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

Shakir Showkat Sofi; Charlotte Vermeylen; Lieven De Lathauwer

quant-phFeb 16, 2026

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

Authors: Shakir Showkat Sofi, Charlotte Vermeylen, Lieven De Lathauwer

Organizations: Dept. Electrical Engineering (ESAT), KU Leuven, Kortrijk/Leuven, Belgium · Leuven.AI - KU Leuven institute for AI, B-3000, Leuven, Belgium · Dept. Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium

Abstract

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.

Figures & tables

Fig. 1: Overlapping block diagonal pattern

Panel of Figure 2 — Fig. 2: The proposed method yields higher accuracy than both references while also enabling significantly faster reconstruction. Moreover, as $d$ grows, more measurements become available per parameter (with $D$ and $R$ fixed), leading to improved accuracy. For fixed $R$ , $d$ (especially when $d$ is small), and a constant noise level, increasing $N$ slightly weakens the subspace perturbation bound due to its dependence on the ratio of $L\propto D\!-\!R$ to $\sigma_{\min}^{+}(\tilde{{\mathbf{{N}}}})$ ; see Eqn. ( 8 ). We observe in our experiments that this ratio increases; consistently, we see a slight increase in subspace error and, consequently, in reconstruction error.

Quantum State Tomography

Quantum state tomography reconstructs unknown quantum states from measurement data, enabling characterization and verification of quantum systems.

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Sep 8, 2026J/K move · Enter open · S save

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

Shakir Showkat Sofi, Charlotte Vermeylen, Fatemeh Mohammadi, Lieven De Lathauwer

Leuven.AI - KU Leuven institute for AI, B-3000, Leuven, Belgium · Department of Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium; Group Science, Engineering and Technology, KU Leuven Kulak, Kortrijk, Belgium · Departments of Computer Science and Mathematics, KU Leuven, Leuven, Belgium

Quantum State Tomography TT Decomposition Low-Rank Approximation

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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.

Abstract

Quantum State Tomography

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