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.
The state of a quantum system is fully described by its density matrix-a Hermitian, positive-semidefinite (PSD) operator with unit trace-that provides a unified representation of both pure and mixed quantum states (probabilistic mixtures of pure states). Its diagonal entries represent state populations (probabilities), while its off-diagonal entries encode quantum coherences between states [1]. Accurately determining this matrix is a key challenge in quantum science, as many tasks, such as benchmarking and verification of quantum hardware, and fidelity estimation, essentially involve its estimation. This matrix is estimated by a technique known as quantum state tomography (QST) [1]- [4]. QST works by performing an informationally complete set of measurements on a large ensemble of identically prepared quantum systems. The collected data is subsequently used to reconstruct the density matrix. However, a major obstacle is the exponential growth in parameters (and correspondingly in measurements and reconstruction complexity) as the system grows, an obstacle known as the "curse of dimensionality."
Physically relevant states occupy only a tiny, structured subset of Hilbert space, constrained by locality and finitecomplexity dynamics that limit correlations and entanglement [3]- [5]. This subset includes pure or low-entropy states, lowenergy states of local Hamiltonians, product states, and structured states in which only a subset of entries carry valuable information. Exploiting these low-rank or structural constraints can tame the curse of dimensionality.
This work was supported by the Flemish Government’s AI Research Program and KU Leuven Internal Funds (iBOF/23/064, C14/22/096). Shakir Showkat Sofi, Charlotte Vermeylen, and Lieven De Lathauwer are affiliated with Leuven.AI -KU Leuven institute for AI, B-3000, Leuven, Belgium.
When the density matrix is known to be low rank or close to a low-rank matrix, low-rank QST exploits the structure to reconstruct the state from fewer measurements. Rank information can be incorporated either implicitly, by promoting low rank through a rank-minimization objective, implemented via the nuclear norm as a convex surrogate, under measurement constraints [3], [4], [6], or explicitly, by optimizing a fixed-rank factorized model, as in Burer-Monteiro-type formulations [7], [8]. If the density matrix is instead viewed as a high-order tensor of low rank in the tensor sense, efficient and scalable tensor network representations-such as matrix product states (tensor trains)-can be used to estimate it [2], [9]- [12].
Another line of research avoids full QST and focuses on learning only specific aspects of the underlying density matrix. For example, permutationally invariant QST efficiently reconstructs the permutationally invariant part of the underlying quantum state, with the number of measurement settings scaling quadratically, and is well suited for states that are close to being permutationally symmetric, such as Dicke or spin-squeezed states [13], [14]. As another example, shadow tomography constructs a compact classical representation known as “classical shadow”-a collection of randomized sketches of the density matrix-obtained via random unitary transformations, allowing many observables (or arbitrary linear functions) to be estimated from the compact representation with measurement complexity that scales logarithmically in the number of observables [15], [16].
Extending the idea of learning partial information, selective QST focuses on directly estimating only a chosen subset of density-matrix entries, enabling targeted estimation without reconstructing the full state [17]- [20]. This approach can exploit additional structure: for instance, if the selected entries follow a specific low-rank-exploitable pattern, they can uniquely determine the global low-rank density matrix. Furthermore, these entries can be obtained via shadow tomography, allowing scalable estimation even of the full state [21].
Contributions: In this work, we further develop selective QST, where we focus on characterizing structured entries that enable unique and efficient recovery of the full density matrix using an algebraic method (based only on standard numerical linear algebra (NLA) operations), and provide deterministic recovery guarantees and subspace error bounds. We compare our method with state-of-the-art (SOTA) techniques and show that it is computationally efficient while achieving competitive accuracy in reconstructing mixed low-rank quantum states. Finally, we touch on how scalability can be improved by estimating structured entries via shadow tomography, combining low-rank and shadow tomography.
Scalars, vectors, and matrices are denoted by lowercase letters, bold lowercase letters, and bold uppercase letters, respectively; that is, x, x, and X. For X ∈ C D×D , the (r, c)-th entry is x rc = X(r, c), and the k-th column is denoted by x :k = X(:, k). The trace, Frobenius norm, and nucle
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