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Journal of Experimental and Theoretical Physics Letters

Solution of the Schrödinger Equation on a Quantum Computer by the Zalka–Wiesner Method Including Quantum Noise

Yu. I. Bogdanov, Nadezhda A. Bogdanova, Dmitriy V. Fastovets, Vladimir F. Lukichev

Published Sep 1, 2021• 6 citations
Publisher

Abstract

The simulation of quantum systems on a quantum computer using the Zalka–Wiesner algorithm including quantum noise has been considered. The efficiency of developed methods and algorithms has been demonstrated by example of the solution of the time-dependent Schrödinger equation for a particle in the Pöschl–Teller potential. The developed analytical theory of the effect of quantum noise on the accuracy of simulation has been compared with Monte Carlo simulations. The accuracy of the solution of the Schrödinger equation for a many-electron system has been forecasted as a function of the number of electrons and for different levels of noise.

DOI
10.1134/s0021364021180065

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The simulation of quantum systems on a quantum computer using the Zalka–Wiesner algorithm including quantum noise has been considered. The efficiency of developed methods and algorithms has been demonstrated by example of the solution of the time-dependent Schrödinger equation for a particle in the Pöschl–Teller potential. The developed analytical theory of the effect of quantum noise on the accuracy of simulation has been compared with Monte Carlo simulations. The accuracy of the solution of the Schrödinger equation for a many-electron system has been forecasted as a function of the number of electrons and for different levels of noise.
The study of the effect of quantum noise on the accuracy of modeling quantum systems on a quantum computer using the Zalka-Wiesner method is carried out. The efficiency of the developed methods and algorithms is demonstrated by the example of solving the nonstationary Schrödinger equation with allowance for quantum noise for a particle moving in the Pöschl–Teller potential. The analysis and prediction of accuracy for the Zalka-Wiesner method are carried out taking into account the complexity of the quantum system and the strength of quantum noise. In our opinion, the considered problem provides a useful test for assessing the quality and efficiency of quantum computing devices on various physical platforms currently being developed. The obtained results are essential for the development of high-precision methods for controlling the technologies of quantum computations.
The study of the effect of quantum noise on the accuracy of modeling quantum systems on a quantum computer using the Zalka-Wiesner method is carried out. The efficiency of the developed methods and algorithms is demonstrated by the example of solving the nonstationary Schrodinger equation with allowance for quantum noise for a particle moving in the Poschl-Teller potential. The analysis and prediction of accuracy for the Zalka-Wiesner method are carried out taking into account the complexity of the quantum system and the strength of quantum noise. In our opinion, the considered problem provides a useful test for assessing the quality and efficiency of quantum computing devices on various physical platforms currently being developed. The obtained results are essential for the development of high-precision methods for controlling the technologies of quantum computations.
We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.
Quantum computers could potentially simulate the dynamics of systems such as polyatomic molecules on a much larger scale than classical computers. One of the central unresolved problems in developing quantum simulation algorithms has been constructing the necessary time evolution operators in enough detail for implementation in a quantum computer. This work presents a complete and very general solution to this problem: namely, a quantum computational algorithm for simulating an arbitrary Coulombic many-body system acting under the three-dimensional Schrödinger equation. We use the simplest discretized model and discuss detailed constructions of the operators involved. The algorithm is also implemented in MAT-LAB and tested against both analytical solutions and classical methods. Our algorithm can determine an arbitrary system’s dynamics rather than merely associated constants, as investigated in previous literature. It can hence be adapted for electronic structure calculations, chemical reaction simulations, etc. 1
This paper presents the numerical and variational solutions of the 1D Schrödinger Equation submitted to the Pöschl-Teller potential. The methods used were the Variational Method and the Finite Difference Method. They were presented in a didactic and detailed way with the purpose of instructing both undergraduate and graduate students, about the applicability and effectiveness of the aforementioned methods. We use the Pöschl-Teller potential due to the fact that it is little explored in the books of Quantum Mechanics used in undergraduation courses and also because of its diverse applications, such as in Bose-Einstein condensates, waveguides, topological defects in field theory and so on. We conclude this paper comparing the variational and numerical solutions with the analytical solution and present the advantages of each method.