FTQC quantum block encoding for the partial differential equations

il y a 11 heures

Palaiseau, Île-de-France EDF Temps plein

As computational demands for simulating large-scale infrastructures grow, new approaches are needed to overcome the limitations of classical simulation technologies. The Finite Element Method (FEM) remains a cornerstone in industrial applications, especially for ensuring the safety and stability of complex structures. However, quantum computing offers promising avenues to address challenges such as scalability and mesh dependency.

A key challenge in quantum computing is efficiently encoding classical data into quantum systems via quantum gates and extracting meaningful outputs through measurements. This is particularly relevant for Variational Quantum Algorithms (VQAs), which introduce data additively and process it iteratively.

At EDF R&D's ERMES department, research began in late 2021 to apply VQAs to solve Partial Differential Equations (PDEs), starting with linear problems like fracture mechanics and heat diffusion [3-4]. Initial studies successfully reformulated FEM problems within the VQA framework and executed them on various quantum processing units (QPUs), demonstrating the feasibility of quantum-enhanced engineering simulations.

While VQAs are suitable for Noisy Intermediate-Scale Quantum (NISQ) devices, they face limitations such as the barren plateau phenomenon, which impedes convergence and lacks formal scalability guarantees. In contrast, Fault-Tolerant Quantum Computing (FTQC) algorithms—like the Harrow-Hassidim-Lloyd (HHL) algorithm [1]—offer more robust solutions by avoiding these issues.

To address the bottleneck in data encoding, this internship will explore block encoding techniques, as introduced by Lin Lin [2], which represent classical information as unitary matrix multiplications within a quantum system. This builds on previous results [3,4] and aims to enhance both expressivity and computational efficiency in quantum simulations.

The proposed scheme will be tested on EDF's high-performance computing (HPC) emulator of a QPU, enabling benchmarking and scalability analysis under realistic conditions. Additionally, the student may contribute to a long-term research goal: the quantum reformulation of non-linear PDEs, potentially forming the basis for a future PhD proposal.

[1] Harrow, Aram W., Hassidim, Avinatan, and Lloyd, Seth.

Quantum Algorithm for Linear Systems of Equations. Physical Review Letters, 103,

[2] Lin Lin. Lecture notes on quantum algorithms for scientific computation. arXiv: , 2022

[3] U Remond, PE Emeriau, L Lysaght, J Ruel, J Mikael, K Kazymyrenko. Quantum remeshing and efficient encoding for fracture mechanics; arXiv:

[4] S Donachie, U Remond, A Mathorel, K Kazymyrenko. Solving the 3D Heat Equation with VQA via Remeshing-Based Warm Starts; arXiv:

Qualifications

Master 2 level or equivalent, in applied math, quantum physics or mechanics

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