Trotter transition in BCS pairing dynamics

TL;DR

This paper investigates the transition from weak to strong chaos in Trotterized BCS pairing dynamics, revealing a critical Trotter step size and deriving scaling laws for chaos indicators, with implications for quantum computing.

Contribution

It identifies a Trotter transition in BCS dynamics and derives scaling laws for chaos measures, connecting quantum chaos with Trotterization parameters.

Findings

A Trotter transition occurs at a finite step size  au_c  \, \, \\sqrt{N}

Different scaling laws are derived for weak and strong chaos regimes

Large  au  \, \, \\gg au_c  \, \, \\) match the kicked top map predictions

Abstract

We study universal aspects of thermalization induced by Trotterization, a procedure routinely used in gate-based quantum computation. We use the reduced-BCS model -- quantum integrable with a classically integrable mean-field limit -- where the effects of Trotter chaos are expected to be particularly stark. The resulting Trotterized chaotic dynamics is characterized by its Lyapunov spectrum and rescaled Kolmogorov-Sinai entropy. The chaos quantifiers depend on the Trotterization time step $\tau$. We observe a Trotter transition at a finite step value $\tau_c \approx \sqrt{N}$. While the dynamics is weakly chaotic for time steps $\tau \ll \tau_c$, the regime of large Trotterization steps is characterized by short temporal correlations. We derive two different scaling laws for the two different regimes by numerically fitting the maximum Lyapunov exponent data. The scaling law of the large \(\tau\) limit agrees well with the one derived from the kicked top map. Beyond its relevance to current quantum computers, our work opens new directions -- such as probing observables like the Loschmidt echo, which lie beyond standard mean-field description -- across the Trotter transition we uncover.