Conditioned quantum-assisted deep generative surrogate for particle-calorimeter interactions

TL;DR

The paper tackles the computational bottleneck of HL-LHC calorimeter simulations by introducing Calo4pQVAE, a conditioned quantum-assisted deep generative surrogate that couples a 4-partite RBM prior to a VAE and is conditioned on incidence energy. It advances conditioning via flux biases and an adaptive inverse-temperature mapping to enable quantum annealer sampling, and validates the approach on CaloChallenge Dataset 2 with competitive FPD and KPD metrics, reporting substantial potential speedups over Geant4. Key innovations include the 4-partite RBM prior, hierarchical encoder/decoder, discrete latent space with the Gumbel trick, and practical conditioned QA sampling, all integrated into a cylindrical 3D calorimeter surrogate. The results suggest that quantum-assisted priors can yield high-fidelity shower generation orders of magnitude faster than traditional simulation, with clear pathways for hardware- and architecture-driven improvements in future work.

Abstract

Particle collisions at accelerators such as the Large Hadron Collider, recorded and analyzed by experiments such as ATLAS and CMS, enable exquisite measurements of the Standard Model and searches for new phenomena. Simulations of collision events at these detectors have played a pivotal role in shaping the design of future experiments and analyzing ongoing ones. However, the quest for accuracy in Large Hadron Collider (LHC) collisions comes at an imposing computational cost, with projections estimating the need for millions of CPU-years annually during the High Luminosity LHC (HL-LHC) run \cite{collaboration2022atlas}. Simulating a single LHC event with \textsc{Geant4} currently devours around 1000 CPU seconds, with simulations of the calorimeter subdetectors in particular imposing substantial computational demands \cite{rousseau2023experimental}. To address this challenge, we propose a conditioned quantum-assisted deep generative model. Our model integrates a conditioned variational autoencoder (VAE) on the exterior with a conditioned Restricted Boltzmann Machine (RBM) in the latent space, providing enhanced expressiveness compared to conventional VAEs. The RBM nodes and connections are meticulously engineered to enable the use of qubits and couplers on D-Wave's Pegasus-structured \textit{Advantage} quantum annealer (QA) for sampling. We introduce a novel method for conditioning the quantum-assisted RBM using \textit{flux biases}. We further propose a novel adaptive mapping to estimate the effective inverse temperature in quantum annealers. The effectiveness of our framework is illustrated using Dataset 2 of the CaloChallenge \cite{calochallenge}.

Figures (10)

• Figure 1: (a) Calochallenge dataset showers are voxelized using cylindrical coordinates $(r,\varphi,z)$ such that the showers evolve in the $z$ direction. For any given event, each voxel value corresponds to the energy (MeV) in that vicinity. Dataset 2 contains $100$k events and the voxelized cylinder has 45 stacked layers. Each layer has 144 voxels composed of $16$ angular bins and $9$ radial bins. The data set is parsed onto a 1D vector following the common way to eat a pizza, i.e., grab a slice and start from the inside towards the crust. Each 1D vector has $45\times 9 \times 16 = 6480$ voxels per each event. (b) Visualization of the voxels in an event in the dataset.
• Figure 2: Sketch of Calo4pQVAE. (a) The input data is composed by the energy per voxel, $x$ and the incidence energy, $e$. During training, the data flows through the encoder gets encoded into a latent space, $z$, it then goes through the decoder and generates a reconstruction of the voxels per energy, while the incidence energy is the label of the event and conditions the encoder and decoder. The decoder outputs the activation vector, $\chi$ and the hits vector $\xi$. The model is trained via the optimization of the mean squared error between the input shower and the reconstructed shower, the binary cross entropy (hit loss) between the hits vector and the input shower and the Kulbach-Liebler divergence which is composed by the entropy of the encoded sample and the restricted Boltzmann machine log-likelihood. (b) For inference, we sample from the RBM or the QA conditioned to an incidence energy, afterwards the sample goes through the decoder to generate a shower.
• Figure 3: Diagram of the encoding framework. (a) We unwrap the cylindrical shower into a tensor of rank 3 with indices. We account the angular periodicity of the cylindrical geometry by padding the tensor in $theta$ dimension, such that the size becomes $45 \times 18 \times 9$. To account for the neighboring voxels in the center of the cylinder, we pad the tensor in the corresponding radial dimension. We pad it by taking the centermost voxels, splitting it in half and permuting the two halves. (b) These operations are performed several times, ech prior to a 3D convolution operation for feature extraction. (c) The encoder embeds hierarchy levels, i.e., the first encoder generates a fraction of the encoded data, which is then fed to the second encoder (together with the input) to generate the remaining fraction of the encoded data. The encoded data is used to train the QPU RBM. The encoded data and the incidence energy is passed to the decoder to reconstruct the energy per voxel. (d) The Calo4pQVAE uses a discrete binary latent space and assumes a Boltzmann distribution for prior. The energy function in the Boltzmann distribution corresponds to a sparse 4-partite graph, which allows the direct mapping to Pegasus-structured Advantage quantum annealer.
• Figure 4: Quadripartite RBM weight matrices. Each panel correspond to the histogram of connections between partition A and partition B (see legend).
• Figure 5: $\lambda(\delta)$ (see Eq. \ref{['eq:stability_maintext']}) vs$\beta$ for different $\delta$ values (see legend). The stability region is shaded in light blue. Different $\delta$ values affect the stability depending on the $\beta$ values. For instance, for low $\beta$ values, large $\delta$ parameters leads to better stability; conversely, large $\beta$ values with large $\delta$ parameter leads to instabilities. The purple pentagons correspond to the average energy ratio between QPU samples and classical samples. The intersection between the black dashed line and the purple curve define the fixed point.