Quantum Diffusion Model for Quark and Gluon Jet Generation

Venue: NeurIPS_cc_2024_Workshop_MLNCP
Published: 2024-09-12 13:46:17

Abstract: In this paper, we introduce a denoising diffusion model that benefits from quantum compute techniques for generative tasks within high energy physics. The fully quantum diffusion model leverages random unitary matrices in the forward process and incorporates a variational quantum circuit within the U-Net architecture. This approach aims to address the computational challenges and resource demands typically associated with generative models, particularly when dealing with extensive datasets. We evaluate our model on the quark and gluon jets dataset from the Large Hadron Collider. The results demonstrate that the fully quantum and hybrid models are competitive with a similar classical model for jet generation, highlighting the potential of using quantum techniques for machine learning problems.

Main paper: 
 introduction: 
 The emersion of denoising diffusion models (DDMs) has revolutionized the field of generative modeling by demonstrating their ability to generate high-quality [5] . They overcome the drawbacks of generative adversarial networks (GANs), which are prone to mode collapse, becoming a new state-of-the-art architecture for image generation [10, 24] . Consequently, DDMs have been applied in many generative tasks for science from molecular biology to medical image synthesis to gravitational lensing [25, 2, 26, 17, 18] .Despite their successes, DDMs face significant challenges, particularly concerning the extensive computational resources required for training [23, 15] . Quantum machine learning (QML) offers a promising solution [20] . By cleverly incorporating quantum components into classical algorithms, quantum computers can efficiently solve problems that are difficult for classical computers with accelerated computations [7] . This paradigm shift has the potential to overcome the bottlenecks of classical computing that currently limit the performance of DDMs. 
denoising diffusion model: 
 There have been multiple variants of a DDPM proposed, such as denoising diffusion probabilistic model (DDPM) [22, 14] and denoising diffusion implicit model (DDIM) [23] , but we generalize these into DDMs with the common factors being a noising scheduling algorithm and a learned denoising process.The forward diffusion process gradually adds Gaussian noise to the data, leading to a series of latent variables x 1 , x 2 , . . . , x T . The forward process is defined as:EQUATIONwhere β t is the variance schedule that controls the amount of noise added at each step t.Starting from the original data distribution q(x 0 ), the joint distribution over the sequence of latent variables is given by:EQUATIONand the marginal distribution of any x t given x 0 is:EQUATIONwhere ᾱt = t s=1 (1 -β s ).The reverse diffusion process is modeled as:EQUATIONwhere µ θ (x t , t) is a neural network parameterized by θ that is trained to predict the mean, and σ 2 t is typically set to β t or some other small variance.The training objective is to minimize the variational lower bound on the negative log-likelihood:EQUATIONwhere D KL denotes the Kullback-Leibler divergence, though the loss is often simplified to the mean squared error (MSE).Quantum bits, or qubits, can exist in superpositions of states, allowing them to represent more information and process it more efficiently than classical bits [16] . Unlike a classical bit that can be either 0 or 1, a qubit can be in a superposition of both states simultaneously. Mathematically, the state of a qubit can be written as:EQUATIONwhere |0⟩ and |1⟩ are the basis states, and α and β are complex numbers representing the probability amplitudes. These amplitudes are normalized so that:EQUATIONWhen a measurement is made on a qubit, the superposition collapses to one of the basis states. The probability of measuring the state |0⟩ is |α| 2 , and the probability of measuring the state |1⟩ is |β| 2 .One method of measurement relevant in quantum computing and QML is the Haar measurement.Haar measurement involves sampling unitary operations uniformly according to the Haar measure, which is the unique, invariant measure on the group of unitary matrices. This type of measurement is useful in QML because it provides a way to generate random quantum states and operations, which is important for algorithms that require randomization [12] .Variational quantum circuits (VQCs) are a central tool in QML used in various architectures [6, 11, 19, 21, 8] . A VQC is a parameterized quantum circuit where some of the gates depend on adjustable parameters represented by quantum gates in the unitary operations. The general form of a VQC can be represented as:EQUATIONwhere U (θ) is a unitary operation that depends on the parameters θ.In the context of machine learning, these parameters are optimized using classical optimization techniques to minimize the cost function that measures the performance of the circuit throughout training. The power of VQCs in machine learning arises from their ability to exploit superposition and entanglement to potentially represent and solve problems more efficiently than classical algorithms [4] .In this work, we generate quark and gluon jet data from the open-source LHC Compact Muon Solenoid (CMS) detector data. The data contains two classes that follow different distributions: hits from quark and gluon jets. Each sample is captured by three CMS subdetectors: electromagnetic calorimeter (ECAL), hadronic calorimeter (HCAL), and the reconstructed tracks as described in [1] .The dataset is preprocessed to crop a subset of 1,000 ECAL-detected jets of 125x125 pixels to 16x16 pixels for faster inference. 
related work: 
 Multiple works have combined the classical DDM and quantum algorithms. The paper in [27] proposed a fully quantum DDPM (QuDDPM) for generating an unknown quantum state distribution. Their method consists of applying random unitaries to quantum states, thus scrambling them into noise, and summing three error functions to optimize during training. The QuDDPM performed best compared to a quantum GAN and a quantum direct transport model.Another work in [9] proposed a hybrid quantum-classical DDM. The algorithm involved a quantum denoising U-Net and classical noising and optimization. Good results were achieved on the MNIST dataset. However, tests on MNIST are not generalizable to other data, and a physics-conscious approach is required for the more complex Quark-Gluon data. 
methodology: 
 In this section, we outline the methods used to construct fully quantum, hybrid, and fully classical models. The pipeline in Figure 1 shows all the models and their combinations.The first step of the pipeline is embedding classical data into quantum. We implemented angle encoding with Rx rotation gates in groups of four pixels. Each sample is split into four channels as shown in Figure 2 .The forward diffusion process was inspired by scrambling implementation in [27] using Haar random unitaries to mimic random noise application as shown in Figure 3 . The final unitary is applied to each encoded channel, avoiding costly calculations associated with each timestep.Figure 1 : The pipeline that the data goes through with all the possible classical-quantum combinations of the forward and backward diffusion process.Similar to classical DDMs, the denoising neural network is trained on learned parameters with a MSE loss function. The hybrid model uses a quantum strongly entangling layer surrounded by fully convolutional layers, and the fully quantum model relies only on the quantum layers. The circuit in the quantum layer, consisting of rotation and strongly entangling gates, is kept uniform across all models, with the number of layers treated as a tunable parameter. 
experiments: 
 In each experiment, we compare the performance using the Fréchet Inception Distance (FID) function, originally introduced in [13] for evaluating GANs, along with the loss function. The FID is defined as:FID(x, g) = ∥µ x -µ g ∥ 2 2 + Tr(Σ x + Σ g -2(Σ x Σ g ) 1/2 ),where µ x and µ g are the mean feature vectors of the real and generated images, respectively, and Σ x and Σ g are their corresponding covariance matrices. Though we do not currently have access to quantum computers, we can simulate the behavior of quantum systems using simulators like Pennylane [3] and compare the results to classical models.First, we present the loss and FID functions of the classical, hybrid, and quantum models, respectively, in Figure 5 . Training on 50 epochs seems to be sufficient to reach convergence on all models. Some qualitative results in the form of generated samples are visualized in Figure ??. The generated images correspond to the best-performing model, which is the fully quantum architecture.  
discussion: 
 All models exhibited a significant decrease in loss, approaching near-zero values, confirming effective learning. The FID scores followed a similar downward trajectory, with final values of 1.8169, 1.8123, and 1.6211. These results imply the superior performance of the models that leverage quantum circuits, suggesting that all or some parts of deep neural network computations can be offloaded to faster quantum processors to reduce training time and improve performance.A potential reason for the plateau in FID scores across all models could be the sparsity of the data, where each sample contains only a few non-zero points. This sparsity may prevent the complete removal of noise in some of the encoded channels. A possible solution is a post-processing step where only the most prominent values are retained in the decoded data, increasing the confidence in jet locations. As a result, the generated samples would align more closely with the originals, as shown in Figure 6 .  
conclusion and future direction: 
 This work marks significant progress in leveraging quantum computing for machine learning applications, specifically for a generative DDM. In the future, we aim to extend the models to generate all three jet channels for every sample. Additionally, experimenting with different quantum scrambling techniques and schedules, such as using Gaussian transformations instead of Haar unitaries, may provide further insights into the impact of noise on quantum generative learning. Finally, testing the quantum model on hardware with a limited number of qubits under practical constraints will be crucial in evaluating its scalability, real-world performance, and resilience to quantum noise. 
 
 Appendix: 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 



Reviews: review: title: title: Very interesting approach of diffusion models introducing quantum computing techniques though practical implementations in hardware are not much clear.
main_review: main_review: Summary:
This paper introduces an innovative diffusion model leveraging quantum computing techniques for quark and gluon jet generation in high-energy physics. Their model integrates quantum random unitary matrices in the forward process and a variational quantum circuit within a U-Net architecture.  They are trying to address computational and resource challenges in generative modeling though its performance on real quantum hardware remains to be seen.

Strengths:
- The use of quantum circuits within diffusion is an exciting new approach for complex tasks like quark and gluon jet generation. This integration of quantum methods could significantly reduce computational resource demands compared to classical methods.
- They have introduced a clear and well-written methodology, from quantum embedding to the denoising process along with the underlying physics. 
- Their simulations show promising results with performance metrics of FID score for both the hybrid and fully quantum models where quantum models perform slightly better than classical counterparts in terms of FID scores.


Weaknesses:
- They have used quantum simulators like Pennylane, but do not test the model on actual quantum hardware. While the results from simulators are encouraging, the real-world performance and scalability of the model under practical quantum constraints are not clear.
- Their dataset is with reduced dimensions (from 125x125 to 16x16 pixels), but it remains unclear how the model would scale to larger datasets or more complex structures. Further exploration could enhance the clarity.
- Do you think there is any overhead in managing quantum circuits and gates that may offset some of the computational gains unless quantum hardware improves significantly?
-The figure quality could be improved significantly, and the FID scores from all three methods would be easier to compare. Reference to the figure is missing on page 4 having a "question mark".
title: title: Numerical experiments not fully specified; architecture seems to be unscalable
main_review: main_review: The authors propose using quantum circuits---potentially in conjunction with classical models---as a diffusion-based model for a generative modeling task relevant to high energy physics. This sort of problem---where the data is generated by a quantum process---is a good prospect for quantum machine learning to achieve a relevant advantage over classical machine learning models. The quantum circuit the authors consider to achieve this begins by scrambling the inputs using a Haar random circuit, followed by a "denoiser" of a trained quantum circuit.

However, the authors do not make any attempt to consider the scaling of their proposed implementation. Given the input to the denoising half of the quantum circuit is Haar random, from the arguments of, e.g., [arXiv:2312.09121](https://arxiv.org/abs/2312.09121) these networks will always have exponentially-vanishing gradients. This means variationally training such a network using a quantum computer would take time exponential in the problem size due to the exponential concentration of the loss landscape.

Furthermore, the authors' numerics are somewhat opaque; how many qubits were they performed using? The authors mention some form of channel-splitting procedure, but it is not clear what the final resource costs are of the network. More details on these numerics would be required for fully understanding how fair the comparison with the corresponding classical model is.