arXiv subjects

Marc K. Ritter

Publications and source records attributed to Marc K. Ritter.

8 recordsLinked to original sources

arXiv2026

Parallelized contraction of tensor trains or matrix product operators

Tensor Trains (TT), also known as Matrix Product States (MPS) and Matrix Product Operators (MPO), provide a compact and structured representation for high-dimensional data and operators. One of the most expensive manipulations involving tensor trains is the contraction of two MPOs. A popular and accurate method for mitigating this cost is the fit algorithm. However, it is still comparatively costly since it involves 2-site updates. Moreover, the parallelization of the fit algorithm when used for MPO-MPO contractions has received comparatively little attention. In this work, we present two strategies for accelerating the fit algorithm, usable in combination: (1) We use MPI-based distributed-memory parallelization tailored for MPO-MPO contractions, employing one of two MPO gauge choices: (1a) the inverse canonical gauge, which yields near-ideal parallelization speedup across all problem sizes; and (1b) the site-canonical gauge, which avoids the inversion of singular values but requires extra computations to ensure global consistency, thus yielding excellent parallelization speedup only for large problems requiring several sweeps before convergence. (2) We use randomized projections to reduce the cost of local updates from 2-site to 1-site costs while retaining 2-site accuracy, and to speed up contractions of environment tensors with MPO tensors.

physics.comp-ph↗

arXiv2026

Fast elementwise operations on tensor trains with alternating cross interpolation

Tensor trains (TTs), also known as matrix product states (MPS), are compressed representations of high-dimensional data that can be efficiently manipulated to perform calculations on the data. In many applications, such as TT-based solvers for nonlinear partial differential equations, the most expensive step is an elementwise multiplication or similar elementwise operation on multiple TTs. Known error-controlled algorithms for such operations scale as $O(\chi^4)$, where $\chi$ is the TT rank. If the rank of the output is smaller than $\chi^2$, it is possible to formulate algorithms with better scaling. In this work, we present the alternating cross interpolation (ACI) algorithm that performs such operations in $O(\chi^3)$, while maintaining error control. We demonstrate these properties on benchmark problems, achieving a significant speedup for TT ranks that are commonly encountered in practical applications.

math.NA↗

arXiv2026

Adaptive Patching for Tensor Train Computations

Quantics Tensor Train (QTT) operations such as matrix product operator contractions are prohibitively expensive for large bond dimensions. We propose an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs through divide-and-conquer, adaptively partitioning tensors into smaller patches with reduced bond dimensions. We demonstrate substantial improvements for sharply localized functions and show efficient computation of bubble diagrams and Bethe-Salpeter equations, opening the door to practical large-scale QTT-based computations previously beyond reach.

physics.comp-ph↗

arXiv2025

Two-particle calculations with quantics tensor trains: Solving the parquet equations

We present the first application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe--Salpeter equations, parquet equation and Schwinger--Dyson equation) can be decomposed into basic operations on the QTT-TCI (QTCI) compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. As examples we take the Hubbard model in the atomic limit and the single impurity Anderson model, where the basic objects in parquet equations, the two-particle vertices, depend on three frequencies, but not on momenta. The results show that this approach is able to overcome major computational bottlenecks of standard numerical methods. The applied methods allow for an exponential increase of the number of grid points included in the calculations leading to an exponentially improving computational error for a linear increase in computational cost.

cond-mat.str-el↗

arXiv2024

Learning tensor networks with tensor cross interpolation: new algorithms and libraries

The tensor cross interpolation (TCI) algorithm is a rank-revealing algorithm for decomposing low-rank, high-dimensional tensors into tensor trains/matrix product states (MPS). TCI learns a compact MPS representation of the entire object from a tiny training data set. Once obtained, the large existing MPS toolbox provides exponentially fast algorithms for performing a large set of operations. We discuss several improvements and variants of TCI. In particular, we show that replacing the cross interpolation by the partially rank-revealing LU decomposition yields a more stable and more flexible algorithm than the original algorithm. We also present two open source libraries, xfac in Python/C++ and TensorCrossInterpolation.jl in Julia, that implement these improved algorithms, and illustrate them on several applications. These include sign-problem-free integration in large dimension, the superhigh-resolution quantics representation of functions, the solution of partial differential equations, the superfast Fourier transform, the computation of partition functions, and the construction of matrix product operators.

physics.comp-ph↗

arXiv2024

Quantics Tensor Cross Interpolation for High-Resolution, Parsimonious Representations of Multivariate Functions in Physics and Beyond

Multivariate functions of continuous variables arise in countless branches of science. Numerical computations with such functions typically involve a compromise between two contrary desiderata: accurate resolution of the functional dependence, versus parsimonious memory usage. Recently, two promising strategies have emerged for satisfying both requirements: (i) The quantics representation, which expresses functions as multi-index tensors, with each index representing one bit of a binary encoding of one of the variables; and (ii) tensor cross interpolation (TCI), which, if applicable, yields parsimonious interpolations for multi-index tensors. Here, we present a strategy, quantics TCI (QTCI), which combines the advantages of both schemes. We illustrate its potential with an application from condensed matter physics: the computation of Brillouin zone integrals.

physics.comp-ph↗

arXiv2022

Benchmark Calculations of Multiloop Pseudofermion fRG

The pseudofermion functional renormalization group (pffRG) is a computational method for determining zero-temperature phase diagrams of frustrated quantum magnets. In a recent methodological advance, the commonly employed Katanin truncation of the flow equations was extended to include multiloop corrections, thereby capturing additional contributions from the three-particle vertex [ arXiv:2011.01268 , arXiv:2011.01269 ]. This development has also stimulated significant progress in the numerical implementation of pffRG, allowing one to track the evolution of pseudofermion vertices under the renormalization group flow with unprecedented accuracy. However, cutting-edge solvers differ in their integration algorithms, heuristics to discretize Matsubara frequency grids, and more. To lend confidence in the numerical robustness of state-of-the-art multiloop pffRG codes, we present and compare results produced with two independently developed and algorithmically distinct solvers for Heisenberg models on three-dimensional lattice geometries. Using the cubic lattice Heisenberg (anti)ferromagnet with nearest and next-nearest neighbor interactions as a generic benchmark model, we find the two codes to quantitatively agree, often up to several orders of magnitude in digital precision, both on the level of spin-spin correlation functions and renormalized fermionic vertices for varying loop orders. These benchmark calculations further substantiate the usage of multiloop pffRG solvers to tackle unconventional forms of quantum magnetism.

cond-mat.str-el↗

arXiv2020

Multiloop pseudofermion functional renormalization for quantum spin systems: Application to the spin-$\frac{1}{2}$ kagome Heisenberg model

We present a multiloop pseudofermion functional renormalization group (pffRG) approach to quantum spin systems. As a test case, we study the spin-$\tfrac{1}{2}$ Heisenberg model on the kagome lattice, a prime example of a geometrically frustrated magnet believed to host a quantum spin liquid. Our main physical result is that, at pure nearest-neighbor coupling, the system shows indications for an algebraic spin liquid through slower-than-exponential decay with distance for the static spin susceptibility, while the pseudofermion self-energy develops intriguing low-energy features. Methodologically, the pseudofermion representation of spin models inherently yields a strongly interacting system, and the quantitative reliability of a truncated fRG flow is \textit{a priori} unclear. Our main technical result is the demonstration of convergence in loop number within multiloop pffRG. Through correspondence with the self-consistent parquet equations, this provides further evidence for the internal consistency of the approach. The loop order required for convergence of the pseudofermion vertices is rather large, but the spin susceptibility is more benign, appearing almost fully converged for loop orders $\ell \geq 5$. The multiloop flow remains stable as the infrared cutoff $Λ$ is reduced relative to the microscopic exchange interaction $J$, allowing us to reach values of $Λ/J$ on the subpercent level in the spin-liquid phase. By contrast, solving the parquet equations via direct fixed-point iteration becomes increasingly difficult for low $Λ/J$. We also scrutinize the pseudofermion constraint of single occupation per site, which is only fulfilled on average in pffRG, by explicitly computing fermion-number fluctuations. Although the latter are not entirely suppressed, we find that they do not affect the qualitative conclusions drawn from the spin susceptibility.

cond-mat.str-el↗