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Alessandro Luongo

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4 papers

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AAAI Conference 2026 Conference Paper

Quantum Algorithms for Spectral Sums

Alessandro Luongo
Changpeng Shao

We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. For a matrix A and a function f, the spectral sum is the trace of f(A), equivalently the sum over eigenvalues of A of f applied to each eigenvalue. Typical examples of spectral sums are the von Neumann entropy, the trace of the inverse of A, the log-determinant, and the Schatten p-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees in a graph.

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ICML Conference 2025 Conference Paper

The Generalized Skew Spectrum of Graphs

Armando Bellante
Martin Plávala
Alessandro Luongo

This paper proposes a family of permutation-invariant graph embeddings, generalizing the Skew Spectrum of graphs of Kondor & Borgwardt (2008). Grounded in group theory and harmonic analysis, our method introduces a new class of graph invariants that are isomorphism-invariant and capable of embedding richer graph structures - including attributed graphs, multilayer graphs, and hypergraphs - which the Skew Spectrum could not handle. Our generalization further defines a family of functions that enables a trade-off between computational complexity and expressivity. By applying generalization-preserving heuristics to this family, we improve the Skew Spectrum’s expressivity at the same computational cost. We formally prove the invariance of our generalization, demonstrate its improved expressiveness through experiments, and discuss its efficient computation.

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ICML Conference 2020 Conference Paper

Quantum Expectation-Maximization for Gaussian mixture models

Iordanis Kerenidis
Alessandro Luongo
Anupam Prakash

We define a quantum version of Expectation-Maximization (QEM), a fundamental tool in unsupervised machine learning, often used to solve Maximum Likelihood (ML) and Maximum A Posteriori (MAP) estimation problems. We use QEM to fit a Gaussian Mixture Model, and show how to generalize it to fit mixture models with base distributions in the exponential family. Given quantum access to a dataset, our algorithm has convergence and precision guarantees similar to the classical algorithm, while the runtime is polylogarithmic in the number of elements in the training set and polynomial in other parameters, such as the dimension of the feature space and the number of components in the mixture. We discuss the performance of the algorithm on a dataset that is expected to be classified successfully by classical EM and provide guarantees for its runtime.

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NeurIPS Conference 2019 Conference Paper

q-means: A quantum algorithm for unsupervised machine learning

Iordanis Kerenidis
Jonas Landman
Alessandro Luongo
Anupam Prakash

Quantum information is a promising new paradigm for fast computations that can provide substantial speedups for many algorithms we use today. Among them, quantum machine learning is one of the most exciting applications of quantum computers. In this paper, we introduce q-means, a new quantum algorithm for clustering. It is a quantum version of a robust k-means algorithm, with similar convergence and precision guarantees. We also design a method to pick the initial centroids equivalent to the classical k-means++ method. Our algorithm provides currently an exponential speedup in the number of points of the dataset, compared to the classical k-means algorithm. We also detail the running time of q-means when applied to well-clusterable datasets. We provide a detailed runtime analysis and numerical simulations for specific datasets. Along with the algorithm, the theorems and tools introduced in this paper can be reused for various applications in quantum machine learning.

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