Fig. 7: Diagram of the encoding framework.

a We unwrap the cylindrical shower into a tensor of rank 3 with indices (z,θ,r). We account the angular periodicity of the cylindrical geometry by padding the tensor in θ dimension, such that the size becomes 45 × 18 × 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, each 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 incident 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 the prior. The energy function in the Boltzmann distribution corresponds to a sparse 4-partite graph, which allows the direct mapping to the Pegasus-structured advantage quantum annealer. e The RBM energy histogram from the encoded showers and the MCMC-generated samples converge during training.