection techniques used observation sheets and tests were then analyzed descriptively qualitatively and quantitatively. The results of this study indicate that the increase in student learning outcomes in the first cycle obtained mastery learning reached 54.55% with an average value of 64.45. In the second cycle, the learning completeness was obtained by 66.7% with an average value of 66.6. From the results of this study, it was concluded that the STAD Type Cooperative approach learning could increase the activity and learning outcomes of class VII.2 students of SMP Negeri 3 Kayangan, North Lombok.

Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

QUBO Formulations for a System of Linear Equations

With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all of science and engineering. Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on the gate model quantum computers, was invented and several advanced variations have been developed. For a given square matrix A∈R(n×n) and a vector b∈R(n), we will find unconstrained binary optimization (QUBO) models for a vector x∈R(n) that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate those QUBO models on the D-Wave system and discuss the results. For a simple system, We provide a python code to calculate the matrix characterizing the relationship between the variables and to print the test code that can be used directly in D-Wave system.

Penerapan Metode E-Tutor Sebaya dalam Pembelajaran Matematika di Masa Pandemi Covid-19

Learning mathematics during the Covid-19 pandemic required a lot of adjustments, in terms of materials, methods, media, strategies, and so on. This paper describes the results of one best practice in learning mathematics by implementing e-peer tutoring method as a form of distance learning during the Covid-19 pandemic. The aim of this writing is to describe the implementation of e-peer tutoring method in learning mathematics, especially in the material of Linear Equations and Inequalities of One Variable for grade VII Junior High School 1 Wonosari, Gunungkidul. The stages of this method includes the preparation, implementation, and evaluation. Analysis of initial ability, tutor selection, group formation, and preparation of learning program were carried out in the preparation stage. As for the activities at the implementation stage, the teacher uploads material and discussion sheets via Google Classroom, students discuss in their groups through Whatsapp group with the help of the tutor, students send assignments online, and the tutor reports on discussion activities. At the evaluation stage, students take test and fill out the questionnaires. The results at the evaluation stage indicate that this method can work well and has a positive impact on cognitive and affective aspects. In terms of cognitive aspects, the average score and percentage of students’ mastery increased compared to the previous basic competencies. In terms of affective aspect, this method has a positive impact on students’ motivation and self-confidence.