Both tensor trains (TTs) and quantum states provide compressed representations of grid-structured data with potentially exponential compression power. We present a unified framework for upsampling data encoded in vector amplitudes, with efficient realizations in both classical TT and quantum settings. Starting from an $n$ -core TT or an $n$ -qubit state on a coarse grid with $2^n$ points, the construction produces an $(n+m)$ -core TT or $(n+m)$ -qubit state on a finer grid with $2^{n+m}$ points. In the TT setting, it supports interpolation, quasi-interpolation, augmentation, and synthesis through efficient low-rank contractions, with the added $m$ cores retaining constant rank. For function-value encodings, the resulting interpolation satisfies an $\ell^2$ -error bound independent of the number of added grid points, achieves exponential compression at fixed accuracy, and has a logarithmic complexity in the number of grid points. In the quantum setting, the refined state is prepared by a $\mathrm{poly}(n,m)$ -size circuit using $\log(p+1)$ ancillas, where $p$ controls the smoothness of the quasi-interpolant; the corresponding error scales quadratically with the initial grid spacing. We validate our framework for tensor networks in one-, two-, and three-dimensional examples, including functions, derivatives, airfoil masks, and synthetic random fields such as three-dimensional turbulence. In particular, fractal fields can be generated directly in TT format with logarithmic memory and runtime. These results open a practical route to multiscale solvers, generative models, and geometry-aware algorithms on tensor-network and quantum platforms, with potential applications in scientific simulation, imaging, and real-time graphics.Both tensor trains (TTs) and quantum states provide compressed representations of grid-structured data with potentially exponential compression power. We present a unified framework for upsampling data encoded in vector amplitudes, with efficient realizations in both classical TT and quantum settings. Starting from an $n$-core TT or an $n$-qubit state on a coarse grid with $2^n$ points, the construction produces an $(n+m)$-core TT or $(n+m)$-qubit state on a finer grid with $2^{n+m}$ points. In the TT setting, it supports interpolation, quasi-interpolation, augmentation, and synthesis through efficient low-rank contractions, with the added $m$ cores retaining constant rank. For function-value encodings, the resulting interpolation satisfies an $\ell^2$-error bound independent of the number of added grid points, achieves exponential compression at fixed accuracy, and has a logarithmic complexity in the number of grid points. In the quantum setting, the refined state is prepared by a $\mathrm{poly}(n,m)$-size circuit using $\log(p+1)$ ancillas, where $p$ controls the smoothness of the quasi-interpolant; the corresponding error scales quadratically with the initial grid spacing. We validate our framework for tensor networks in one-, two-, and three-dimensional examples, including functions, derivatives, airfoil masks, and synthetic random fields such as three-dimensional turbulence. In particular, fractal fields can be generated directly in TT format with logarithmic memory and runtime. These results open a practical route to multiscale solvers, generative models, and geometry-aware algorithms on tensor-network and quantum platforms, with potential applications in scientific simulation, imaging, and real-time graphics.

math.NACross-listedarXiv:2601.03885Jan 8, 20260 citations

Geometry vs Binary: Quantum Physics Puzzles — geometric algebra quantum entanglement interpretat — E8 Intelligence Research

Andrew Stewart Caldin

## Binary Failure → Geometry Solution # Analysis: Tensor Networks, Entanglement Simulation, and the Geometric Wall ## Where the Conventional Approach Hits the Wall Tensor networks are the *best* conventional tool for simulating quantum entanglement — and that's exactly why they're the perfect case study. They work by representing a quantum state as a network of contracted tensors, with **bond dimension** controlling how much entanglement you can capture. The wall is precise: **1. The bond-dimension explosion.** To exactly represent a generic entangled state of N qubits, bond dimension grows exponentially (χ ~ 2^(N/2)). Every talk in this list — Ran's "simulators inspired by tensor network," Dziarmaga's "2D tensor networks," Lucia's "entanglement structures" — is dancing around the same fact: **the geometry of the tensor network is chosen by hand, and the entanglement geometry of the actual state doesn't match it.** You pick a 2D grid, or a tree, or a MERA, and hope the state's enta Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com## Binary Failure → Geometry Solution # Analysis: Tensor Networks, Entanglement Simulation, and the Geometric Wall ## Where the Conventional Approach Hits the Wall Tensor networks are the *best* conventional tool for simulating quantum entanglement — and that's exactly why they're the perfect case study. They work by representing a quantum state as a network of contracted tensors, with **bond dimension** controlling how much entanglement you can capture. The wall is precise: **1. The bond-dimension explosion.** To exactly represent a generic entangled state of N qubits, bond dimension grows exponentially (χ ~ 2^(N/2)). Every talk in this list — Ran's "simulators inspired by tensor network," Dziarmaga's "2D tensor networks," Lucia's "entanglement structures" — is dancing around the same fact: **the geometry of the tensor network is chosen by hand, and the entanglement geometry of the actual state doesn't match it.** You pick a 2D grid, or a tree, or a MERA, and hope the state's enta Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 26, 20260 citations

Tensor Network Lattice Boltzmann Method for Data-Compressed Fluid Simulations

Lukas Gross, Elie Mounzer, David M. Wawrzyniak +2 more

Resolving unsteady transport phenomena in geometrically complex domains is traditionally constrained by polynomial scaling of computational cost with spatial resolution. While methods based on tensor-network data representations or matrix-product states (MPS) data encodings have emerged as a technique to systematically reduce degrees of freedom, existing formulations do not extend to complex geometries and complex flow physics. Both capabilities are offered by lattice Boltzmann methods, for which we develop a generalized MPS formulation. This development marks a paradigm shift from classical methods that rely on explicit grid refinement for data reduction. Instead, our approach exploits non-local correlations in the MPS representation to systemically compress the global fluid state directly without modifying the underlying grid. We benchmark the proposed solver against classical LBM using three-dimensional flows through structured media and vascular geometries. The results confirm that the MPS formulation reproduces the reference solution with high fidelity while achieving compression ratios exceeding two orders of magnitude, positioning tensor networks or MPS encodings as a scalable paradigm for continuum mechanics on high-performance GPU hardware.Resolving unsteady transport phenomena in geometrically complex domains is traditionally constrained by polynomial scaling of computational cost with spatial resolution. While methods based on tensor-network data representations or matrix-product states (MPS) data encodings have emerged as a technique to systematically reduce degrees of freedom, existing formulations do not extend to complex geometries and complex flow physics. Both capabilities are offered by lattice Boltzmann methods, for which we develop a generalized MPS formulation. This development marks a paradigm shift from classical methods that rely on explicit grid refinement for data reduction. Instead, our approach exploits non-local correlations in the MPS representation to systemically compress the global fluid state directly without modifying the underlying grid. We benchmark the proposed solver against classical LBM using three-dimensional flows through structured media and vascular geometries. The results confirm that the MPS formulation reproduces the reference solution with high fidelity while achieving compression ratios exceeding two orders of magnitude, positioning tensor networks or MPS encodings as a scalable paradigm for continuum mechanics on high-performance GPU hardware.

physics.flu-dynCross-listedarXiv:2512.07615Dec 9, 2025

n

-core TT or an

n

-qubit state on a coarse grid with

2^n

points, the construction produces an

(n+m)

-core TT or

(n+m)

-qubit state on a finer grid with

2^{n+m}

points. In the TT setting, it supports interpolation, quasi-interpolation, augmentation, and synthesis through efficient low-rank contractions, with the added

m

cores retaining constant rank. For function-value encodings, the resulting interpolation satisfies an

\ell^2

-error bound independent of the number of added grid points, achieves exponential compression at fixed accuracy, and has a logarithmic complexity in the number of grid points. In the quantum setting, the refined state is prepared by a

\mathrm{poly}(n,m)

-size circuit using

\log(p+1)

ancillas, where

p

controls the smoothness of the quasi-interpolant; the corresponding error scales quadratically with the initial grid spacing. We validate our framework for tensor networks in one-, two-, and three-dimensional examples, including functions, derivatives, airfoil masks, and synthetic random fields such as three-dimensional turbulence. In particular, fractal fields can be generated directly in TT format with logarithmic memory and runtime. These results open a practical route to multiscale solvers, generative models, and geometry-aware algorithms on tensor-network and quantum platforms, with potential applications in scientific simulation, imaging, and real-time graphics.