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arXiv:2601.03885

Local Interpolation via Low-Rank Tensor Trains

Siddhartha E. Guzman, Egor Tiunov, Leandro Aolita

math.NAcs.GRquant-phSubmitted Jan 7, 2026• Announced Jan 8, 2026• 0 citations
v1 · Jan 7, 2026v2 · Jun 17, 2026
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Abstract

Tensor Train (TT) decompositions provide a powerful framework to compress grid-structured data, such as sampled function values, on regular Cartesian grids. Such high compression, in turn, enables efficient high-dimensional computations. Exact TT representations are only available for simple analytic functions. Furthermore, global polynomial or Fourier expansions typically yield TT-ranks that grow proportionally with the number of basis terms. State-of-the-art methods are often prohibitively expensive or fail to recover the underlying low-rank structure. We propose a low-rank TT interpolation framework that, given a TT describing a discrete (scalar-, vector-, or tensor-valued) function on a coarse regular grid with $n$ cores, constructs a finer-scale version of the same function represented by a TT with $n+m$ cores, where the last $m$ cores maintain constant rank. Our method guarantees a $\ell^{2}$-norm error bound independent of the total number of cores, achieves exponential compression at fixed accuracy, and admits logarithmic complexity with respect of the number of grid points. We validate its performance through numerical experiments, including 1D, 2D, and 3D applications such as: 2D and 3D airfoil mask embeddings, image super-resolution, and synthetic noise fields such as 3D synthetic turbulence. In particular, we generate fractal noise fields directly in TT format with logarithmic complexity and memory. This work opens a path to scalable TT-native solvers with complex geometries and multiscale generative models, with implications from scientific simulation to imaging and real-time graphics.

Comments
22 pages, 16 figures

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Data encoding remains a fundamental bottleneck in quantum machine learning, where amplitude encoding of high-dimensional classical vectors into quantum states incurs exponential cost. In this work, we propose a pre-trained tensor-train (TT) encoding network (Pre-TT-Encoder) that significantly reduces the computational complexity of amplitude encoding while preserving essential data structure. The Pre-TT-Encoder exploits low-rank TT decompositions learned from classical data, enabling polynomial-time state preparation in the number of qubits and TT-ranks. We provide a theoretical analysis of the encoding complexity and establish fidelity bounds that quantify the trade-off between TT-rank and approximation error. Empirical evaluations on classical (MNIST) and quantum-native (semiconductor quantum dot) datasets demonstrate that our approach achieves substantial gains in encoding efficiency over direct amplitude encoding and PCA-based dimensionality reduction, while maintaining competitive performance in downstream variational quantum circuit classification tasks. The proposed method highlights the role of tensor networks as scalable intermediaries between classical data and quantum processors.
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Tensor networks are a compressed format for multi-dimensional data. One dimensional tensor networks—often referred to as tensor trains (TT) or matrix product states (MPS)—are increasingly being used as a numerical Ansatz for continuum functions by ``quantizing’’ the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks (TTNs) for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient Ansatz than the commonly used tensor train. Finally, we demonstrate how the methods introduced in this work can be used to realize a TTN-based solver for multi-dimensional, non-linear Fredholm equations.
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Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose tensor network-based quantum data encoding (TNQE), a circuit-efficient framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as 0.04×0.04\times0.04× the depth of amplitude encoding, while naturally scaling to high-resolution images (256×256256 \times 256256×256) and demonstrating practical feasibility on real quantum hardware.
Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to 2702^{70}270 configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.