Luca Nutricati

Abstract:

Abstract We present a method to construct approximate analytic expressions for Ricci-flat K"ahler metrics on Calabi–-Yau threefolds with explicit dependence on the K"ahler moduli. Our strategy combines numerical data obtained from machine learning with an explicit analytic Ansatz for the K"ahler potential and symbolic regression methods. Specifically, we use neural networks to learn the K"ahler potential at selected points in K"ahler moduli space, fit this data to analytic expressions with K"ahler moduli-dependent parameters, and determine an analytic form of these coefficients as functions of the K"ahler moduli using symbolic regression. In this way, we reconstruct closed-form approximations to the Ricci-flat metric that retain explicit K"ahler-moduli dependence. We apply this method to two Calabi–-Yau threefolds with $h^{1,1}=2$, namely a bicubic hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$ and a bi-degree $(2,4)$ hypersurface in $\mathbb{P}^1 \times \mathbb{P}^3$, both of which admit nontrivial discrete symmetry groups that simplify the structure of the metric. In both cases, the resulting analytic expressions reproduce the numerically learned K"ahler potentials with percent-level accuracy and yield a Ricci-flatness measure that remains sufficiently small across the sampled region. Our results represent a concrete bridge between purely numerical results for Calabi–-Yau metrics and analytic constructions, opening the door to a systematic study of their dependence on K"ahler moduli.

Abstract:

Quantum annealing is a computational paradigm in which optimisation problems are encoded in the energy landscape of an interacting quantum system and explored through its dynamical evolution. By continuously transforming an initial Hamiltonian into one whose ground state represents the solution, the system navigates complex energy landscapes through a combination of quantum fluctuations, tunnelling processes, and dissipative dynamics. Quantum annealing is primarily designed for discrete optimisation and sampling tasks and provides a physically motivated heuristic for exploring rugged landscapes that arise across science and engineering. Modern quantum annealers realise programmable spin systems with thousands of qubits, making them among the largest controllable quantum devices currently available. Beyond optimisation, they also serve as experimental platforms for studying non-equilibrium many-body quantum dynamics in regimes that are challenging to access classically. In this review we introduce the principles of quantum annealing, describe the main hardware platforms and algorithmic techniques, and analyse the roles of tunnelling, spectral gaps, and open-system effects in determining performance. We survey applications ranging from optimisation and machine learning to quantum simulation and many-body physics, and discuss the central challenges of benchmarking, scaling, and control. These developments position quantum annealing at the interface of optimisation, stochastic sampling, and programmable quantum dynamics.

Abstract:

One of the main challenges in solving combinatorial optimisation problems with quantum annealers is the emergence of extremely small energy gaps between the ground state and the first excited state of the annealing Hamiltonian. These small gaps may be symptoms of an underlying first-order phase transition, which, according to the adiabatic theorem, can significantly extend the required anneal time, making practical implementation effectively infeasible. In this paper we demonstrate that attaching an XX-catalyst on all the edges of a graph upon which a MWIS (Maximum Weighted Independent Set) problem is defined, significantly enhances the minimum energy gap. Remarkably, our analysis shows that the smaller the energy gap, the more effective the catalyst is in opening it. This result is based on a detailed statistical analysis performed on a large number of randomly generated MWIS problem instances on both Erdõs–Rényi and Barabáasi–Albert graphs. We perform the analysis using both stoquastic and non-stoquastic catalysts.