Researchers at INO-CNR and the Universita di Trento are formulating a tangent-space method for algebraic varieties of Matrix Product States (MPS) to study excitation spectra of non-uniform quantum many-body systems with open boundary conditions, framing MPS as projective varieties defined by rank constraints on tensor flattenings. This algebro-geometric perspective allows Otto T.P. Schmidt and Iacopo Carusotto to construct a tangent-space method now incorporating open boundary conditions, a crucial step towards modeling realistic, finite-sized quantum systems without the limitations of periodic boundaries. Using the Bose-Hubbard model as a test case, the method reproduces low-lying excitations and captures finite-size precursors of the Mott-insulator to superfluid transition. The researchers detail their approach to characterizing its expressivity through a novel rank tomography of the MPS tangent space.
Recent advances allow quantum simulations to move beyond idealized conditions and model more realistic, finite-sized systems. Researchers are formulating a tangent-space method for algebraic varieties of matrix product states (MPS) to study excitation spectra in complex quantum materials, by incorporating open boundary conditions. This step moves beyond the simplifying assumption of periodic boundaries, enabling the study of systems with edges and finite dimensions. Schmidt and Carusotto define the MPS variety as a variational space, emphasizing its algebraic structure, and introduce a rank tomography of the MPS tangent space, characterizing its expressivity in terms of particle-sector rank profiles. Using the Bose, Hubbard model as a benchmark, the researchers illustrate that the method reproduces low-lying excitations and captures finite-size precursors of the Mott-insulator to superfluid transition. This approach characterizes the expressivity of the tangent space, revealing a trade-off between bond dimension size, computational cost, and spectral reconstruction quality.
Current approaches to simulating quantum many-body systems increasingly rely on Matrix Product States (MPS), a technique offering efficient variational descriptions of ground states in one dimension. Recent work is shifting the focus from MPS as merely a computational tool to understanding its underlying mathematical structure. Researchers formulate a tangent-space method for algebraic varieties of matrix product states (MPS) to study excitation spectra of non-uniform quantum many-body systems with open boundary conditions. This approach defines MPS of a fixed bond dimension as a projective variety defined by rank constraints on tensor flattenings. By focusing on the smooth full-rank stratum of this variety, the team has derived an explicit representation of the tangent space, crucial for calculating excitation spectra. The method extends beyond ground states; linear perturbations are represented within this MPS tangent space, allowing reconstruction of low-energy excitations.
Rank Tomography of MPS Tangent Space Expressivity
Researchers at the INO-CNR Pitaevskii BEC Center and the University of Trento, alongside colleagues at the Max-Planck-Institute for the Mathematics in the Sciences, are formulating a tangent-space method for algebraic varieties of Matrix Product States (MPS) to study excitation spectra of non-uniform quantum many-body systems with open boundary conditions. This work details the introduction of a rank tomography of the MPS tangent space, which characterizes its expressivity in terms of particle-sector rank profiles of the underlying MPS variety. The method calculates excitation spectra and provides insights into the accuracy of the tangent-space approach itself. The researchers define a particle-number resolved parametric deficiency, which quantifies the number of missing independent directions in each sector. They resolve the usual MPS bond dimensions by decomposing each ground-state flattening into particle-number blocks and recording their ranks.
The resulting particle-resolved Schmidt-rank distribution defines particle-resolved rank profiles inside the fixed coarse MPS rank stratum. The researchers illustrate that the method reproduces low-lying excitations and captures finite-size precursors of the Mott-insulator to superfluid transition. The researchers emphasize that this work highlights the algebraic structure of the MPS tangent-space variational ansatz and suggests its use for practical calculations of quantum many-body systems.
A key component of this work is a detailed analysis of the MPS tangent space, offering a novel way to characterize its capabilities. Two states with identical bond dimensions can exhibit different PRSR distributions and, consequently, different tangent spaces, highlighting the subtlety of MPS expressivity. This detailed analysis of rank profiles and tangent space structure promises a more nuanced understanding of the strengths and limitations of the tangent-space method, paving the way for more accurate and efficient simulations of quantum materials.
Researchers are now leveraging this algebro-geometric perspective to characterize MPS expressivity. The implications extend to understanding how internal ground-state structure governs the expressivity of linear tangent methods. By defining a “particle-number resolved parametric deficiency,” researchers define the missing independent directions within each sector of the tangent space. This detailed analysis highlights a trade-off: larger bond dimensions enhance tangent space directions but increase computational cost, while smaller dimensions offer approximations at reduced expense.
Validation of the technique came through application to the Bose, Hubbard model, a cornerstone of condensed matter physics. The method reproduces low-lying excitations and captures finite-size precursors of the Mott-insulator to superfluid transition. The example numerical calculations highlight the trade-off introduced with the tangent-space method: larger bond dimensions increase the available directions on the tangent space and can improve the spectral reconstruction, at higher computational cost, while small bond dimensions only give an approximation to the excitations, at lower computational cost. This detailed analysis suggests potential for practical calculations of excitation spectra in a variety of quantum many-body systems, offering a powerful new tool for exploring the quantum world.
The pursuit of accurately simulating quantum systems has led to increasing sophistication in the methods used to represent their complex wavefunctions. Matrix Product States (MPS) have long been a cornerstone of this effort, offering a computationally tractable approach to approximating solutions for one-dimensional systems. This work emphasizes the algebraic structure underlying the variational space. Careful consideration of dimensionality and computational demands is vital for applying MPS tangent spaces to increasingly complex quantum systems, and understanding the limits of their expressivity.
The core of their method lies in characterizing MPS as projective varieties defined by rank constraints on tensor flattenings, allowing for a rigorous mathematical description of the variational space. They define an “affine MPS contraction map” which provides a polynomial relationship between the tensors defining the MPS and the overall quantum state. This allows researchers to analyze the inherent structure of the variational space.
Recent advancements are reshaping how scientists approach quantum simulations, moving beyond merely calculating approximations to exploring the fundamental mathematical structure of the systems themselves. Otto T.P. Schmidt and Iacopo Carusotto formulate a tangent-space method for algebraic varieties of matrix product states (MPS) to study excitation spectra of non-uniform quantum many-body systems with open boundary conditions. This allows for a precise definition of the MPS variety, essentially a geometric space defined by the constraints imposed on the tensors within the MPS. They emphasize the importance of moving away from solely focusing on ground states; linear perturbations around these states can be represented within the MPS tangent space, enabling the reconstruction of excitation spectra. Previously, many simulations relied on periodic boundaries, simplifying calculations but limiting realism. This new approach allows researchers to study systems without that assumption, paving the way for modeling more complex, finite-sized quantum systems.
Source: https://arxiv.org/abs/2607.05269
