Fermionic Born Machines: Classical training of quantum generative models based on Fermion Sampling

TL;DR

This work identifies trainability as a key bottleneck for quantum generative models and introduces Fermionic Born Machines (FBMs), a restricted quantum model using parameterized magic input states and fermionic linear-optical (FLO) transformations. FBMs can be classically trained by efficiently computing local Pauli-$Z$ expectations through a Gaussian decomposition of the magic inputs, enabling a squared maximum mean discrepancy loss $\,\mathcal{L}_{\mathrm{MMD}^2}$ without quantum gradient evaluations; sampling from the learned distribution remains classically hard under standard complexity assumptions and is delegated to quantum hardware via FLO circuits. The authors prove and demonstrate that the expectation values of fixed-length observables can be computed in time $\mathcal{O}(\ell^3 4^{\ell} N^{\lfloor \ell/2 \rfloor})$, and show favorable loss landscapes with overparametrization; numerical experiments on datasets up to 160 qubits—including molecular fingerprints and gene sequences—illustrate FBMs' capacity to capture structured correlations where classical models (e.g., Chow-Liu trees, RBMs) struggle. Overall, FBMs offer a scalable, trainable, and practically relevant avenue toward quantum-assisted generative modeling for data with local structure, while delineating a clear separation between classical trainability and quantum-sampling hardness.

Abstract

Quantum generative learning is a promising application of quantum computers, but faces several trainability challenges, including the difficulty in experimental gradient estimations. For certain structured quantum generative models, however, expectation values of local observables can be efficiently computed on a classical computer, enabling fully classical training without quantum gradient evaluations. Although training is classically efficient, sampling from these circuits is still believed to be classically hard, so inference must be carried out on a quantum device, potentially yielding a computational advantage. In this work, we introduce Fermionic Born Machines as an example of such classically trainable quantum generative models. The model employs parameterized magic states and fermionic linear optical (FLO) transformations with learnable parameters. The training exploits a decomposition of the magic states into Gaussian operators, which permits efficient estimation of expectation values. Furthermore, the specific structure of the ansatz induces a loss landscape that exhibits favorable characteristics for optimization. The FLO circuits can be implemented, via fermion-to-qubit mappings, on qubit architectures to sample from the learned distribution during inference. Numerical experiments on systems up to 160 qubits demonstrate the effectiveness of our model and training framework.

Figures (5)

• Figure 1: Framework for the classical training of Fermionic Born Machines (FBMs). The FBM ansatz consists of a parametrized magic input state followed by a fermionic linear optical (FLO) transformation. The parameters are classically optimized by first estimating expectation values of Pauli-Z strings for the training set and the FBM model, then computing the squared maximum mean discrepancy loss function. After training, the corresponding quantum circuit is run on a quantum hardware with the optimized parameters.
• Figure 2: Training performance of small FBMs on graph-structured problems. FBMs are trained on $10$ datasets sampled from different Markov networks of the same grid topology (shown in left inset). The training $\rm{MMD}^2$ loss is approximated using different cutoffs. (left) The mean total variation distance (TVD) is shown along the training. (right) The final TVD improvement over the Chow-Liu approximation of the dataset is presented.
• Figure 3: The effect of overparametrization on the performance FBMs. The models were trained on the equilibrium states of a cellular automaton over a $6\times 7$ grid. FBMs were constructed with an increasing number of layers, and we trained an RBM with $40$ hidden units for comparison.
• Figure 4: Test results on the molecular fingerprint dataset. The models were trained on $60$-bit long molecular fingerprints. FBM have $7$ trainable layers, and we trained an RBM with $44$ hidden units for comparison. The FBM with magic input states captures the structure present in the dataset.
• Figure 5: Test results on the gene sequence dataset. The models were trained on $120$-bit long bitstrings obtained from $60$-long gene sequences. The used FBMs have $10$ trainable layers, and we trained an RBM with $62$ hidden units for comparison.

Theorems & Definitions (16)

• Definition 1: Classically trainable quantum generative model
• Proposition 1: Informal version of Theorem 3 from Ref. oszmaniec2022fermion
• Definition 2: Fermionic Born Machine
• proof
• Definition 3: Covariance
• Definition 4: $\rm{MMD}^2$ to a test set
• Proposition 2
• proof
• Proposition 3
• proof