A Readout-Aware Theory of Approximate Heisenberg Propagation
Abstract
Classical methods for computing quantum expectation values often propagate observables in the Heisenberg picture. To remain tractable, these methods rely on approximations that discard operator components during evolution. Yet the magnitude of these components alone does not determine the resulting expectation-value error, which also depends on their subsequent evolution and the input state. We develop \emph{generalized Pauli propagation}, a common algebraic framework encompassing Pauli propagation, Lie-algebraic simulation, and Krylov subspace methods. Using weighted commutator graphs, we derive exact identities that relate approximation errors to the signed contributions of omitted and discarded operator components. Our central result establishes \emph{delayed observability}, identifying conditions under which these contributions vanish at low orders in propagation time. We use this structure to develop finite-order error estimates and state-dependent selection rules. We further prove that retaining the symmetry-preserving part of a weak perturbation eliminates first-order expectation-value errors under suitable symmetry conditions. By revealing how approximation errors affect the desired expectation value, our framework provides a basis for more informative error analysis and the development of readout-aware simulation strategies.
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BibTeX
@misc{bhandari2026readoutaware,
title = {{A Readout-Aware Theory of Approximate Heisenberg Propagation}},
author = {Yubraj Bhandari and Jakob S. Kottmann and Adelina Bärligea},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26222},
eprint = {2610.26222},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26222},
}
RIS
TY - UNPB
TI - A Readout-Aware Theory of Approximate Heisenberg Propagation
AU - Yubraj Bhandari
AU - Jakob S. Kottmann
AU - Adelina Bärligea
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26222
UR - https://arxiv.org/abs/2610.26222
AB - Classical methods for computing quantum expectation values often propagate observables in the Heisenberg picture. To remain tractable, these methods rely on approximations that discard operator components during evolution. Yet the magnitude of these components alone does not determine the resulting expectation-value error, which also depends on their subsequent evolution and the input state. We develop \emph{generalized Pauli propagation}, a common algebraic framework encompassing Pauli propagation, Lie-algebraic simulation, and Krylov subspace methods. Using weighted commutator graphs, we derive exact identities that relate approximation errors to the signed contributions of omitted and discarded operator components. Our central result establishes \emph{delayed observability}, identifying conditions under which these contributions vanish at low orders in propagation time. We use this structure to develop finite-order error estimates and state-dependent selection rules. We further prove that retaining the symmetry-preserving part of a weak perturbation eliminates first-order expectation-value errors under suitable symmetry conditions. By revealing how approximation errors affect the desired expectation value, our framework provides a basis for more informative error analysis and the development of readout-aware simulation strategies.
ER -