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Simulating a Quantum Harmonic Oscillator by Introducing it to a Bosonic System

Rajdeep Tah

Published Aug 23, 2020• 1 citation
Publisher

Abstract

The goal of this paper is to associate a Quantum Harmonic Oscillator to a Bosonic System, try to simulate it in IBMQ-Experience (at 8192 shots) and further study it. We associated the concept of Pauli Matrix equivalent to Bosonic Particles and used it to calculate the Unitary Operators which helped us to theoretically visualize each Quantum states and further simulate our system.

DOI
10.35543/osf.io/xh8wf

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The goal of this paper is to associate a Quantum Harmonic Oscillator to a Bosonic System, try to simulate it in IBMQ-Experience (at 8192 shots) and further study it. We associated the concept of Pauli Matrix equivalent to Bosonic Particles and used it to calculate the Unitary Operators which helped us to theoretically visualize each Quantum states and further simulate our system.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
Bosonic quantum devices offer a novel approach to realize quantum computations, where the quantum two-level system ( qubit ) is replaced with the quantum (an)harmonic oscillator ( qumode ) as the fundamental building block of the quantum simulator. The simulation of chemical structure and dynamics can then be achieved by representing or mapping the system Hamiltonians in terms of bosonic operators. In this Perspective, we review recent progress and future potential of using bosonic quantum devices for addressing a wide range of challenging chemical problems, including the calculation of molecular vibronic spectra, the simulation of gas-phase and solution-phase adiabatic and nonadiabatic chemical dynamics, the efficient solution of molecular graph theory problems, and the calculations of electronic structure.
A bstract According to the pioneering work of Nielsen and collaborators, the length of the minimal geodesic in a geometric realization of a suitable operator space provides a measure of the quantum complexity of an operation. Compared with the original concept of complexity based on the minimal number of gates required to construct the desired operation as a product, this geometrical approach amounts to a more concrete and computable definition, but its evaluation is nontrivial in systems with a high-dimensional Hilbert space. The geometrical formulation can more easily be evaluated by considering the geometry associated with a suitable finite-dimensional group generated by a small number of relevant operators of the system. In this way, the method has been applied in particular to the harmonic oscillator, which is also of interest in the present paper. However, subtle and previously unrecognized issues of group theory can lead to unforeseen complications, motivating a new formulation that remains on the level of the underlying Lie algebras for most of the required steps. Novel insights about complexity can thereby be found in a low-dimensional setting, with the potential of systematic extensions to higher dimensions as well as interactions. Specific examples include the quantum complexity of various target unitary operators associated with a harmonic oscillator, inverted harmonic oscillator, and coupled harmonic oscillators. The generality of this approach is demonstrated by an application to an anharmonic oscillator with a cubic term.