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  • Open Access

Out-of-Distribution Generalization for Learning Quantum Channels with Low-Energy Coherent States

Jason L. Pereira1,2,3,*, Quntao Zhuang2,4, and Leonardo Banchi5,1

  • *Contact author: Jason.Pereira@lip6.fr
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PRX Quantum 6, 040306 – Published 8 October, 2025

DOI: https://doi.org/10.1103/5m7p-kbf3

Abstract

When experimentally learning the action of a continuous-variable quantum process by probing it with inputs, there will often be some restriction on the input states used. One experimentally simple way to probe a quantum channel is to use low-energy coherent states. Learning a quantum channel in this way presents difficulties, due to the fact that two channels may act similarly on low-energy inputs but very differently for high-energy inputs. They may also act similarly on coherent-state inputs but differently on nonclassical inputs. Extrapolating the behavior of a channel for more general input states from its action on the far more limited set of low-energy coherent states is a case of out-of-distribution generalization. To be sure that such generalization gives meaningful results, one needs to relate error bounds for the training set to bounds that are valid for all inputs. We show that for any pair of channels that act sufficiently similarly on low-energy coherent-state inputs, one can bound how different the input-output relations are for any (high-energy or highly nonclassical) input. This proves that out-of-distribution generalization is always possible for learning quantum channels using low-energy coherent states, as long as enough samples are used.

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Popular Summary

Learning how physical processes transform inputs into outputs is an important task in a wide variety of fields, such as quantum metrology, process tomography, and quantum machine learning. When working in continuous variables, the set of inputs used to probe a quantum process will be finite but the dimension of the system is infinite. Any real experiment will also have energy constraints. To understand the outputs for more general inputs, we therefore have to extrapolate from limited data. It is important to understand how reliable our knowledge of the quantum process is for inputs that are very different from those we originally used to learn the input-output relations.

The low-energy coherent states are a particularly useful set of probes for continuous-variable quantum processes. They are experimentally simple to generate with an ordinary laser, and if we perfectly understand how the quantum process acts on such inputs, we can extrapolate this knowledge to all states (even much more complicated ones). However, in any real experiment, we will only learn the input-output relations up to some error. In this paper, we show how a bound on the output error for low-energy coherent-state inputs can be used to formulate a bound on the output error for any state. We therefore show that we can get as low an output error as we like by probing with just low-energy coherent states.

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PRX Quantum 6, 040306– Published 8 October, 2025
  • Received 12 February 2025
  • Revised 9 July 2025
  • Accepted 13 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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