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QTCI free energy density and numerical evaluations of the superconducting gap equation

Mio Tammi

Published Apr 22, 2026• 0 citations
Publisher

Abstract

Jupyter notebooks used for computational analysis of the self-consistent gap equation. The gap quation was solved iteratively, with the free energy density functional, and with the quantics tensor-network cross interpolated free energy density. References:https://xfac.readthedocs.io/en/latest/tutorial-python/quantics2d.htmlhttps://github.com/tensor4all/xfachttps://github.com/joselado/qtcipy

DOI
10.5281/zenodo.19665753

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Jupyter notebooks used for computational analysis of the self-consistent gap equation. The gap quation was solved iteratively, with the free energy density functional, and with the quantics tensor-network cross interpolated free energy density. References:https://xfac.readthedocs.io/en/latest/tutorial-python/quantics2d.htmlhttps://github.com/tensor4all/xfachttps://github.com/joselado/qtcipy
Tensor cross interpolation (TCI) is a powerful technique for learning a tensor train (TT) by adaptively sampling a target tensor based on an interpolation formula. However, when the tensor evaluations contain random noise, optimizing the TT is more advantageous than interpolating the noise. Here, we propose a new method that starts with an initial guess of TT and optimizes it using non-linear least-squares by fitting it to measured points obtained from TCI. We use quantics TCI (QTCI) in this method and demonstrate its effectiveness on sine and two-time correlation functions, with each evaluated with random noise. The resulting QTT exhibits increased robustness against noise compared to the QTCI method. Furthermore, we employ this optimized QTT of the correlation function in quantum simulation based on pseudo-imaginary-time evolution, resulting in ground-state energy with higher accuracy than the QTCI or Monte Carlo methods.
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.
We present trainsum, a versatile Python package for doing computations with multidimensional quantics tensor trains: https://github.com/fh-igd-iet/trainsum. Using the Array API standard together with opt_einsum, trainsum allows the effortless approximation of tensors or functions by tensor trains independent of their shape or dimensionality. Once approximated, our package can perform normal arithmetic operations with quantics tensor trains, including addition, Einstein summations and element-wise transformations. It can be therefore used for generic computations with applications in simulation, data compression, machine learning and data analysis.