Complexity of Quantum Harmonic Oscillator in External Magnetic Field

TL;DR

This paper explores how an external magnetic field affects the circuit complexity of a quantum harmonic oscillator, revealing temperature and magnetic field-dependent behaviors and confirming the Lloyd bound on complexity growth.

Contribution

It introduces a novel analysis of quantum harmonic oscillator complexity under magnetic fields using Nielsen's approach within the TFD framework.

Findings

Complexity oscillates more with higher temperature.

Low temperatures lead to constant complexity.

Strong magnetic fields cause periodic oscillations.

Abstract

In this paper, we investigate the circuit complexity of a quantum harmonic oscillator subjected to an external magnetic field. Utilizing the Nielsen approach within the thermofield dynamics (TFD) framework, we determine the complexity of thermofield double states as functions of time, temperature, and the external magnetic field. Our subsequent analysis reveals various features of this complexity. For instance, as temperature increases, the amplitude of complexity oscillations also rises, while at low temperatures, complexity stabilizes at a constant positive value. Furthermore, the magnetic field creates two distinct sectors: strong magnetic fields exhibit periodic complexity oscillations, whereas weak magnetic fields induce a beating effect. Finally, we confirm that the rate of complexity obeys the Lloyd bound.