Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions

arXiv:2407.09609 · quant-ph, cs.NA, math.NA · Submitted 2024-07-12 · Read on arXiv

quant-ph, cs.NA, math.NA

Submitted: 2024-07-12

Updated: 2026-09-16

Terminology

• Core — 45× in this paper · explained in 2 episodes
• Benchmark — 18× in this paper · explained in 2 episodes
• Accuracy — 12× in this paper · explained in 1 episode
• Convergence Guarantees — 4× in this paper · explained in 3 episodes
• Chebyshev Polynomials — 2× in this paper · explained in 2 episodes
• Matrix Product States (MPS) — 2× in this paper · explained in 2 episodes
• Quantization — 2× in this paper · explained in 12 episodes
• Adaptive Sampling — 1× in this paper · explained in 1 episode
• Curse of Dimensionality — 1× in this paper · explained in 6 episodes
• Generalization — 1× in this paper · explained in 13 episodes
• Hilbert Space — 1× in this paper · explained in 2 episodes
• Quantum Circuits — 1× in this paper · explained in 2 episodes
• Robustness — 1× in this paper · explained in 5 episodes
• Trace — 1× in this paper · explained in 2 episodes
• Tucker Decomposition — 1× in this paper · explained in 2 episodes
• Uncertainty Quantification — 1× in this paper · explained in 8 episodes

Sources

• Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations
• Quantum Fourier analysis for multivariate functions and applications to a class of Schr\"odinger-type partial differential equations
• A Quantum Inspired Approach to Exploit Turbulence Structures
• Parallel cross interpolation for high-precision calculation of high-dimensional integrals
• Tensorization of neural networks for improved privacy and interpretability
• Multiscale interpolative construction of quantized tensor trains
• Direct interpolative construction of the discrete Fourier transform as a matrix product operator
• Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations
• Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
• Efficient MPS representations and quantum circuits from the Fourier modes of classical image data
• Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functions
• Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions
• Quasioptimality of maximum-volume cross interpolation of tensors
• Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations
• SeeMPS: A Python-based Matrix Product State and Tensor Train Library
• Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format
• Multigrid Renormalization
• Black box approximation in the tensor train format initialized by ANOVA decomposition

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