Multi-product Zeno effect achieving higher order convergence rates

Tim Möbus

doi:10.22331/q-2026-06-30-2148

Multi-product Zeno effect achieving higher order convergence rates

Tim Möbus

Department of Mathematics, University of Tübingen, Tübingen, Germany
Department of Mathematics, Technical University of Munich, Munich, Germany

Published:	2026-06-30, volume 10, page 2148
Editor:	Di Fang
Eprint:	arXiv:2410.16260v3
Doi:	https://doi.org/10.22331/q-2026-06-30-2148
Citation:	Quantum 10, 2148 (2026).

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Abstract

The quantum Zeno effect is a fundamental mechanism for implementing the effective dynamics of projected Hamiltonian and Lindbladian systems. It approximates the target projected evolution by interleaving Hamiltonian or Lindblad dynamics with quantum operations associated with the desired subspace. In contrast to the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is typically limited to order $1/n$. In this work, we improve this convergence rate by employing a multi-product formula, thereby achieving arbitrarily high-order convergence of the form $1/n^{K+1}$. This yields an improved approximation scheme for Zeno-like expectation values via an efficient post-processing method. The approach combines a modified Chernoff lemma, an adapted Dunford-Segal approximation, holomorphic functional calculus, and Chebyshev interpolation. We illustrate the method with the bosonic cat code and also consider the broader class of systems governed by the Bang-Bang decoupling method.

Popular summary

The paper shows how to make the quantum Zeno effect converge much faster. Normally, repeatedly interrupting a quantum system can force it to remain inside a chosen subspace, or to follow an effective “Zeno dynamics,” but the approximation error usually shrinks only linearly with the step size $1/n$. The key idea is to run several Zeno sequences with different step sizes and combine their measured outcomes using carefully chosen weights. These weights cancel the leading error terms, improving the convergence to $1/n^{K+1}$ for any chosen order $K$, under suitable assumptions. Thus, the method is mainly a post-processing trick: it does not require a fundamentally new Zeno operation, but instead uses a higher-order expansion of the Zeno-effect error to cancel it in post-processing. Applications include Bang-Bang dynamical decoupling, where frequent kicks protect a system from its environment, and bosonic cat codes, where Zeno dynamics can help implement logical quantum gates.

► BibTeX data

@article{Mobus2026multiproductzeno,
  doi = {10.22331/q-2026-06-30-2148},
  url = {https://doi.org/10.22331/q-2026-06-30-2148},
  title = {Multi-product {Z}eno effect achieving higher order convergence rates},
  author = {M{\"{o}}bus, Tim},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2148},
  month = jun,
  year = {2026}
}

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[1] Kasra Rajabzadeh Dizaji, Leeseok Kim, Milad Marvian, and Christian Arenz, "Higher-order Zeno sequences", Physical Review Research 8 2, 023051 (2026).

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