arXiv subjects

Adam Wesołowski

Publications and source records attributed to Adam Wesołowski.

7 recordsLinked to original sources

arXiv2026

Torsion detection in clique complexes is conditionally $QMA_1$-hard

Quantum algorithms for topological data analysis compute Betti numbers, the ranks of the homology groups of a simplicial complex, which can be read off from the kernel of a combinatorial Laplacian. Deciding whether a Betti number of a clique complex is nonzero is $QMA_1$-hard, and remains so under a spectral gap promise on vertex-weighted graphs. Integral homology, however, contains information inaccessible to the Laplacian spectrum. A new part that appears in integral homology is torsion: cycles that become boundaries only after being traversed several times, as in a projective plane or a Klein bottle. We ask how hard it is to detect torsion, and we answer with a simple reduction. We attach to an arbitrary clique complex a fixed $31$-vertex triangulation of the projective plane. The $k$th mod-$2$ Betti number of the input then reappears as $2$-torsion two degrees up, while all rational homology disappears and every combinatorial Laplacian acquires a constant spectral gap. We conclude that detecting torsion in clique complexes of unweighted graphs is $NP$-hard, even under a constant gap promise, and that it is $QMA_1$-hard if mod-$2$ clique homology is.

quant-ph↗

arXiv2026

Average-case hardness of Betti number estimation

We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below $\tfrac12$ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.

quant-ph↗

arXiv2026

Quantum Topological Data Encoding

Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.

quant-ph↗

arXiv2026

Advances in quantum algorithms for the shortest path problem

Given an undirected, weighted graph, with $n$ vertices and $m$ edges, and two special vertices $s$ and $t$, the problem is to find the shortest path between them. We give two bounded-error quantum algorithms with improved runtime in the adjacency list model that solve the problem on special classes of graphs defined via pathfinding probabilities of classical random walks and the electrical network framework. Firstly, we give a simple quantum algorithm based on sampling edges from a graph via the quantum flow state and running a classical algorithm on the sampled edges. It runs in $\tilde{O}(l^2\sqrt{m})$ expected time and uses $O(\log{n})$ space on graphs where the shortest $s$-$t$ path is also a minimum resistance $s$-$t$ subgraph. Our main algorithm can be thought of as a divide and conquer version of this approach and works on a special class of graphs where classical loop-erased random walk has a probability $q>0.537$ of finding the shortest $s$-$t$ path. In such cases the quantum algorithm outputs the shortest $s$-$t$ path with high probability in $\widetilde{O}(\ell\sqrt{m})$ expected time and $O(\log{n})$ space, where $l$ is the length (or total weight, in case of weighted graphs) of the shortest $s$-$t$ path. This algorithm can be parallelised to $\tilde{O}(\sqrt{lm})$ circuit depth when using $O(l\log{n})$ space. With the latter we partially resolve with an affirmative answer the open problem of whether a path between two vertices can be found in the number of steps required to detect it.

quant-ph↗

arXiv2025

Fast, Accurate and Interpretable Graph Classification with Topological Kernels

We introduce a novel class of explicit feature maps based on topological indices that represent each graph by a compact feature vector, enabling fast and interpretable graph classification. Using radial basis function kernels on these compact vectors, we define a measure of similarity between graphs. We perform evaluation on standard molecular datasets and observe that classification accuracies based on single topological-index feature vectors underperform compared to state-of-the-art substructure-based kernels. However, we achieve significantly faster Gram matrix evaluation -- up to $20\times$ faster -- compared to the Weisfeiler--Lehman subtree kernel. To enhance performance, we propose two extensions: 1) concatenating multiple topological indices into an \emph{Extended Feature Vector} (EFV), and 2) \emph{Linear Combination of Topological Kernels} (LCTK) by linearly combining Radial Basis Function kernels computed on feature vectors of individual topological graph indices. These extensions deliver up to $12\%$ percent accuracy gains across all the molecular datasets. A complexity analysis highlights the potential for exponential quantum speedup for some of the vector components. Our results indicate that LCTK and EFV offer a favourable trade-off between accuracy and efficiency, making them strong candidates for practical graph learning applications.

cs.LG↗

arXiv2025

Quantum algorithms and lower bounds for eccentricity, radius, and diameter in undirected graphs

The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter and radius of undirected, weighted graphs in the adjacency list model. The algorithms output diameter and radius with the corresponding paths in $\widetilde{O}(n\sqrt{m})$ time. Additionally, for the diameter, we present a quantum algorithm that approximates the diameter within a $2/3$ ratio in $\widetilde{O}(\sqrt{m}n^{3/4})$ time. We also establish quantum query lower bounds of $Ω(\sqrt{nm})$ for all the aforementioned problems through a reduction from the minima finding problem.

quant-ph↗

arXiv2025

Benchmark of the Full and Reduced Effective Resistance Kernel for Molecular Classification

We present a comprehensive study of the commute time kernel method via the effective resistance framework analyzing the quantum complexity of the originally classical approach. Our study reveals that while there is a trade-off between accuracy and computational complexity, significant improvements can be achieved in terms of runtime efficiency without substantially compromising on precision. Our investigation highlights a notable quantum speedup in calculating the kernel, which offers a quadratic improvement in time complexity over classical approaches in certain instances. In addition, we introduce methodical improvements over the original work on the commute time kernel and provide empirical evidence suggesting the potential reduction of kernel queries without significant impact on result accuracy. Benchmarking our method on several chemistry-based datasets: $\tt{AIDS}$, $\tt{NCL1}$, $\tt{PTC-MR}$, $\tt{MUTAG}$, $\tt{PROTEINS}$ - data points previously unexplored in existing literature, shows that while not always the most accurate, it excels in time efficiency. This makes it a compelling alternative for applications where computational speed is crucial. Our results highlight the balance between accuracy, computational complexity, and speedup offered by quantum computing, promoting further research into efficient algorithms for kernel methods and their applications in chemistry-based datasets.

quant-ph↗