publications

2026

1. preprint

   

   A PAC-Bayesian approach to generalization for quantum models

   Pablo Rodriguez-Grasa, Matthias C Caro, Jens Eisert, Elies Gil-Fuster, Franz J Schreiber, and Carlos Bravo-Prieto

   arXiv:2603.22964, 2026

   Abs DOI arXiv

   Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model’s overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.

2. PRX Quantum

   

   Double descent in quantum kernel methods

   Marie Kempkes, Aroosa Ijaz, Elies Gil-Fuster, Carlos Bravo-Prieto, Jakob Spiegelberg, Evert Nieuwenburg, and Vedran Dunjko

   PRX Quantum, 2026

   Abs DOI arXiv

   The double descent phenomenon challenges traditional statistical learning theory by revealing scenarios where larger models do not necessarily lead to reduced performance on unseen data. While this counterintuitive behavior has been observed in a variety of classical machine learning models, particularly modern neural network architectures, it remains elusive within the context of quantum machine learning. In this work, we analytically demonstrate that linear regression models in quantum feature spaces can exhibit double descent behavior by drawing on insights from classical linear regression and random matrix theory. Additionally, our numerical experiments on quantum kernel methods across different real-world datasets and system sizes further confirm the existence of a test error peak, a characteristic feature of double descent. Our findings provide evidence that quantum models can operate in the modern, overparameterized regime without experiencing overfitting, potentially opening pathways to improved learning performance beyond traditional statistical learning theory.

2025

1. preprint

   

   Prospects for quantum advantage in machine learning from the representability of functions

   Sergi Masot-Llima, Elies Gil-Fuster, Carlos Bravo-Prieto, Jens Eisert, and Tommaso Guaita

   arXiv:2512.15661, 2025

   Abs DOI arXiv

   Demonstrating quantum advantage in machine learning tasks requires navigating a complex landscape of proposed models and algorithms. To bring clarity to this search, we introduce a framework that connects the structure of parametrized quantum circuits to the mathematical nature of the functions they can actually learn. Within this framework, we show how fundamental properties, like circuit depth and non-Clifford gate count, directly determine whether a model’s output leads to efficient classical simulation or surrogation. We argue that this analysis uncovers common pathways to dequantization that underlie many existing simulation methods. More importantly, it reveals critical distinctions between models that are fully simulatable, those whose function space is classically tractable, and those that remain robustly quantum. This perspective provides a conceptual map of this landscape, clarifying how different models relate to classical simulability and pointing to where opportunities for quantum advantage may lie.

2024

1. preprint

   

   Learning complexity gradually in quantum machine learning models

   Erik Recio-Armengol, Franz J Schreiber, Jens Eisert, and Carlos Bravo-Prieto

   arXiv:2411.11954, 2024

   Abs DOI arXiv

   Quantum machine learning is an emergent field that continues to draw significant interest for its potential to offer improvements over classical algorithms in certain areas. However, training quantum models remains a challenging task, largely because of the difficulty in establishing an effective inductive bias when solving high-dimensional problems. In this work, we propose a training framework that prioritizes informative data points over the entire training set. This approach draws inspiration from classical techniques such as curriculum learning and hard example mining to introduce an additional inductive bias through the training data itself. By selectively focusing on informative samples, we aim to steer the optimization process toward more favorable regions of the parameter space. This data-centric approach complements existing strategies such as warm-start initialization methods, providing an additional pathway to address performance challenges in quantum machine learning. We provide theoretical insights into the benefits of prioritizing informative data for quantum models, and we validate our methodology with numerical experiments on selected recognition tasks of quantum phases of matter. Our findings indicate that this strategy could be a valuable approach for improving the performance of quantum machine learning models.

2. Nat. Comms.

   

   Understanding quantum machine learning also requires rethinking generalization

   Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto

   Nature Communications, 2024

   Abs DOI arXiv

   Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.

2023

1. Quantum

   

   Variational quantum linear solver

   Carlos Bravo-Prieto, Ryan LaRose, Marco Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J Coles

   Quantum, 2023

   Abs DOI arXiv

   Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare |x⟩ such that A|x⟩∝|b⟩. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that C ⩾ ϵ²/κ², where C is the VQLS cost function and κ is the condition number of A. We present efficient quantum circuits to estimate C, while providing evidence for the classical hardness of its estimation. Using Rigetti’s quantum computer, we successfully implement VQLS up to a problem size of 1024×1024. Finally, we numerically solve non-trivial problems of size up to 250×250. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in ϵ, κ, and the system size N.

2022

1. Quantum

   

   Style-based quantum generative adversarial networks for Monte Carlo events

   Carlos Bravo-Prieto, Julien Baglio, Marco Cè, Anthony Francis, Dorota M Grabowska, and Stefano Carrazza

   Quantum, 2022

   Abs DOI arXiv

   We propose and assess an alternative quantum generator architecture in the context of generative adversarial learning for Monte Carlo event generation, used to simulate particle physics processes at the Large Hadron Collider (LHC). We validate this methodology by implementing the quantum network on artificial data generated from known underlying distributions. The network is then applied to Monte Carlo-generated datasets of specific LHC scattering processes. The new quantum generator architecture leads to a generalization of the state-of-the-art implementations, achieving smaller Kullback-Leibler divergences even with shallow-depth networks. Moreover, the quantum generator successfully learns the underlying distribution functions even if trained with small training sample sets; this is particularly interesting for data augmentation applications. We deploy this novel methodology on two different quantum hardware architectures, trapped-ion and superconducting technologies, to test its hardware-independent viability.

2. JPhysA: Math. Theor.

   

   Variational quantum eigensolver for SU (N) fermions

   Mirko Consiglio, Wayne J Chetcuti, Carlos Bravo-Prieto, Sergi Ramos-Calderer, Anna Minguzzi, José I Latorre, Luigi Amico, and Tony JG Apollaro

   Journal of Physics A: Mathematical and Theoretical, 2022

   Abs DOI arXiv

   Variational quantum algorithms aim at harnessing the power of noisy intermediate-scale quantum (NISQ) computers, by using a classical optimizer to train a parameterized quantum circuit to solve tractable quantum problems. The variational quantum eigensolver (VQE) is one of the aforementioned algorithms designed to determine the ground-state of many-body Hamiltonians. Here, we apply the VQE to study the ground-state properties of N-component fermions. With such knowledge, we study the persistent current of interacting SU(N) fermions, which is employed to reliably map out the different quantum phases of the system. Our approach lays out the basis for a current-based quantum simulator of many-body systems that can be implemented on NISQ computers.

3. preprint

   

   Quantum computing for data analysis in high energy physics

   Andrea Delgado, Kathleen E Hamilton, Jean-Roch Vlimant, Duarte Magano, Yasser Omar, Pedrame Bargassa, Anthony Francis, Alessio Gianelle, Lorenzo Sestini, Donatella Lucchesi, and 14 more authors

   arXiv:2203.08805, 2022

   Abs DOI arXiv

   Some of the biggest achievements of the modern era of particle physics, such as the discovery of the Higgs boson, have been made possible by the tremendous effort in building and operating large-scale experiments like the Large Hadron Collider or the Tevatron. In these facilities, the ultimate theory to describe matter at the most fundamental level is constantly probed and verified. These experiments often produce large amounts of data that require storing, processing, and analysis techniques that often push the limits of traditional information processing schemes. Thus, the High-Energy Physics (HEP) field has benefited from advancements in information processing and the development of algorithms and tools for large datasets. More recently, quantum computing applications have been investigated in an effort to understand how the community can benefit from the advantages of quantum information science. In this manuscript, we provide an overview of the state-of-the-art applications of quantum computing to data analysis in HEP, discuss the challenges and opportunities in integrating these novel analysis techniques into a day-to-day analysis workflow, and whether there is potential for a quantum advantage.

4. Phys. Rev. Research

   

   Solving systems of Boolean multivariate equations with quantum annealing

   Sergi Ramos-Calderer, Carlos Bravo-Prieto, Ruge Lin, Emanuele Bellini, Marc Manzano, Najwa Aaraj, and José I Latorre

   Physical Review Research, 2022

   Abs DOI arXiv

   Polynomial systems over the binary field have important applications, especially in symmetric and asymmetric cryptanalysis, multivariate-based postquantum cryptography, coding theory, and computer algebra. In this paper, we study the quantum annealing model for solving Boolean systems of multivariate equations of degree 2, usually referred to as the multivariate quadratic problem. We present different methodologies to embed the problem into a Hamiltonian that can be solved by available quantum annealing platforms. In particular, we provide three embedding options, and we highlight their differences in terms of quantum resources. Moreover, we design a machine-agnostic algorithm that adopts an iterative approach to better solve the problem Hamiltonian by repeatedly reducing the search space. Finally, we use D-Wave devices to successfully implement our methodologies on several instances of the multivariate quadratic problem.

5. Quantum Sci. Tech.

   

   Qibo: a framework for quantum simulation with hardware acceleration

   Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego Garcı́a-Martı́n, Artur Garcia-Saez, José Ignacio Latorre, and Stefano Carrazza

   Quantum Science & Technology, 2022

   Abs DOI arXiv

   We present Qibo, a new open-source software for fast evaluation of quantum circuits and adiabatic evolution which takes full advantage of hardware accelerators. The growing interest in quantum computing and the recent developments of quantum hardware devices motivates the development of new advanced computational tools focused on performance and usage simplicity. In this work we introduce a new quantum simulation framework that enables developers to delegate all complicated aspects of hardware or platform implementation to the library so they can focus on the problem and quantum algorithms at hand. This software is designed from scratch with simulation performance, code simplicity and user friendly interface as target goals. It takes advantage of hardware acceleration such as multi-threading Central Processing Unit (CPU), single Graphics Processing Unit (GPU) and multi-GPU devices.

2021

1. Phys. Rev. A

   

   Quantum unary approach to option pricing

   Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego Garcı́a-Martı́n, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planaguma, and José I Latorre

   Physical Review A, 2021

   Abs DOI arXiv

   We present a quantum algorithm for European option pricing in finance, where the key idea is to work in the unary representation of the asset value. The algorithm needs novel circuitry and is divided in three parts: first, the amplitude distribution corresponding to the asset value at maturity is generated using a low-depth circuit; second, the computation of the expected return is computed with simple controlled gates; and third, standard amplitude estimation is used to gain quantum advantage. On the positive side, unary representation remarkably simplifies the structure and depth of the quantum circuit. Amplitude distributions use quantum superposition to bypass the role of classical Monte Carlo simulation. The unary representation also provides a postselection consistency check that allows for a substantial mitigation in the error of the computation. On the negative side, unary representation requires linearly many qubits to represent a target probability distribution, as compared to the logarithmic scaling of binary algorithms. We compare the performance of both unary vs binary option pricing algorithms using error maps, and find that unary representation may bring a relevant advantage in practice for near-term devices.

2. MLST

   

   Quantum autoencoders with enhanced data encoding

   Carlos Bravo-Prieto

   Machine Learning: Science and Technology, 2021

   Abs DOI arXiv

   We present the enhanced feature quantum autoencoder, or EF-QAE, a variational quantum algorithm capable of compressing quantum states of different models with higher fidelity. The key idea of the algorithm is to define a parameterized quantum circuit that depends upon adjustable parameters and a feature vector that characterizes such a model. We assess the validity of the method in simulations by compressing ground states of the Ising model and classical handwritten digits. The results show that EF-QAE improves the performance compared to the standard quantum autoencoder using the same amount of quantum resources, but at the expense of additional classical optimization. Therefore, EF-QAE makes the task of compressing quantum information better suited to be implemented in near-term quantum devices.

2020

1. Phys. Rev. A

   

   Quantum singular value decomposer

   Carlos Bravo-Prieto, Diego Garcı́a-Martı́n, and José I Latorre

   Physical Review A, 2020

   Abs DOI arXiv

   We present a variational quantum circuit that produces the singular value decomposition of a bipartite pure state. The proposed circuit, which we name quantum singular value decomposer or QSVD, is made of two unitaries respectively acting on each part of the system. The key idea of the algorithm is to train this circuit so that the final state displays exact output coincidence from both subsystems for every measurement in the computational basis. Such circuit preserves entanglement between the parties and acts as a diagonalizer that delivers the eigenvalues of the Schmidt decomposition. Our algorithm only requires measurements in one single setting, in striking contrast to the 3ⁿ settings required by state tomography. Furthermore, the adjoints of the unitaries making the circuit are used to create the eigenvectors of the decomposition up to a global phase. Some further applications of QSVD are readily obtained. The proposed QSVD circuit allows us to construct a SWAP between the two parties of the system without the need of any quantum gate communicating them. We also show that a circuit made with QSVD and CNOTs acts as an encoder of information of the original state onto one of its parties. This idea can be reversed and used to create random states with a precise entanglement structure.

2. Quantum

   

   Scaling of variational quantum circuit depth for condensed matter systems

   Carlos Bravo-Prieto, Josep Lumbreras-Zarapico, Luca Tagliacozzo, and José I Latorre

   Quantum, 2020

   Abs DOI arXiv

   We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy improves slowly with the number of layers, and a finite-size regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.

3. Entropy

   

   Measuring the tangle of three-qubit states

   Adrián Pérez-Salinas, Diego Garcı́a-Martı́n, Carlos Bravo-Prieto, and José I Latorre

   Entropy, 2020

   Abs DOI arXiv

   We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.