Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top

Lukas M Sieberer,Markus Heyl,Peter Zoller,Tobias Olsacher,Philipp Hauke,Fritz Haake,Andreas Elben

doi:10.1038/s41534-019-0192-5

Abstract

This work aims at giving Trotter errors in digital quantum simulation (DQS) of collective spin systems an interpretation in terms of quantum chaos of the kicked top. In particular, for DQS of such systems, regular dynamics of the kicked top ensures convergence of the Trotterized time evolution, while chaos in the top, which sets in above a sharp threshold value of the Trotter step size, corresponds to the proliferation of Trotter errors. We show the possibility to analyze this phenomenology in a wide variety of experimental realizations of the kicked top, ranging from single atomic spins to trapped-ion quantum simulators which implement DQS of all-to-all interacting spin-1/2 systems. These platforms thus enable in-depth studies of Trotter errors and their relation to signatures of quantum chaos, including the growth of out-of-time-ordered correlators.

Highlights

In digital quantum simulation (DQS), unitary Hamiltonian evolution is decomposed into a sequence of quantum gates
We consider two measures of Trotter errors: deviations of the expectation values of physical observables[13] and the fidelity of the quantum state obtained in DQS
Interpretation of the proliferation of Trotter errors as quantum chaos As we show in the following, the threshold behavior in the magnetization error and the simulation accuracy can be traced back to the transition from dynamical localization to quantum chaos in the kicked top

Summary

Introduction

In digital quantum simulation (DQS), unitary Hamiltonian evolution is decomposed into a sequence of quantum gates. A common approach to achieve this decomposition utilizes Suzuki-Trotter formulas[1,2] to approximately factorize the time evolution operator.[3,4,5,6,7,8,9,10,11,12] It is a fundamental conceptual question under which conditions this “Trotterization” is a controlled approximation. A recent numerical study[13] by some of us found that Trotter errors in DQS of generic many-body systems remain bounded below and proliferate above a dynamical transition to many-body quantum chaos.[14] Motivated by these findings we revisit the kicked top, a paradigmatic model of single-body quantum chaos.[15] Resorting to this well-studied model system allows us to gain insights into

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Results

Conclusion