Quantum consistent neural/tensor networks for photonic circuits with strongly/weakly entangled states

TL;DR

The paper addresses the challenge of simulating high‑dimensional quantum photonic circuits for metrology by introducing physics‑informed neural and tensor networks that approximate exact unitary evolution with quantum consistency. It develops a quantum consistent neural network (QCNN) that enforces invariances and yields accurate, scalable parameter estimation via backpropagation, and demonstrates superior performance over vanilla networks, especially for strongly entangled states. To handle large Hilbert spaces, it also shows that tensor networks (MPS/MPO) can capture low‑rank entanglement more efficiently, with substantially fewer trainable parameters and competitive accuracy, outperforming the neural approach in the weakly entangled regime. The work further extends to higher photon numbers and boson sampling, outlining how MPO/MPS contractions can scale better than naïve tensor‑ or vector‑based methods, thereby providing a practical framework for classical simulation and evaluation of quantum advantages in photonic systems.

Abstract

Modern quantum optical systems such as photonic quantum computers and quantum imaging devices require great precision in their designs and implementations in the hope to realistically exploit entanglement and reach a real quantum advantage. The theoretical and experimental explorations and validations of these systems are greatly dependent on the precision of our classical simulations. However, as Hilbert spaces increases, traditional computational methods used to design and optimize these systems encounter hard limitations due to the quantum curse of dimensionally. To address this challenge, we propose an approach based on neural and tensor networks to approximate the exact unitary evolution of closed entangled systems in a precise, efficient and quantum consistent manner. By training the networks with a reasonably small number of examples of quantum dynamics, we enable efficient parameter estimation in larger Hilbert spaces, offering an interesting solution for a great deal of quantum metrology problems.

Figures (21)

• Figure 1: Illustration of the random unitary circuit with phase-shifter (triangles) and photon counting detectors (semi-circles)
• Figure 2: Examples of photon coincidence detection patterns with $\vert \psi_i \rangle=|c_{2},c_{3}\rangle_{\mathrm{sym}}$, $d=16$, $p=1000$ and the same Haar matrix and $\vec{\theta}$.
• Figure 3: Examples of photon coincidence detection patterns with $\vert \psi_i \rangle=d^{-1/2}\sum_{m=1}^{d} |m,m\rangle$, $d=16$, $p=1000$ and the same Haar matrix and $\vec{\theta}$.
• Figure 4: Results of optimizations of the exact unitary evolution in $d=16$ with 100 different $\mathbb{P}_{\mathrm{emp}}(p=1000)$ with $\vert \psi_i \rangle=|c_{2},c_{3}\rangle_{\mathrm{sym}}$ (Top) and the $\mathrm{NOON}$ state (bottom): the figures show the convergence of the mean phase shift residuals and their standard deviations over the 100 trials. The difference in precision between the two initial states is evident and explained in the main text.
• Figure 5: Representation of the layers of the quantum consistent neural network. Starting from the vector $\vec{\theta}$ in input, the system performs a sequence of transformations and outputs a coincidence probability matrix $\mathbb{P}_{\vec{\theta}}$. The successive layers are described in the main text.