Quantum Chebyshev Probabilistic Models for Fragmentation Functions

TL;DR

This work introduces Quantum Chebyshev Probabilistic Models (QCPMs) to learn and sample multivariate fragmentation-function distributions in high-energy physics by encoding two correlated variables, momentum fraction $z$ and energy scale $Q$, in a Chebyshev basis. A correlation circuit entangles the two registers, enabling accurate modeling and high-resolution sampling via inverse Quantum Chebyshev Transforms, with extended registers ($S$ qubits per variable) enabling exponentially finer grids. Application to LHAPDF6 NNFF datasets, especially $D_g^{K^\pm}(z,Q)$, shows improved predictive performance when correlations are included, quantified via $R^2$ and cross-register nonpurity $C=|1-\gamma_{\mathcal{Z}}|$, and reveals how entanglement patterns relate to learned dependencies. The approach demonstrates the potential of quantum generative modeling for FF analysis, offering efficient interpolation on dense grids with modest training cost and scalable sampling for dataset augmentation in high-energy physics.

Abstract

Quantum generative modeling is emerging as a powerful tool for advancing data analysis in high-energy physics, where complex multivariate distributions are common. However, efficiently learning and sampling these distributions remains challenging. We propose a quantum protocol for a bivariate probabilistic model based on shifted Chebyshev polynomials, trained as a circuit-based representation of two correlated variables, with sampling performed via quantum Chebyshev transforms. As a key application we study fragmentation functions (FFs) of charged pions and kaons from single-inclusive hadron production in electron-positron annihilation. We learn the joint distribution of momentum fraction $z$ and energy scale $Q$, and infer their correlations from the entanglement structure. Building on the generalization capabilities of the quantum model and extended register architecture, we perform fine-grid multivariate sampling for FF dataset augmentation. Our results highlight the growing potential of quantum generative modeling to advance data analysis and scientific discovery in high-energy physics.

Figures (11)

• Figure 1: Workflow for describing fragmentation functions with quantum probabilistic models based on shifted Chebyshev polynomials. (a) The input data for the fragmentation function (FF) $D_i^\textrm{h}(z,Q)$ is produced for a grid of $z$ and $Q$. (b) Quantum probabilistic model is composed of two Chebyshev feature maps for encoding $z$ and $Q$, a correlation circuit that entangles both registers, and basis adaptation circuits to be trained on $D_i^\textrm{h}(z,Q)$. (c) For sampling we perform the inverse of basis adaptation, the correlation circuit, followed by parallel inverse quantum Chebyshev transforms for mapping the model into the bit basis. (d) Sampling results assembled in a 2-dimensional plot that represents $D_i^\textrm{h}(z,Q)$.
• Figure 2: Applying linear maps and setting up quantum Chebyshev probabilistic models (QCPM). Application of shifted Chebyshev polynomials to map between (a) the problem and (b) Chebyshev spaces. (c) Visualization of training (black) and sampling (color) grids for different extended registers.
• Figure 3: Sampling of $D_{g}^{K^\pm}(z,Q)$, the fragmentation function (FF) of a gluon fragmenting into kaons. In panels (a)-(d) the sampling is performed with $S=0,1,2$ additional qubits for each variable to interpolate in untrained regions. (a) Target distribution $D_{g}^{K^\pm}(z,Q)$. (b) Samples from trained Quantum Chebyshev Probabilistic Model (QCPM) of $D_{g}^{K^\pm}(z,Q)$ with the same number of qubits ($S=0$). (c,d) Samples from $D_{g}^{K^\pm}(z,Q)$ with extended register ($S=1,2$). (e) Contour overlay of the FF projected onto variable $Q$ with $S = 0$, $1$, and $2$ and the target function. The histogram bins represent an average of the $D_{g}^{K^\pm}$ value for a range of values of $z$.
• Figure 4: Training performance and nonpurity evaluation of the models for different fragmentation functions (FF). (a) Accuracy ($R^2$) comparison of Quantum Chebyshev Probabilistic Models (QCPMs) for learning FFs $D_i^\textrm{h}(z, Q)$ with (w/ CC) and without (w/o CC) correlations between $z$ and $Q$. (b) Nonpuriry coefficient $C = |1-\gamma_{\mathcal{Z}}|$ of system $\mathcal{Z}$ for 100 training epochs with the fixed correlation circuit (solid curves, log-log curves). Dashed curves represent $C$ when the system starts in a product state without entanglement between registers (no correlations). Solid curves highlight the values of nonpurity after training.
• Figure 5: Quantum circuits for implementation of QCPMs. (a) Quantum circuit used to train the multivariate distribution in the QCPM latent space, where a correlation circuit $\hat{\mathcal{C}}$ is sandwiched between two identical sets of feature map circuits and variational Ansätze. Measured observable is defined as $\hat{\mathcal{O}} = |0_a \mathrm{\o} 0_a \mathrm{\o} \rangle \langle 0_a \mathrm{\o} 0_a \mathrm{\o}|$, where $|\mathrm{\o}\rangle \equiv |0\rangle^{\otimes N}$. Here, $\alpha$ and $\beta$ are trainable scaling and bias parameters. (b) Quantum circuit used to sample the multivariate distribution from the trained model, where the inverse versions of the same parameterized circuits are applied with $\theta^*$ and $\vartheta^*$ being retrieved after the optimization procedure, followed by the inverse versions of the same correlation and two identical sets of quantum Chebyshev transform circuits associated with extended registers of $S$ qubits ($|0\rangle^{\otimes S}$) in parallel, for fine sampling in the computational basis $|u_s u_\textrm{a} u_j v_s v_\textrm{a} v_j\rangle$. The quantum state prior to measurement is denoted as $|\psi\rangle$.