arXiv subjects

Soumyadip Das

Publications and source records attributed to Soumyadip Das.

11 recordsLinked to original sources

arXiv2025

Image Denoising via Quantum Reservoir Computing

Quantum Reservoir Computing (QRC) leverages the natural dynamics of quantum systems for information processing, without requiring a fault-tolerant quantum computer. In this work, we apply QRC within a hybrid quantum classical framework for image denoising. The quantum reservoir is implemented using a Rydberg atom array, while a classical neural network serves as the readout layer. To prepare the input, images are first compressed using Principal Component Analysis (PCA), reducing their dimensionality to match the size of the atom array. Each feature vector is encoded into local detuning parameters of a time-dependent Hamiltonian governing the Rydberg system. As the system evolves, it generates nonlinear embeddings through the measurement of observables across multiple time steps. These temporal embeddings capture complex correlations, which are fed into a classical neural network to reconstruct the denoised images. To evaluate performance, we compare this QRC-assisted model against a baseline architecture consisting of PCA followed by a dense neural network, trained under identical conditions. Our results show that the QRC-based approach achieves improved image sharpness and similar structural recovery compared to the PCA-based model. We demonstrate the practical viability of this framework through experiments on QuEra's Aquila neutral-atom processor, leveraging its programmable atom arrays to physically realize the reservoir dynamics.

quant-ph↗

arXiv2025

Grid-Partitioned MWIS Solving with Neutral Atom Quantum Computing for QUBO Problems

Quadratic Unconstrained Binary Optimization (QUBO) problems are prevalent in real-world applications, such as portfolio optimization, but pose significant computational challenges for large-scale instances. We propose a hybrid quantum-classical framework that leverages neutral atom quantum computing to address QUBO problems by mapping them to the Maximum Weighted Independent Set (MWIS) problem on unit disk graphs. Our approach employs spatial grid partitioning to decompose the problem into manageable subgraphs, solves each subgraph using Analog Hamiltonian Simulation (AHS), and merges solutions greedily to approximate the global optimum. We evaluate the framework on a 50-asset portfolio optimization problem using historical S&P 500 data, benchmarking against classical simulated annealing. Results demonstrate competitive performance, highlighting the scalability and practical potential of our method in the Noisy Intermediate-Scale Quantum (NISQ) era. As neutral atom quantum hardware advances, our framework offers a promising path toward solving large-scale optimization problems efficiently.

quant-ph↗

arXiv2024

Construction of the Moduli Space of Vector Bundles on an Orbifold Curve

Let $k$ be an algebraically closed field of any characteristic, and let $(X,P)$ be an orbifold curve over $k$. We construct the moduli space $\mathrm{M}_{(X,P)}^{\mathrm{ss}}(n, Δ)$ of $P$-semistable bundles on $(X,P)$ of rank $n$ and determinant $Δ$. In the characteristic zero case, this result is well known and follows from GIT techniques. Our construction follows a different approach inspired by a GIT-free construction of Faltings. We show that when the moduli space is non-empty, it is a finite disjoint union of irreducible projective varieties.

math.AG↗

arXiv2024

On the Covers of Orbifold Curves Preserving the Slope Stability under Pullback

We completely characterize the covers of connected orbifold curves which preserve slope stability of vector bundles under the pullback morphism. More precisely, given a cover $f \colon (Y,Q) \longrightarrow (X,P)$ of connected orbifold curves, we show that the maximal destabilizing sub-bundle of the pushforward sheaf $f_*\mathcal{O}_{(Y,Q)}$ defines the maximal étale sub-cover of $f$. The cover $f$ is said to be genuinely ramified if $f$ does not factor through any non-trivial étale sub-cover. Our main result states that the class of covers $f$ that preserves the stable bundles under a pullback are precisely the class of genuinely ramified covers $f$. Further, we establish equivalent conditions for the cover $f$ to be genuinely ramified, generalizing earlier works on covers of curves. We thoroughly study the slope stability conditions of bundles on an orbifold curve, their properties under the pushforward and pullback maps under covers with a stand point of Deligne-Mumford stacks, hence giving a solid foundation of the subject. As a consequence, we also answer the question of descent of stable bundles under genuinely ramified covers.

math.AG↗

arXiv2022

Towards the Generalized Purely Wild Inertia Conjecture for product of Alternating and Symmetric Groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi $p$-groups (groups generated by their Sylow $p$-subgroups) if the conjecture is established for individual groups.

math.AG↗

arXiv2022

Stability of pullback of orbifold bundles

In this article, we study the behavior of the stability of pullback of a vector bundle under a finite morphism from a (not necessarily smooth) stacky curve to an orbifold curve. We establish a categorical equivalence between proper formal orbifold curves and proper orbifold curves in the sense of Deligne-Mumford stacks. Using this identification, we define the notion of slope $P$-(semi)stability of vector bundles on proper formal orbifold curves $(X,P)$. We establish some equivalent conditions for a stacky genuinely ramified morphism, analogous to the case of curves. Finally, we show that for a cover of an orbifold curve arising as a cartesian pullback via a genuinely ramified morphism of smooth projective connected curves, the orbifold slope stability is preserved under the pullback.

math.AG↗

arXiv2022

Galois Covers of Singular Curves in Positive Characteristics

We study the étale fundamental groups of singular reduced connected curves defined over an algebraically closed field of arbitrary prime characteristic. It is shown that when the curve is projective, the étale fundamental group is a free product of the étale fundamental group of its normalization with a free finitely generated profinite group whose rank is well determined. As a consequence of this result and the known results for the smooth case, necessary conditions are given for a finite group to appear as a quotient of the étale fundamental group. Next, we provide similar results for an affine integral curve $U$. We provide a complete group theoretic classification on which finite groups occur as the Galois groups for Galois étale connected covers of $U$. In fact, when $U$ is a seminormal curve embedded in a connected seminormal curve $X$ such that $X - U$ consists of smooth points, the tame fundamental group $π_1^t(U \subset X)$ is shown to be isomorphic to a free product of the tame fundamental group of the normalization of $U$ with a free finitely generated profinite group whose rank is known. An analogue of the Inertia Conjecture is also posed for certain singular curves.

math.AG↗

arXiv2022

Genuinely ramified maps and stable vector bundles

Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups $f_* : π_1^{\rm et}(X)\rightarrowπ_1^{\rm et}(Y)$ is surjective. Fix a polarization on $Y$ and equip $X$ with the pullback, by $f$, of this polarization on $Y$. Given a stable vector bundle $E$ on $X$, we prove that there is a vector bundle $W$ on $Y$ with $f^*W$ isomorphic to $E$ if and only if the direct image $f_*E$ contains a stable vector bundle $F$ such that $$ \frac{{\rm degree}(F)}{{\rm rank}(F)}= \frac{1}{{\rm degree}(f)}\cdot \frac{{\rm degree}(E)}{{\rm rank}(E)} $$ We also prove that $f^*V$ is stable for every stable vector bundle $V$ on $Y$.

math.AG↗

arXiv2020

On the Inertia Conjecture and its generalizations

Studying two point branched Galois covers of the projective line we prove the Inertia Conjecture for the Alternating groups $A_{p+1}$, $A_{p+3}$, $A_{p+4}$ for any odd prime $p \equiv 2 \pmod{3}$ and for the group $A_{p+5}$ when additionally $4 \nmid (p+1)$ and $p \geq 17$. We obtain a generalization of a patching result by Raynaud which reduces these proofs to showing the realizations of the étale Galois covers of the affine line with a fewer candidates for the inertia groups above $\infty$. We also pose a general question motivated by the Inertia Conjecture and obtain some affirmative results. A special case of this question, which we call the Generalized Purely Wild Inertia Conjecture, is shown to be true for the groups for which the purely wild part of the Inertia Conjecture is already established. In particular, we show that if this generalized conjecture is true for the groups $G_1$ and $G_2$ which do not have a common quotient, then the conjecture is also true for the product $G_1 \times G_2$.

math.AG↗

arXiv2020

Local Oort groups and the isolated differential data criterion

It is conjectured that if k is an algebraically closed field of characteristic p > 0, then any branched G-cover of smooth projective k-curves where the "KGB" obstruction vanishes and where a p-Sylow subgroup of G is cyclic lifts to characteristic 0. Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on P_1^k with behavior determined by the ramification data of the cover. We give a more efficient computational procedure to compute these forms than was previously known. As a consequence, we show that all D_25- and D_27-covers lift to characteristic zero.

math.AG↗

arXiv2018

On The Inertia Conjecture for Alternating group Covers

The wild part of Abhyankar's Inertia Conjecture for a product of certain Alternating groups is shown for any algebraically closed field of odd characteristic. For $d$ a multiple of the characteristic of the base field, a new étale $A_d$-cover of the affine line is obtained using an explicit equation and it is shown that it has the minimal possible upper jump.

math.AG↗