Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator

Federico Girotti; Alfred Godley; Madalin Guta

doi:10.22331/q-2025-08-27-1835

Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator

Federico Girotti^1,2,3, Alfred Godley^1,2, and Madalin Guta^1,2

¹School of Mathematical Sciences, University of Nottingham, United Kingdom
²Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
³Department of Mathematics, Polytechnic University of Milan, Milan, Piazza L. da Vinci 32, 20133, Italy

Published:	2025-08-27, volume 9, page 1835
Editor:	Philip Taranto
Eprint:	arXiv:2408.00626v3
Doi:	https://doi.org/10.22331/q-2025-08-27-1835
Citation:	Quantum 9, 1835 (2025).

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Abstract

We propose a two step strategy for estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a "pattern counting'' estimator computed as a simple additive functional of the outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cramer-Rao bound in the limit of large output size. Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs' quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of [1] this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of [2].

Featured image: Basic elements of the pattern counting estimator. a) A quantum Markov chain as a system interacting sequentially with the environment via a parameter dependent unitary $U_\theta$. The first stage estimator is obtained by performing a standard sequential measurement on the output. b) Post-processing the output using a coherent absorber. c) After the first estimation stage the absorber is fixed at a value $\theta_{\rm abs} = \tilde{\theta}_n -\delta_n$ where $\tilde{\theta}_n$ is the preliminary estimator and $\delta_n$ is the parameter shift required by the displaced-null measurement theory. The output generated by the system and absorber dynamics with unitary $V_{\theta_{\rm abs}}U_\theta$ is measured sequentially in the standard basis. d) Given a measurement trajectory, excitation patterns are identified as binary sequences starting and ending with a $1$ separated by long sequences of $0$s. The final estimator is a correction to the preliminary estimator which depends only on the total number of patterns $\sum_{\alpha} N_{\alpha, n}$, the QFI $f$ at $\tilde{\theta}_n$ and the displacement parameter $\tau_n$.

Popular summary

We propose a two step strategy for estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a "pattern counting'' estimator computed as a simple additive functional of the outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cramer-Rao bound in the limit of large output size. Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs' quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of J. Phys. A: Math. Theor. 57 245304 2024 this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of Phys. Rev. X 13, 031012 2023.

► BibTeX data

@article{Girotti2025estimatingquantum,
  doi = {10.22331/q-2025-08-27-1835},
  url = {https://doi.org/10.22331/q-2025-08-27-1835},
  title = {Estimating quantum {M}arkov chains using coherent absorber post-processing and pattern counting estimator},
  author = {Girotti, Federico and Godley, Alfred and Guta, Madalin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1835},
  month = aug,
  year = {2025}
}

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