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EPJ Quantum Technology

Application of the QDST algorithm for the Schrödinger particle simulation in the infinite potential well

Marcin Ostrowski

Published Feb 22, 2024• 1 citation
Publisher

Abstract

Abstract This paper examines whether a quantum computer can efficiently simulate the time evolution of the Schrödinger particle in a one-dimensional infinite potential well. In order to solve the Schrödinger equation in the quantum register, an algorithm based on the Quantum Discrete Sine Transform (QDST) is applied. The paper compares the results obtained in this way with the results given by the previous method (based on the QFT algorithm).

DOI
10.1140/epjqt/s40507-024-00223-3

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Abstract This paper examines whether a quantum computer can efficiently simulate the time evolution of the Schrödinger particle in a one-dimensional infinite potential well. In order to solve the Schrödinger equation in the quantum register, an algorithm based on the Quantum Discrete Sine Transform (QDST) is applied. The paper compares the results obtained in this way with the results given by the previous method (based on the QFT algorithm).
This paper examines whether a quantum computer can efficiently simulate the evolution of the Schrödinger particle in a one-dimensional infinite potential well.The present study involves simulation of the change of the particle's state due to instant enlarging of the well's size.In order to solve the Schrödinger equation in the quantum register, an algorithm based on the Quantum Fourier Transform is applied.The paper presents an algorithm for simulating time evolution of such a system, implemented on standard two-input gates.
In this paper we examine whether a quantum computer can efficiently simulate time evolution of a one dimensional Schrödinger particle. Two cases are considered: free particle and particle dispersed on a rectangular potential. In order to simulate the Schrödinger particle, we use a quantum algorithm based on the procedure of diagonalisation of time evolution operator. The procedure of diagonalisation is based on the Quantum Fourier Transform (QFT) algorithm. The effects of simulation are presented in the form of figures. We also compare the results obtained from quantum algorithm with the results of classical simulations (Cayley's method).
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
This paper investigates whether a quantum computer can efficiently simulate the non-elastic scattering of the Schrödinger particle on a stationary excitable shield. The return of the shield to the ground state is caused by photon emission. An algorithm is presented for simulating the time evolution of such a process, implemented on standard two-input gates. The algorithm is used for the computation of elastic and non-elastic scattering probabilities. The results obtained by our algorithm are compared with those obtained using the standard Cayley’s method.
This paper adds up how the matrix method can be used for solving the one-dimensional time-independent Schrodinger's equation for some specific potential energy variation like half-infinite potential well.The matrix method is illustrated to obtain solution of the time-independent Schrodinger's equation for half-infinite potential well, which is generally done by ordinary algebraic and analytical methods.The transcendental equation determining the discrete eigenvalues for bound state and the corresponding eigenwave functions are obtained by the time-independent Schrodinger's equation for half-infinite potential well via the application of matrix method.