arXiv subjects

Charlotte Vermeylen

Publications and source records attributed to Charlotte Vermeylen.

8 recordsLinked to original sources

arXiv2026

A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography

Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.

quant-ph↗

arXiv2026

Tomography by Design: An Algebraic Approach to Low-Rank Quantum States

We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.

quant-ph↗

arXiv2025

Tensor Train Quantum State Tomography using Compressed Sensing

Quantum state tomography (QST) is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become impractical due to the exponential growth of parameters in the state representation. In this work, we address this challenge by parameterizing the state using a low-rank block tensor train decomposition and demonstrate that our approach is both memory- and computationally efficient. This framework applies to a broad class of quantum states that can be well approximated by low-rank decompositions, including pure states, nearly pure states, and ground states of Hamiltonians.

quant-ph↗

arXiv2025

Generalized cyclic symmetric decompositions for the matrix multiplication tensor

A new generalized cyclic symmetric structure in the factor matrices of polyadic decompositions of matrix multiplication tensors for non-square matrix multiplication is proposed to reduce the number of variables in the optimization problem and in this way improve the convergence. The structure is implemented in an existing numerical optimization algorithm. Extensive numerical experiments are given that the proposed structure indeed finds more (practical) decompositions.

math.NA↗

arXiv2024

A Riemannian rank-adaptive method for higher-order tensor completion in the tensor-train format

In this paper a new Riemannian rank adaptive method (RRAM) is proposed for the low-rank tensor completion problem (LRTCP) formulated as a least-squares optimization problem on the algebraic variety of tensors of bounded tensor-train (TT) rank. The method iteratively optimizes over fixed-rank smooth manifolds using a Riemannian conjugate gradient algorithm from Steinlechner (2016) and gradually increases the rank by computing a descent direction in the tangent cone to the variety. Additionally, a numerical method to estimate the amount of rank increase is proposed based on a theoretical result for the stationary points of the low-rank tensor approximation problem and a definition of an estimated TT-rank. Furthermore, when the iterate comes close to a lower-rank set, the RRAM decreases the rank based on the TT-rounding algorithm from Oseledets (2011) and a definition of a numerical rank. We prove that the TT-rounding algorithm can be considered as an approximate projection onto the lower-rank set which satisfies a certain angle condition to ensure that the image is sufficiently close to that of an exact projection. Several numerical experiments are given to illustrate the use of the RRAM and its subroutines in {\Matlab}. Furthermore, in all experiments the proposed RRAM outperforms the state-of-the-art RRAM for tensor completion in the TT format from Steinlechner (2016) in terms of computation time.

math.OC↗

arXiv2023

Stability Improvements for Fast Matrix Multiplication

We implement an Augmented Lagrangian method to minimize a constrained least-squares cost function designed to find polyadic decompositions of the matrix multiplication tensor. We use this method to obtain new discrete decompositions and parameter families of decompositions. Using these parametrizations, faster and more stable matrix multiplication algorithms can be discovered.

math.NA↗

arXiv2023

Rank Estimation for Third-Order Tensor Completion in the Tensor-Train Format

We propose a numerical method to obtain an adequate value for the upper bound on the rank for the tensor completion problem on the variety of third-order tensors of bounded tensor-train rank. The method is inspired by the parametrization of the tangent cone derived by Kutschan (2018). A proof of the adequacy of the upper bound for a related low-rank tensor approximation problem is given and an estimated rank is defined to extend the result to the low-rank tensor completion problem. Some experiments on synthetic data illustrate the approach and show that the method is very robust, e.g., to noise on the data.

math.OC↗

arXiv2023

An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank

An approximate projection onto the tangent cone to the variety of third-order tensors of bounded tensor-train rank is proposed and proven to satisfy a better angle condition than the one proposed by Kutschan (2019). Such an approximate projection enables, e.g., to compute gradient-related directions in the tangent cone, as required by algorithms aiming at minimizing a continuously differentiable function on the variety, a problem appearing notably in tensor completion. A numerical experiment is presented which indicates that, in practice, the angle condition satisfied by the proposed approximate projection is better than both the one satisfied by the approximate projection introduced by Kutschan and the proven theoretical bound.

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