Unraveling Rodeo Algorithm Through the Zeeman Model

TL;DR

This paper presents a novel approach to determine eigenstates and eigenvalues of Hamiltonians using the Rodeo Algorithm, leveraging quantum computing platforms and optimizing performance for single and multi-qubit systems.

Contribution

It introduces a new methodology for the Rodeo Algorithm that does not require prior eigenstate knowledge and extends it to bipartite systems with degeneracy and entanglement.

Findings

Optimized parameters improve algorithm performance.

Successful application to Zeeman model scenarios.

Effective execution on real superconducting quantum devices.

Abstract

We unravel the Rodeo Algorithm to determine the eigenstates and eigenvalues spectrum for a general Hamiltonian considering arbitrary initial states. By presenting a novel methodology, we detail the original method and show how to define all properties without having prior knowledge regarding the eigenstates. To this end, we exploit Pennylane and Qiskit platforms resources to analyze scenarios where the Hamiltonians are described by the Zeeman model for one and two spins. We also introduce strategies and techniques to improve the algorithm's performance by adjusting its intrinsic parameters and reducing the fluctuations inherent to data distribution. First, we explore the dynamics of a single qubit on Xanadu simulators to set the parameters that optimize the method performance and select the best strategies to execute the algorithm. On the sequence, we extend the methodology for bipartite systems to discuss how the algorithm works when degeneracy and entanglement are taken into account. Finally, we compare the predictions with the results obtained on a real superconducting device provided by the IBM Q Experience program, establishing the conditions to increase the protocol efficiency for multi-qubit systems.