Efficient Construction of Canonical Polyadic Approximations of Tensor Networks

Pierce, Karl; Valeev, Edward F.

doi:10.1021/acs.jctc.2c00861

Cited by 4 publications

(2 citation statements)

References 69 publications

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“…Use of this doubly-factorized form has also been previously suggested in the use of a reduced-scaling particle-particle RPA scheme 80 . This form for the integrals can be directly constructed with controllable errors in O[N 3 ] time 81,82 . Once found, the Z P Q can be symmetrically decomposed as Z P Q = Y P R Y QR .…”

Section: Reduction To Cubic-scaling Gwmentioning

confidence: 99%

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A 'moment-conserving' reformulation of GW theory

Scott¹,

Backhouse²,

Booth³

2023

Preprint

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We show how to construct an effective Hamiltonian whose dimension scales linearly with system size, and whose eigenvalues systematically approximate the excitation energies of GW theory. This is achieved by rigorously expanding the self-energy in order to exactly conserve a desired number of frequency-independent moments of the self-energy dynamics. Recasting GW in this way admits a low-scaling O[N 4 ] approach to build this Hamiltonian, with a proposal to reduce this further to O [N 3 ]. This relies on exposing a novel recursive framework for the density response moments of the random phase approximation (RPA), where the efficient calculation of its starting point mirrors the low-scaling approaches to compute RPA correlation energies. The frequency integration of GW which distinguishes so many different GW variants can be performed directly and cheaply in this moment representation. Furthermore, the solution to the Dyson equation can be performed exactly, avoiding analytic continuation, diagonal approximations or iterative solutions to the quasiparticle equation, with the full-frequency spectrum of all solutions obtained in a complete diagonalization of this effective static Hamiltonian. We show how this approach converges rapidly with respect to the order of the conserved self-energy moments, and is applied across the GW 100 benchmark dataset to obtain accurate GW spectra in comparison to traditional implementations. We also show the ability to systematically converge all-electron full-frequency spectra and high-energy features beyond frontier excitations, as well as avoiding discontinuities in the spectrum which afflict many other GW approaches.

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Section: Reduction To Cubic-scaling Gwmentioning

confidence: 99%

“…B QR (p) = X iQ X aQ i e − a p e ip X iR X aR (82) can be evaluated in O[N 3 ] cost. Further contractions in the evaluation of Eq.…”

Section: Reduction To Cubic-scaling Gwmentioning

confidence: 99%

A 'moment-conserving' reformulation of GW theory

Scott¹,

Backhouse²,

Booth³

2023

Preprint

View full text Add to dashboard Cite

show abstract

A “moment-conserving” reformulation of GW theory

Scott

Backhouse

Booth

2023

The Journal of Chemical Physics

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We present a reformulation of $GW$ theory, where the self-energy is systematically expanded in order to exactly conserve a truncated set of frequency-independent moments of the full spectral dynamics. Recasting $GW$ in this way admits a low-scaling $\mathcal{O}[N^4]$ implementation, with a proposal to reduce this further to $\mathcal{O}[N^3]$. This relies on exposing a novel recursive framework for the density response moments of the random phase approximation (RPA), where the efficient calculation of its starting point mirrors the low-scaling approaches to compute RPA correlation energies. The frequency integration of $GW$ which distinguishes so many different $GW$ variants can be performed directly and cheaply in this moment representation. Furthermore, the solution to the Dyson equation can be performed exactly, avoiding diagonal approximations or iterative solutions to the quasiparticle equation, with the full-frequency spectrum of all solutions obtained in a single diagonalization of an effective static Hamiltonian. We show how this approach converges rapidly with respect to the order of the conserved self-energy moments, and is applied across the $GW100$ benchmark dataset to obtain accurate $GW$ spectra in comparison to traditional implementations. We also show the ability to systematically converge all-electron full-frequency spectra and high-energy features beyond frontier excitations, as well as avoiding discontinuities in the spectrum which afflict many other $GW$ approaches.

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Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

Kaye,

Huang,

Strand

et al. 2024

Phys. Rev. X

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We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an Mth-order diagram at inverse temperature β and spectral width ωmax from O((βωmax)2M−1) for a direct quadrature to O(M(log(βωmax))M+1), with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca2RuO4, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024

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Efficient Construction of Canonical Polyadic Approximations of Tensor Networks

Cited by 4 publications

References 69 publications

A 'moment-conserving' reformulation of GW theory

A 'moment-conserving' reformulation of GW theory

A “moment-conserving” reformulation of GW theory

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

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