Learning unitaries with quantum statistical queries

Armando Angrisani

doi:10.22331/q-2025-07-30-1817

Learning unitaries with quantum statistical queries

Armando Angrisani

LIP6, Centre National de la Recherche Scientifique (CNRS), Sorbonne Universit´e, 75005 Paris, France
Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne CH-1015, Switzerland
Center for Quantum Science and Engineering, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

Published:	2025-07-30, volume 9, page 1817
Editor:	Daniel Grier
Eprint:	arXiv:2310.02254v3
Doi:	https://doi.org/10.22331/q-2025-07-30-1817
Citation:	Quantum 9, 1817 (2025).

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Abstract

We propose several algorithms for learning unitary operators from quantum statistical queries with respect to their Choi-Jamiolkowski state. Quantum statistical queries capture the capabilities of a learner with limited quantum resources, which receives as input only noisy estimates of expected values of measurements. Our approach leverages quantum statistical queries to estimate the Fourier mass of a unitary on a subset of Pauli strings, generalizing previous techniques developed for uniform quantum examples. Specifically, we show that the celebrated quantum Goldreich-Levin algorithm can be implemented with quantum statistical queries, whereas the prior version of the algorithm involves oracle access to the unitary and its inverse. As an application, we prove that quantum Boolean functions with constant total influence or with constant degree are efficiently learnable in our model. Moreover, we prove that $\mathcal{O}(\log n)$-juntas are efficiently learnable and constant-depth circuits are learnable query-efficiently with quantum statistical queries. On the other hand, all previous algorithms for these tasks demand significantly greater resources, such as oracle access to the unitary or direct access to the Choi-Jamiolkowski state. We also demonstrate that, despite these positive results, quantum statistical queries lead to an exponentially larger query complexity for certain tasks, compared to separable measurements to the Choi-Jamiolkowski state. In particular, we show an exponential lower bound for learning a class of phase-oracle unitaries and a double exponential lower bound for testing the unitarity of channels. Taken together, our results indicate that quantum statistical queries offer a unified framework for various unitary learning tasks, with potential applications in quantum machine learning, many-body physics and benchmarking of near-term devices.

Popular summary

This work extends the theory of quantum statistical query learning to the task of learning unitary operators, which are central to quantum computing. A key advantage of this learning model is its robustness to noise: the proposed algorithms can tolerate both shot noise and certain systematic errors in measurement outcomes.
Importantly, the model is designed to reflect realistic experimental conditions: it assumes access only to noisy, binary measurements on non-entangled copies of the dual state of the unitary. This makes it particularly relevant for near-term and early fault-tolerant quantum devices, where such constraints are common.
The paper presents both new algorithms and fundamental limitations within this framework, and highlights several promising applications. These include building surrogate models for quantum machine learning, estimating correlation functions in many-body physics, and performing cross-platform verification of quantum devices.

► BibTeX data

@article{Angrisani2025learningunitaries,
  doi = {10.22331/q-2025-07-30-1817},
  url = {https://doi.org/10.22331/q-2025-07-30-1817},
  title = {Learning unitaries with quantum statistical queries},
  author = {Angrisani, Armando},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1817},
  month = jul,
  year = {2025}
}

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[9] Alexander Nietner, "Unifying (Quantum) Statistical and Parametrized (Quantum) Algorithms", arXiv:2310.17716, (2023).

[10] Pragatheeswaran Vipulananthan, Kamal Premaratne, Dilip Sarkar, and Manohar N. Murthi, "A Quantum Tensor Network-Based Viewpoint for Modeling and Analysis of Time Series Data", arXiv:2511.13514, (2025).

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