arXiv subjects

Jérémie Messud

Publications and source records attributed to Jérémie Messud.

12 recordsLinked to original sources

arXiv2026

Orthogonal Quantum Krylov Diagonalisation

Quantum subspace-diagonalization methods, particularly Quantum Krylov Diagonalization (QKD), provide a promising route for computing low-energy spectra of quantum many-body Hamiltonians. However, existing quantum Krylov approaches rely on non-orthogonal Krylov bases, requiring overlap-matrix regularization that limits numerical stability and accuracy. In this work, we introduce an Orthogonal Quantum Krylov Diagonalization (OQKD) framework that reformulates the classical Lanczos recursion at the operator level, enabling an orthogonal quantum implementation of Krylov-subspace diagonalization. By expressing Lanczos vectors as polynomial transformations of the Hamiltonian, OQKD reproduces the orthogonality, tridiagonal structure, and convergence behavior of the classical Lanczos algorithm thus eliminating the need for overlap-matrix regularization. We further show that the required Lanczos polynomials can be implemented using block encoding and Generalized Quantum Signal Processing with the same asymptotic query complexity as Chebyshev-based QKD methods. Numerical simulations of the $J_1$--$J_2$ Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, while the measurement-complexity scaling is established analytically. Building upon the OQKD framework, we then introduce a restarted state-preparation protocol that replaces a single high-degree polynomial transformation with a sequence of fixed low-degree transformations, maintaining an affordable block encoding success probability while retaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state-preparation strategy for Quantum Phase Estimation.

quant-ph↗

arXiv2026

Refining Quantum Phase Estimation Precision Conditions on Unitaries for Many-Electron Systems

Beyond ground state energy estimation, quantum phase estimation (QPE) applied to many-electron systems has the potential to output an approximation of the ground state, enabling in a second step an evaluation of observables other than the energy. We here focus on the impact of approximate controlled-unitaries implementations on QPE precision. After recalling the role of the QPE free parameters, we derive first-order and unified conditions on the unitaries that are necessary to control the QPE energy estimation precision together with the QPE output state precision, important in case we want to leverage the full potential of QPE. We apply these conditions to a Trotterization case, leading to tighter or more general bounds than in previous works. The main results in this article are formal. First numerical illustrations on the H2 molecule provide useful insights.

quant-ph↗

arXiv2026

On the robustness of Quantum Phase Estimation to compute ground properties of many-electron systems

We propose an analysis of the Quantum Phase Estimation (QPE) algorithm applied to many-electron systems by investigating its free parameters such as the time step, number of phase qubits, initial state preparation, number of measurement shots, and other parameters related to the unitary operators implementation. A deep understanding of these parameters and their impact on QPE probability of success and precision of the results is important to pave the way towards more automation of QPE applied to predictive computational chemistry and material science. We here explicit a constructive method to set the QPE free parameters for ground energy estimation and ground state projection, gathering disseminated results from previous works, refining these results and developing new conditions for achieving target performance. We detail the impact of the QPE `blurring function', related to discretization effects, and propose a method to overcome corresponding pathologies. We finally demonstrate that, using the conditions gathered here, the complexity of the Trotterized version of QPE tends to depend mostly on physical system properties and weakly on the number of phase qubits. Various numerical results illustrate the impact of QPE free parameters on success probability and discretization effects. The impact of Trotterization and other features on the precision of the results are illustrated by first numerical simulations on the H2 molecule, that allows us to derive useful insights.

quant-ph↗

arXiv2026

From quantum to quantum-inspired: the LogQ algorithm as a non-linear continuous relaxation of variables method

The LogQ algorithm encodes Quadratic Unconstrained Binary Optimization (QUBO) problems, which are often encountered in the industry (portfolio optimization, fleet optimization, charging stations, etc.). It was developed within the framework of quantum computing, designed as a pragmatic approach to quantum combinatorial optimization that drastically reduces the number of required qubits and quantum circuit depth. While LogQ has recently been made compliant with gradient-inspired methods, greatly improving parameter optimization efficiency, it still faced hurdles regarding Pauli decomposition and measurement overhead. We here demonstrate that LogQ can be fully reformulated within a classical framework, which effectively eliminates the need for Pauli decomposition and bypasses the measurement challenges altogether. This finally leads to a classical heuristic based on a non-linear continuous relaxation of variables and is, to the best of our knowledge, novel. The LogQ story illustrates how quantum computing can inspire classical algorithms, leading to so-called "quantum-inspired" methods.

math.OC↗

arXiv2026

Portfolio Optimization with 'Physical' Decision Variables and Non-Linear Performance Metrics: Diversification Challenge and Proposals

Portfolio optimization (PO) is a core tool in financial and operational decision-making, typically balancing expected profit and risk. In real-world applications, particularly in the energy sector, decision variables can be expressed as physical quantities (e.g., production volumes), and nonlinear performance metrics such as Return on Investment (ROI) may be requested. These modeling choices introduce challenges, including the non-additivity of the objective function. This often results in highly concentrated optimized portfolios and thus limited diversification, which can be problematic for decision-makers seeking balanced investment strategies. This paper proposes two strategies to enhance diversification in ROI-based PO models, both based on the Herfindahl-Hirschman Index (HHI). The first incorporates an HHI term directly into the objective function, with its corresponding weight allowing control over diversification. The second directly maximizes diversification while controlling expected profit and risk degradation around the optimum portfolio (obtained through conventional PO). Both strategies are evaluated using synthetic data (energy assets) to illustrate their behavior and practical trade-offs. The results highlight how each method can support different decision-making needs and enhance portfolio robustness.

math.OC↗

arXiv2025

Unified Approach to Portfolio Optimization using the `Gain Probability Density Function' and Applications

This article proposes a unified framework for portfolio optimization (PO), recognizing an object called the `gain probability density function (PDF)' as the fundamental object of the problem from which any objective function could be derived. The gain PDF has the advantage of being 1-dimensional for any given portfolio and thus is easy to visualize and interpret. The framework allows us to naturally incorporate all existing approaches (Markowitz, CVaR-deviation, higher moments...) and represents an interesting basis to develop new approaches. It leads us to propose a method to directly match a target PDF defined by the portfolio manager, giving them maximal control on the PO problem and moving beyond approaches that focus only on expected return and risk. As an example, we develop an application involving a new objective function to control high profits, to be applied after a conventional PO (including expected return and risk criteria) and thus leading to sub-optimality w.r.t. the conventional objective function. We then propose a methodology to quantify a cost associated with this optimality deviation in a common budget unit, providing a meaningful information to portfolio managers. Numerical experiments considering portfolios with energy-producing assets illustrate our approach. The framework is flexible and can be applied to other sectors (financial assets, etc).

q-fin.PM↗

arXiv2025

Towards solving large QUBO problems using quantum algorithms: improving the LogQ scheme

The LogQ algorithm encodes Quadratic Unconstrained Binary Optimization (QUBO) problems with exponentially fewer qubits than the Quantum Approximate Optimization Algorithm (QAOA). The advantages of conventional LogQ are accompanied by a challenge related to the optimization of its free parameters, which requires the usage of resource intensive evolutionary or even global optimization algorithms. We propose a new LogQ parameterization that can be optimized with a gradient-inspired method, which is less resource-intensive and thus strengthens the advantage of LogQ over QAOA for large/industrial problems. We illustrate the features of our method on an analytical model and present larger scale numerical results on MaxCut problems.

quant-ph↗

arXiv2022

A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform

We consider the high-resolution seismic imaging method called full-waveform inversion (FWI). FWI is a data fitting method aimed at inverting for subsurface mechanical parameters. Despite the large adoption of FWI by the academic and industrial communities, and many successful results, FWI still suffers from severe limitations. From a mathematical standpoint, FWI is a large scale PDE-constrained optimization problem. The misfit function that is used, which measures the discrepancy between observed seismic data and data calculated through the solution of a wave propagation problem, is non-convex. After discretization, the size of the FWI problem requires the use of local optimization solvers, which are prone to converge towards local minima. Thus the success of FWI strongly depends on the choice of the initial model to ensure the convergence towards the global minimum of the misfit function. This limitation has been the motivation for a large variety of strategies. Among the different methods that have been investigated, the use of optimal transport (OT) distances-based misfit functions has been recently promoted. The leading idea is to benefit from the inherent convexity of OT distances with respect to dilation and translation to render the FWI problem more convex. However, the application of OT distances in the framework of FWI is not straightforward, as seismic data is signed, while OT has been developed for the comparison of probability measures. The purpose of this study is to review two methods that were developed to overcome this difficulty. Both have been successfully applied to field data in an industrial framework. Both make it possible to better exploit the seismic data, alleviating the sensitivity to the initial model and to various conventional workflow steps, and reducing the uncertainty attached to the subsurface mechanical parameters inversion.

physics.geo-ph↗

arXiv2021

Notes on open questions within density functional theory (existence of a derivative of the Lieb functional in a restricted sense and non-interacting $v$-representability)

Density functional theory together with the Kohn-Sham scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary ``non-interacting'' system (one-body with a local potential). However, theoretical questions remain open. An important one is related to the existence of a derivative of the Lieb functional in some restricted (weak) sense (as pointed out by Lammert). Then, a further one would be related to the validity of the ``non-interacting $v$-representability'' conjecture. We gather here elements on these questions, providing reminders and thoughts that are non-conclusive but hopefully contribute to the reflection. This document must be considered as notes (it does not represent an article).

quant-ph↗

arXiv2021

Optimal transport in full-waveform inversion: Analysis and practice of the multidimensional Kantorovich-Rubinstein norm

In the last ten years, full-waveform inversion has emerged as a robust and efficient high-resolution velocity model-building tool for seismic imaging, with the unique ability to recover complex subsurface structures. Originally based on a data fitting process using a least-squares cost function, it suffered from high sensitivity to cycle-skipping and was therefore of poor efficiency in handling large time shifts between observed and modelled seismic events. To tackle this problem, a common practice is to start the inversion using the low temporal frequencies of the data and selecting diving wave events. Complementary to this, the use of other cost functions has been investigated. Among these, cost functions based on optimal transport appeared appealing to possibly handle large time shifts between seismic events. Several strategies inspired by optimal transport have been proposed, taking into account the specificities of seismic data. Among them, the approach based on the Kantorovich-Rubinstein norm offers the possibility of the direct use of seismic data and an efficient numerical implementation allowing for a multidimensional (data coordinate space) application. We present here an analysis of the Kantorovich-Rubinstein norm, discussing its theoretical and practical aspects. A key component of our analysis is the back-propagated adjoint-source. We highlight its piecewise linearity, analyze its frequency content and amplitude balancing. We also emphasize the benefit of having a multidimensional implementation. Furthermore, we give practical rules for setting the tuning parameters of the numerical implementation. Our set of synthetic and field data examples demonstrate the improvements brought by the use of the Kantorovich-Rubinstein norm over least-squares full-waveform inversion, and highlight the improvements brought by the multidimensional approach over the one-dimensional one.

physics.geo-ph↗

arXiv2020

On tomography velocity uncertainty in relation with structural imaging

Evaluating structural uncertainties associated with seismic imaging and target horizons can be of critical importance for decision-making related to oil and gas exploration and production. An important breakthrough for industrial applications has been made with the development of industrial approaches to velocity model building. We propose an extension of these approaches, sampling an equi-probable contour of the tomography posterior probability density function (pdf) rather than the full pdf, and using non-linear slope tomography (rather than standard tomographic migration velocity analysis as in previous publications). Our approach allows to assess the quality of uncertainty-related assumptions (linearity and Gaussian hypothesis within the Bayesian theory) and estimate volumetric migration positioning uncertainties (a generalization of horizon uncertainties), in addition to the advantages in terms of efficiency. We derive the theoretical concepts underlying this approach and unify our derivations with those of previous publications. As the method works in the full model space rather than in a preconditioned model space, we split the analysis into the resolved and unresolved tomography spaces. We argue that the resolved space uncertainties are to be used in further steps leading to decision-making and can be related to the output of methods that work in a preconditioned model space. The unresolved space uncertainties represent a qualitative byproduct specific to our method, strongly highlighting the most uncertain gross areas, thus useful for QCs. These concepts are demonstrated on a synthetic data. Complementarily, the industrial viability of the method is illustrated on two different 3D field datasets.

physics.geo-ph↗

arXiv2020

Adversarial training for predictive tasks: theoretical analysis and limitations in the deterministic case

To train a deep neural network to mimic the outcomes of processing sequences, a version of Conditional Generalized Adversarial Network (CGAN) can be used. It has been observed by others that CGAN can help to improve the results even for deterministic sequences, where only one output is associated with the processing of a given input. Surprisingly, our CGAN-based tests on deterministic geophysical processing sequences did not produce a real improvement compared to the use of an $L_p$ loss; we here propose a first theoretical explanation why. Our analysis goes from the non-deterministic case to the deterministic one. It led us to develop an adversarial way to train a content loss that gave better results on our data.

stat.ML↗