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Heisenberg picture tensor network formalism for optical circuits

Dario Cilluffo ^*, Matthias Kost , Nicola Lorenzoni , and Martin B. Plenio^†

Institute of Theoretical Physics, Ulm University, Albert-Einstein-Allee 11, 89081 Ulm, Germany and Center for Integrated Quantum Science and Technology (IQST), 89081 Ulm, Germany

^*Contact author: dario.cillufo@uni-ulm.de
^†Contact author: martin.plenio@uni-ulm.de

Phys. Rev. Research 8, 023098 – Published 30 April, 2026

DOI: https://doi.org/10.1103/wtwl-979d

Abstract

Tensor network formalisms have emerged as powerful tools for simulating quantum state evolution. While widely applied in the study of optical quantum circuits, such as boson sampling, existing tensor network approaches fail to address the complexity mismatch between tensor contractions and the calculation of photon-counting probability amplitudes. Here, we present an alternative tensor network framework that exploits the input-output relations of quantum optical circuits encoded in the unitary interferometer matrix. Our approach bridges the complexity gap by enabling the computation of the permanent—central to boson sampling—with the same computational complexity as the best known classical algorithm based on a graphical representation of the operator-basis matrix product states that we introduce. Furthermore, we exploit the flexibility of tensor networks to extend our formalism to incorporate partial distinguishability and photon loss, two key imperfections in practical interferometry experiments. This work offers a significant step forward in the simulation of large-scale quantum optical systems and the understanding of their computational complexity.

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Phys. Rev. Research 8, 023098– Published 30 April, 2026

Vol. 8, Iss. 2 — April - June 2026

Received 13 February 2025
Revised 3 November 2025
Accepted 27 March 2026

DOI: https://doi.org/10.1103/wtwl-979d

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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Heisenberg picture tensor network formalism for optical circuits

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