Integrable Digital Quantum Simulation: Generalized Gibbs Ensembles and Trotter Transitions

TL;DR

This paper investigates how integrable quantum systems simulated digitally behave differently from non-integrable ones, revealing a novel Trotter transition characterized by abrupt changes in the generalized Gibbs ensemble and local observables.

Contribution

It demonstrates that integrable digital quantum simulations exhibit a distinct Trotter transition with bounded errors, contrasting with chaotic systems, and introduces the concept of discrete GGE as a new nonequilibrium state.

Findings

Bounded discretization errors for small Trotter steps.

Abrupt change in the discrete GGE at a threshold Trotter step.

Emergence of non-zero staggered magnetization at the transition.

Abstract

The Trotter-Suzuki decomposition is a promising avenue for digital quantum simulation (DQS), approximating continuous-time dynamics by discrete Trotter steps of duration $\tau$. Recent work suggested that DQS is typically characterized by a sharp Trotter transition: when $\tau$ is increased beyond a threshold value, approximation errors become uncontrolled at large times due to the onset of quantum chaos. Here we contrast this picture with the case of \emph{integrable} DQS. We focus on a simple quench from a spin-wave state in the prototypical XXZ Heisenberg spin chain, and study its integrable Trotterized evolution as a function of $\tau$. Due to its exact local conservation laws, the system does not heat up to infinite temperature and the late-time properties of the dynamics are captured by a discrete Generalized Gibbs Ensemble (dGGE). By means of exact calculations we find that, for small $\tau$, the dGGE depends analytically on the Trotter step, implying that discretization errors remain bounded even at infinite times. Conversely, the dGGE changes abruptly at a threshold value $\tau_{\rm th}$, signaling a novel type of Trotter transition. We show that the latter can be detected locally, as it is associated with the appearance of a non-zero staggered magnetization with a subtle dependence on $\tau$. We highlight the differences between continuous and discrete GGEs, suggesting the latter as novel interesting nonequilibrium states exclusive to digital platforms.