Generative Invertible Quantum Neural Networks

TL;DR

The study tackles density estimation for complex collider data using invertible networks and introduces a quantum analogue, the QINN, built from circuit-based QNNs with an invertible design and a learnable inverse state preparation. It trains the QINN with a combination of negative log-likelihood and Maximum Mean Discrepancy losses, and demonstrates its application to the LHC process $pp \rightarrow Z j \rightarrow \ell^+ \ell^- j$, achieving comparable or superior performance to larger classical INNs with far fewer trainable parameters. A key finding is that a hybrid QINN with about 2k parameters can match a classical INN of 6k–16k parameters on reconstructing the $M_Z$ distribution and related observables, indicating enhanced expressivity per parameter due to quantum circuitry. The work highlights the potential of QINNs for density estimation in high-energy physics and broader generative tasks, suggesting pathways toward hardware-scale quantum advantages and new insights from access to the full Jacobian of the generative process.

Abstract

Invertible Neural Networks (INN) have become established tools for the simulation and generation of highly complex data. We propose a quantum-gate algorithm for a Quantum Invertible Neural Network (QINN) and apply it to the LHC data of jet-associated production of a Z-boson that decays into leptons, a standard candle process for particle collider precision measurements. We compare the QINN's performance for different loss functions and training scenarios. For this task, we find that a hybrid QINN matches the performance of a significantly larger purely classical INN in learning and generating complex data.

Figures (7)

• Figure 1: Layout of a coupling block. The input is split into two halves along the feature dimension. One half is used as neural network input to predict parameters for either an affine or a spline transformation of the other. In the end, both halves are fused again. The blue lines illustrate example coupling transformations of both block types, based on the network output (black knots). The orange line shows the inverse transformation.
• Figure 2: An overview of the model circuit. We show a three-qubit, two-layer model following the hardware efficient ansatz from Kandala2017. The state preparation uses angle encoding, where each feature is encoded on its qubit. The learnable parameters of the model circuit are the rotation angles $\theta$ in each layer.
• Figure 3: The SWAP-test, comparing a state $\ket{y} = U \ket{x}$ created in the forward direction to $\ket{\tilde{y}} = g(y)$ created by the ISP. The CSWAP gate acts pairwise on the wires, i.e. $y_1$ is swapped with $\tilde{y}_1$, etc.
• Figure 4: A diagram of the learnable ISP method. To map a measurement back to a quantum state, first a NN predicts $3n$ angles, which serve as state preparation for $\ket{\psi}$, which is then transformed further by a separate QNN that recreates $\ket{y}$.
• Figure 5: $M_Z$ and $\Delta R_{\ell^+, \ell^-}$ of the $Z$ reconstructed from the leptons as generated by the QINN with Fidelity and MSE loss for invertibility. True shows the distribution of the entire test set.