Time evolution of quantum spin systems with Qiskit

Faculty Commons material developed by Jarrett Lancaster - Published February 13, 2023

The attached Jupyter notebook contains a set of tutorials for performing time evolution in spin-$\frac{1}{2}$ systems using the [Qiskit](https://qiskit.org/) software development kit. Parts of this were used as lab activities in a junior/senior undergraduate course on quantum mechanics, and other parts have been used to introduce undergraduate research students to using quantum computers to investigate dynamics. Conceptual simplicity is stressed, so only the simplest version of Trotter decomposition is introduced and employed. To make notebook this self-contained, the first section contains some small overlap with a previous Faculty Commons [submission](https://www.compadre.org/PICUP/exercises/Exercise.cfm?I=1531). The dynamics of single spins in a static magnetic field is reviewed. Time-dependent systems are introduced with the dynamics of a single spin in a time-varying magnetic field. The majority of this notebook concerns time evolution of multiple spins with Heisenberg-like interactions. A two-spin system is simulated exactly with the full time-evolution operator represented in terms of single-qubit and two-qubit gates. The Trotter decomposition is introduced as an approximation scheme in the simple context of a single spin system and then used to calculate the dynamical observables (e.g., time-dependent spin component expectation values) in a three spin system. Lastly, it is shown how to simulate larger ($N = \mathcal{O}(10)$) systems with only minor modifications to the $N=3$ circuit. These materials were designed for students who have worked through the previous introduction and have a working knowledge of using Jupyter notebooks for scientific computation (numpy, scipy, etc.).

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