Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components $D$ grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small-$D$ regime, the characteristic function undergoes a sharp bond-dimension collapse for $D \gtrsim 300$ components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of $N = 2^{30}$ frequency modes on standard hardware, far beyond the $N = 2^{24}$ ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.
Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components D grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small-D regime, the characteristic function undergoes a sharp bond-dimension collapse for D≳300 components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of N=230 frequency modes on standard hardware, far beyond the N=224 ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.
Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly.Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; N_v = 128 ×64; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5.Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE.Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv.Related: Paper page · De-risking register · 3D model · Learn more about KronosPart of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689).All numerical values are frozen design-point anchors; see the register.Public research artifact. No proprietary, financial, or supply-chain information is included.Note (REPLACE): This record replaces and supersedes DOI 10.5281/zenodo.22132192; please cite this version.
Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly. Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5. Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE. Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included. Previous version: DOI 10.5281/zenodo.22132192. Please cite this version.
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 2n points, the construction produces an (n+m)-core TT or (n+m)-qubit state on a finer grid with 2n+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 ℓ2-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 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.
We focus on the construction of explicit low-rank representations of matrices in the tensor train (TT) and quantized tensor train (QTT) formats which have been proposed recently for the low-parametric structured representation of large-scale tensors. The matrices under consideration are discretizations of the Laplace operator in a hypercube (in one and many dimensions) and their inverses (in one dimension). For these matrices we derive explicit and exact QTT representations of low QTT ranks independent of the numbers d and n2 of dimensions and entries in each dimension. This implies that for the matrices considered the storage cost is O(dlogn), i.e., logarithmic with respect to the total number n2d of entries. The same applies to the computational complexity of the QTT-structured operations with these matrices, which now depends on the QTT ranks of the other operands. The general result of the paper is the notation and technique we introduce in order to examine the QTT structure of matrices analytically. They prove to be an efficient tool of studying other tensors related to particular computational problems, which are not considered in this paper.
Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as 1010 discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.
In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).