arXiv subjects

Sebastian Krämer

Publications and source records attributed to Sebastian Krämer.

10 recordsLinked to original sources

arXiv2026

Low-rank eigenvalue solvers for block-sparse matrix product states

We consider an iterative eigensolver for Schrödinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schrödinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation. We provide a complete analysis of a solver based on preconditioned inverse iteration combined with rank truncation and propose a generalization to subspace iteration for the joint approximation of several eigenspaces. The practical performance of the method is illustrated by numerical tests for several model problems.

math.NA↗

arXiv2021

Asymptotic Log-Det Sum-of-Ranks Minimization via Tensor (Alternating) Iteratively Reweighted Least Squares

Affine sum-of-ranks minimization (ASRM) generalizes the affine rank minimization (ARM) problem from matrices to tensors. Here, the interest lies in the ranks of a family $\mathcal{K}$ of different matricizations. Transferring our priorly discussed results on asymptotic log-det rank minimization, we show that iteratively reweighted least squares with weight strength $p = 0$ remains a, theoretically and practically, particularly viable method denoted as $\mathrm{IRLS}$-$0\mathcal{K}$. As in the matrix case, we prove global convergence of asymptotic minimizers of the log-det sum-of-ranks function to desired solutions. Further, we show local convergence of $\mathrm{IRLS}$-$0\mathcal{K}$ in dependence of the rate of decline of the therein appearing regularization parameter $γ\searrow 0$. For hierarchical families $\mathcal{K}$, we show how an alternating version ($\mathrm{AIRLS}$-$0\mathcal{K}$, related to prior work under the name $\mathrm{SALSA}$) can be evaluated solely through tensor tree network based operations. The method can thereby be applied to high dimensions through the avoidance of exponential computational complexity. Further, the otherwise crucial rank adaption process becomes essentially superfluous even for completion problems. In numerical experiments, we show that the therefor required subspace restrictions and relaxation of the affine constraint cause only a marginal loss of approximation quality. On the other hand, we demonstrate that $\mathrm{IRLS}$-$0\mathcal{K}$ allows to observe the theoretical phase transition also for generic tensor recoverability in practice. Concludingly, we apply $\mathrm{AIRLS}$-$0\mathcal{K}$ to larger scale problems.

math.NA↗

arXiv2021

Asymptotic Log-Det Rank Minimization via (Alternating) Iteratively Reweighted Least Squares

The affine rank minimization (ARM) problem is well known for both its applications and the fact that it is NP-hard. One of the most successful approaches, yet arguably underrepresented, is iteratively reweighted least squares (IRLS), more specifically $\mathrm{IRLS}$-$0$. Despite comprehensive empirical evidence that it overall outperforms nuclear norm minimization and related methods, it is still not understood to a satisfying degree. In particular, the significance of a slow decrease of the therein appearing regularization parameter denoted $γ$ poses interesting questions. While commonly equated to matrix recovery, we here consider the ARM independently. We investigate the particular structure and global convergence property behind the asymptotic minimization of the log-det objective function on which $\mathrm{IRLS}$-$0$ is based. We expand on local convergence theorems, now with an emphasis on the decline of $γ$, and provide representative examples as well as counterexamples such as a diverging $\mathrm{IRLS}$-$0$ sequence that clarify theoretical limits. We present a data sparse, alternating realization $\mathrm{AIRLS}$-$p$ (related to prior work under the name $\mathrm{SALSA}$) that, along with the rest of this work, serves as basis and introduction to the more general tensor setting. In conclusion, numerical sensitivity experiments are carried out that reconfirm the success of $\mathrm{IRLS}$-$0$ and demonstrate that in surprisingly many cases, a slower decay of $γ$ will yet lead to a solution of the ARM problem, up to the point that the exact theoretical phase transition for generic recoverability can be observed. Likewise, this suggests that non-convexity is less substantial and problematic for the log-det approach than it might initially appear.

math.OC↗

arXiv2019

Stable ALS Approximation in the TT-Format for Rank-Adaptive Tensor Completion

Low rank tensor completion is a highly ill-posed inverse problem, particularly when the data model is not accurate, and some sort of regularization is required in order to solve it. In this article we focus on the calibration of the data model. For alternating optimization, we observe that existing rank adaption methods do not enable a continuous transition between manifolds of different ranks. We denote this characteristic as $\textit{instability (under truncation)}$. As a consequence of this property, arbitrarily small changes in the iterate can have arbitrarily large influence on the further reconstruction. We therefore introduce a singular value based regularization to the standard alternating least squares (ALS), which is motivated by averaging in microsteps. We prove its $\textit{stability}$ and derive a natural semi-implicit rank adaption strategy. We further prove that the standard ALS microsteps for completion problems are only stable on manifolds of fixed ranks, and only around points that have what we define as $\textit{internal tensor restricted isometry property, iTRIP}$. In conclusion, numerical experiments are provided that show improvements of the reconstruction quality up to orders of magnitude in the new Stable ALS Approximation (SALSA) compared to standard ALS and the well known Riemannian optimization RTTC.

math.NA↗

arXiv2019

A Geometric Description of Feasible Singular Values in the Tensor Train Format

Tree tensor networks such as the tensor train format are a common tool for high dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While the behavior of such is as essential as in the matrix case, here the question about the $\textit{feasibility}$ of specific constellations arises: which prescribed tuples can be realized as singular values of a tensor and what is this feasible set? We first show the equivalence of the $\textit{tensor feasibility problem (TFP)}$ to the $\textit{quantum marginal problem (QMP)}$. In higher dimensions, in case of the tensor train (TT-)format, the conditions for feasibility can be decoupled. By present results for three dimensions for the QMP, it then follows that the tuples of squared, feasible TT-singular values form polyhedral cones. We further establish a connection to eigenvalue relations of sums of Hermitian matrices, which in turn are described by sets of interlinked, so called $\textit{honeycombs}$, as they have been introduced by Knutson and Tao. Besides a large class of universal, necessary inequalities as well as the vertex description for a special, simpler instance, we present a linear programming algorithm to check feasibility and a simple, heuristic algorithm to construct representations of tensors with prescribed, feasible TT-singular values in parallel.

math.NA↗

arXiv2018

QuantumOptics.jl: A Julia framework for simulating open quantum systems

We present an open source computational framework geared towards the efficient numerical investigation of open quantum systems written in the Julia programming language. Built exclusively in Julia and based on standard quantum optics notation, the toolbox offers speed comparable to low-level statically typed languages, without compromising on the accessibility and code readability found in dynamic languages. After introducing the framework, we highlight its features and showcase implementations of generic quantum models. Finally, we compare its usability and performance to two well-established and widely used numerical quantum libraries.

quant-ph↗

arXiv2016

Quantum annealing with ultracold atoms in a multimode optical resonator

A dilutely filled $N$-site optical lattice near zero temperature within a high-$Q$ multimode cavity can be mapped to a spin ensemble with tailorable interactions at all length scales. The effective full site to site interaction matrix can be dynamically controlled by the application of up to $N(N+1)/2$ laser beams of suitable geometry, frequency and power, which allows for the implementation of quantum annealing dynamics relying on the all-to-all effective spin coupling controllable in real time. Via an adiabatic sweep starting from a superfluid initial state one can find the lowest energy stationary state of this system. As the cavity modes are lossy, errors can be amended and the ground state can still be reached even from a finite temperature state via ground state cavity cooling. The physical properties of the final atomic state can be directly and almost non-destructively read off from the cavity output fields. As example we simulate a quantum Hopfield associative memory scheme.

quant-ph↗

arXiv2015

Optimized geometries for future generation optical lattice clocks

Atoms deeply trapped in magic wavelength optical lattices provide a Doppler- and collision-free dense ensemble of quantum emitters ideal for high precision spectroscopy. Thus, they are the basis of some of the best optical clock setups to date. However, despite their minute optical dipole moments the inherent long range dipole-dipole interactions in such lattices generate line shifts, dephasing and modified decay. We show that in a perfectly filled lattice these effects are resonantly enhanced depending on lattice constant, lattice geometry and excitation scheme inducing clock shifts of many atomic linewidths and reducing measurement precision via superradiance. However, under optimal conditions collective effects can be exploited to yield zero effective shifts and prolong dipole lifetimes beyond the single atom decay. In particular we identify 2D hexagonal or square lattices with lattice constants below the optical wavelength as most promising configurations for an accuracy and precision well below the independent ensemble limit. This geometry should also be an ideal basis for related applications such as superradiant lasers, precision magnetometry or long lived quantum memories.

quant-ph↗

arXiv2015

Alternating Least Squares Tensor Completion in The TT-Format

We consider the problem of fitting a low rank tensor $A\in\mathbb{R}^{\mathcal I}$, ${\mathcal I} = \{1,\ldots,n\}^{d}$, to a given set of data points $\{M_i\in\mathbb{R}\mid i\in P\}$, $P\subset{\mathcal I}$. The low rank format under consideration is the hierarchical or TT or MPS format. It is characterized by rank bounds $r$ on certain matricizations of the tensor. The number of degrees of freedom is in ${\cal O}(r^2dn)$. For a fixed rank and mode size $n$ we observe that it is possible to reconstruct random (but rank structured) tensors as well as certain discretized multivariate (but rank structured) functions from a number of samples that is in ${\cal O}(\log N)$ for a tensor having $N=n^d$ entries. We compare an alternating least squares fit (ALS) to an overrelaxation scheme inspired by the LMaFit method for matrix completion. Both approaches aim at finding a tensor $A$ that fulfils the first order optimality conditions by a nonlinear Gauss-Seidel type solver that consists of an alternating fit cycling through the directions $μ=1,\ldots,d$. The least squares fit is of complexity ${\cal O}(r^4d\#P)$ per step, whereas each step of ADF is in ${\cal O}(r^2d\#P)$, albeit with a slightly higher number of necessary steps. In the numerical experiments we observe robustness of the completion algorithm with respect to noise and good reconstruction capability. Our tests provide evidence that the algorithm is suitable in higher dimension ($>$10) as well as for moderate ranks. Keywords: MPS, Tensor Completion, Tensor Train, TT, Hierarchical Tucker, HT, ALS.

math.NA↗

arXiv2014

Self-ordering dynamics of ultracold atoms in multicolored cavity fields

We study light induced spatial crystallization of ultracold quantum particles confined along the axis of a high-$Q$ linear cavity via a transverse multicolor pump using numerical simulations. Whenever a pump frequency is tuned close to resonance with a longitudinal cavity mode, the dynamics favors bistable spatial particle ordering into a Bragg grating at a wavelength distance. Simultaneous pumping at several resonant frequencies fosters competition between the different spatial lattice orders, exhibiting complex nonlinear field dynamics involving several metastable atom-field states. For few particles even superpositions of different spatial orders entangled with different light mode amplitudes appear. By a proper choice of trap geometry and pump frequencies a broad variety of many particle Hamiltonians with a nontrivial long range coupling can be emulated in such a setup. When applying quantum Monte Carlo wave function simulations to study time evolution we find simultaneous super radiant scattering into several light modes and the buildup of strong non-classical atom field correlations.

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