Latest update: 04 October 2026

Research

Connes Distance in Discrete-Time Quantum Walks Sept '26 arxiv

  supervised by Pablo Arrighi and A.F. Reyes-Lega, QuaCS team, LMF (Laboratoire Méthodes Formelles)

We study the geometric spread of discrete-time quantum walks using Connes' distance formula. We introduce its discrete analogue by adapting the continuous spectral metric to the lattice, with the Dirac operator replaced by the unitary step operator. Focusing first on a (2+1)-dimensional quantum walk, we evaluate the dynamical spread of localized wavepackets to compare propagation in free space against that under a uniform magnetic field. We find that while the free walk shows linear ballistic growth, gauge-induced phases in the magnetic walk constrain the admissible test operators, suppressing the long-time geometric expansion. Furthermore, we apply this discrete framework to a (1+1)-dimensional plastic quantum walk simulating an inhomogeneous spatial metric. The results demonstrate that the discrete Connes distance naturally captures the underlying spatially varying metric, providing a robust measure of the wavepacket's spread that adapts directly to the local simulated geometry.

You can find here informal slides.

From Symmetric Toeplitz Hamiltonians to Quantum Circuits Aug '25 arxiv

  supervised by Benoît Valiron, QuaCS team, LMF (Laboratoire Méthodes Formelles)

This work introduces a quantum circuit synthesis framework for simulating the unitary time evolution under a subclass of symmetric Toeplitz Hamiltonians by decomposing them into specific diagonal matrices M_k. These matrices are then classified, to achieve significant simplification, into, M_k when k is a power of two, and congruence classes with constant coefficients. Finally, the practical utility of this approach is demonstrated by applying it to a canonical problem in physics: the simulation of the one dimensional Poisson equation.

For more details, read the following report (less rigorous, more intuitive)