Exact Trotterization in the Degenerate Quantum Rabi Model
Abstract
Digital quantum simulation of the quantum Rabi model in the deep strong coupling regime motivates the need for an exact characterization of Trotterization errors, so far known only through qualitative bounds. Here we derive such a characterization in the exactly solvable degenerate limit, valid for arbitrary coupling strength. The first-order Trotter operator admits a closed expression, showing that the generated phase-space trajectory lies exactly on a circle that is rescaled and rigidly rotated with respect to the exact one, rather than merely approximating it. This geometric renormalization yields a closed form for the state fidelity, structurally identical to the physical survival probability governing revivals in the deep strong coupling regime: both are the same geometric question -- the distinguishability of a displaced Fock state -- applied to two different pairs of points in phase space. We further show that this geometric error is exactly correctable by promoting the Trotter-step coupling to a complex amplitude, at no additional circuit cost. These results, obtained in the exactly solvable degenerate limit, provide a benchmark for Trotterization errors in the general Rabi model.
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BibTeX
@misc{lizuain2026exact,
title = {{Exact Trotterization in the Degenerate Quantum Rabi Model}},
author = {I. Lizuain and A. Rodriguez-Prieto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36861},
eprint = {2609.36861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36861},
}
RIS
TY - UNPB
TI - Exact Trotterization in the Degenerate Quantum Rabi Model
AU - I. Lizuain
AU - A. Rodriguez-Prieto
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.36861
UR - https://arxiv.org/abs/2609.36861
AB - Digital quantum simulation of the quantum Rabi model in the deep strong coupling regime motivates the need for an exact characterization of Trotterization errors, so far known only through qualitative bounds. Here we derive such a characterization in the exactly solvable degenerate limit, valid for arbitrary coupling strength. The first-order Trotter operator admits a closed expression, showing that the generated phase-space trajectory lies exactly on a circle that is rescaled and rigidly rotated with respect to the exact one, rather than merely approximating it. This geometric renormalization yields a closed form for the state fidelity, structurally identical to the physical survival probability governing revivals in the deep strong coupling regime: both are the same geometric question -- the distinguishability of a displaced Fock state -- applied to two different pairs of points in phase space. We further show that this geometric error is exactly correctable by promoting the Trotter-step coupling to a complex amplitude, at no additional circuit cost. These results, obtained in the exactly solvable degenerate limit, provide a benchmark for Trotterization errors in the general Rabi model.
ER -