Generative modeling with Gaussian Boson Sampling: classically trainable Bosonic Born Machines

Abstract

Quantum generative modeling has emerged as a promising application of quantum computers, aiming to model complex probability distributions beyond the reach of classical methods. In practice, however, training such models often requires costly gradient estimation performed directly on the quantum hardware. Crucially, for certain structured quantum circuits, expectation values of local observables can be efficiently evaluated on a classical computer, enabling classical training without calls to the quantum hardware in the optimization loop. In these models, sampling from the resulting circuits can still be classically hard, so inference must be performed on a quantum device, yielding a potential computational advantage. In this work, we introduce a photonic quantum generative model built on parametrized Gaussian Boson Sampling circuits. The training is based on the efficient classical evaluation of expectation values enabled by the Gaussian structure of the state, allowing scalable optimization of the model parameters through the maximum mean discrepancy loss function. We demonstrate the effectiveness of the approach through numerical experiments on photonic systems with up to 805 modes and over a million trainable parameters, highlighting its scalability and suitability for near-term photonic quantum devices.

Figures (6)

• Figure 1: Schematic overview of the Gaussian Bosonic Born Machine (GBBM) framework. During classical training, the Born machine is optimized by minimizing an $\mathcal{L}_{\mathrm{MMD}}^2$ loss between model and target expectation values computed from the training set. After classical training, the learned GBBM is deployed for quantum sampling, where measurements on the quantum device produce generated samples.
• Figure 2: The effect of ansatz choice on the performance GBBM s. The models were trained on the equilibrium states of a cellular automaton over a $6\times 12$ grid. GBBM s were constructed with an increasing number of layers and different interferometer layouts (Clements, Chow-Liu and all-to-all layouts).
• Figure 3: Test results on the USPS dataset. The dataset containing $16 \times 16$ grayscale images was converted into $256$-long bitstrings. A GBBM was trained with $3$ layers of the all-to-all interferometer layout and the fine-tuned RBM was trained with $188$ hidden units for comparison. Markers denote mean values from $5$ independent estimates and errorbars denote the corresponding standard deviations.
• Figure 4: Test results on the genomic dataset. The models were trained on $805$-bit long bitstrings. GBBMs have a single layer with $1297660$ trainable parameters, and we trained an RBM with $492$ hidden units for comparison. Markers denote mean values from $5$ independent estimates and errorbars denote the corresponding standard deviations.
• Figure 5: Training parity and threshold GBBM s on the Ising dataset. The models were trained with at most $7$-long operator-string expectation values. (top, left) Distribution of the Hamming weight of training and test samples. (top, right) Truncated $\rm{MMD}^2$ scores of the trained models evaluated up to locality $7$. (bottom) Covariance matrices of the test set and the trained GBBM s.

Theorems & Definitions (5)

• Definition 1: Gaussian Bosonic Born Machine
• Theorem 1: Classical trainability of parity GBBMs
• proof
• Theorem 2: Classical trainability of threshold GBBMs
• proof