A learning theory for quantum photonic processors and beyond

Matteo Rosati

doi:10.22331/q-2024-08-08-1433

A learning theory for quantum photonic processors and beyond

Matteo Rosati

Dipartimento di Ingegneria Civile, Informatica e delle Tecnologie Aeronautiche, Università Roma Tre, Via Vito Volterra 62, Rome, I-00146, Italy

Published:	2024-08-08, volume 8, page 1433
Eprint:	arXiv:2209.03075v4
Doi:	https://doi.org/10.22331/q-2024-08-08-1433
Citation:	Quantum 8, 1433 (2024).

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Abstract

We consider the tasks of learning quantum states, measurements and channels generated by continuous-variable (CV) quantum circuits. This family of circuits is suited to describe optical quantum technologies and in particular it includes state-of-the-art photonic processors capable of showing quantum advantage. We define classes of functions that map classical variables, encoded into the CV circuit parameters, to outcome probabilities evaluated on those circuits. We then establish efficient learnability guarantees for such classes, by computing bounds on their pseudo-dimension or covering numbers, showing that CV quantum circuits can be learned with a sample complexity that scales polynomially with the circuit's size, i.e., the number of modes. Our results show that CV circuits can be trained efficiently using a number of training samples that, unlike their finite-dimensional counterpart, does not scale with the circuit depth.

Featured image: Depiction of the simplest learning problem studied for continuous-variable (CV) architectures. We aim to reproduce the statistics of an unknown state $\rho$ with respect to random couples of CV channels and measurements $(\Phi_i, M_i)$. For each $i$, the state is passed through channel $\Phi_i$ and measured with a binary measurement $M_i$ obtaining outcome $b_i$. Then the learner produces a hypothesis $\sigma$ by tuning the parameters of a CV circuit, e.g., a photonic chip, and obtains its output statistics on the channels and measurements of the training set. A suitable empirical loss evaluates how well $\sigma$ approximates $\rho$ on the training set. This can be used to optimize $\sigma$ iteratively.

Popular summary

Infinite-dimensional quantum systems are an alternative to two-level quantum systems in quantum information processing, combining a potentially unbounded information content with convenient implementation platforms, such as integrated photonics.

While full tomography puts a burden on the number of experiments needed to characterize a quantum system or device, the last few years have seen the rise of a machine-learning mindset, advocating for the use of few experiments to predict a number of properties of the quantum system under study. Unfortunately, these studies have been limited to finite-dimensional quantum systems.

Here we establish a statistical learning theory for infinite-dimensional quantum systems, characterizing the number of experiments needed to approximate the statistics of a quantum device based on bosonic continuous-variable (BCV) information processing. In this setting, we prove that the class of BCV Gaussian devices is learnable with a sample complexity that scales quadratically with the number of bosonic modes, i.e., the system size. Furthermore, if the device contains non-Gaussian components, the sample complexity also scales polynomially with suitable non-Gaussianity quantifiers.

Our results provide learnability guarantees not only for BCV quantum states, but also for general quantum devices, including processing and measurement stages. Finally, we provide learnability guarantees beyond the problem of device characterization, including information-processing tasks such as discrimination, synthesis and compiling.

► BibTeX data

@article{Rosati2024learningtheory,
  doi = {10.22331/q-2024-08-08-1433},
  url = {https://doi.org/10.22331/q-2024-08-08-1433},
  title = {A learning theory for quantum photonic processors and beyond},
  author = {Rosati, Matteo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1433},
  month = aug,
  year = {2024}
}

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[5] Simon Becker, Nilanjana Datta, Ludovico Lami, and Cambyse Rouzé, "Classical shadow tomography for continuous variables quantum systems", arXiv:2211.07578, (2022).

[6] Simon Becker, Nilanjana Datta, Ludovico Lami, and Cambyse Rouze, "Classical Shadow Tomography for Continuous Variables Quantum Systems", IEEE Transactions on Information Theory 70 5, 3427 (2024).

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[11] T. J. Volkoff and Andrew T. Sornborger, "Learning linear optical circuits with coherent states", Journal of Physics A Mathematical General 57 30, 305302 (2024).

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