Hardware-inspired Continuous Variables Quantum Optical Neural Networks

TL;DR

The paper tackles neural-network style computation in continuous-variable quantum optics under practical hardware constraints by introducing a hardware-feasible CV-QONN architecture that encodes inputs as coherent states, applies a Gaussian affine processing step, and uses multi-mode photon subtraction as a tunable nonlinear activation read out via homodyne detection. It establishes closed-form layer expressions and proves universal approximation within a single layer, while cross-mode photon subtraction enriches expressivity. A high-performance QuaNNTO library enables exact classical simulation and gradient-based training without truncating the Hilbert space, and the authors demonstrate strong performance on curve fitting, classification tasks including MNIST-like data, and non-Gaussian gate synthesis. The work highlights potential for scalable photonic quantum machine learning and quantum state engineering, offering practical hardware guidance and a foundation for further exploration of non-Gaussian operations in CV quantum information processing.

Abstract

Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources -- the challenging piece for physical implementation -- originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick--Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.

Figures (8)

• Figure 1: General CV-QONN architecture with single photon subtraction per layer. Top: $N$-mode QONN with $L$ layers where each layer is carried out by the operator $\hat{L} = \hat{a}_1 \hat{G} = \hat{a}_1 \hat{D} \hat{U}_2 \hat{S} \hat{U}_1$. Bottom: Hardware implementation example of a single-layer two-mode QONN using local oscillators (LO), beam splitters (BS), phase shifters (PS or $\hat{R}$), single-mode squeezers ($\hat{S}$), single-photon detectors (SPD) and photodetectors (PD).
• Figure 2: Nonlinearity of the photon subtraction in the QONN: expectation values of Eq. \ref{['ph-sub-expval']} with respect to real displacements $\alpha \in (-2, 2)$ for different factors of (left) momentum squeezing and (right) position squeezing.
• Figure 3: Curve fitting training of $N$-mode single-layer ($L=1$) QONNs with photon subtraction $\hat{a}$ in different modes trained over 100-sample noisy datasets of exponential, polynomial and oscillatory functions. Left: Gaussian, single-neuron and two-neuron QONNs results for $f(x) = x^5$ with $x \in (-3, 3),\ \epsilon = 5$. Center: one, two and three-neuron QONNs results for $f(x) = 2\cosh(x)$ with $x \in (-5, 5),\ \epsilon = 3$. Right: one, two, three and four-neuron QONNs results for $f(x) = \sin (3x) + \cos (5x)$ with $x \in (-1, 2.5),\ \epsilon = 0.1$.
• Figure 4: Moons classification (upper row): two interleaved categories of points. Upper-left: Linear decision boundary with 86.8% accuracy of a Gaussian QONN (no subtractions). Upper-right: S-like decision boundary with 100% of accuracy of a single-layer single-subtraction QONN. Circles classification (lower row): two concentric categories of points. Lower-left: highly-curved decision boundary with 93% of accuracy of a single-layer single-subtraction QONN. Lower-right: radial decision boundary with 100% of accuracy of a single-layer two-subtractions QONN (same accuracy as a 2-layer QONN with one subtraction per layer).
• Figure 5: 5-category MNIST classification showing per-class accuracy (opaque lines) and average precision (transparent horizontal lines) for different QONN models with an input pre-processing of 3-dimensional latent space.