Robust Approximation of Tensor Networks: Application to Grid-Free Tensor Factorization of the Coulomb Interaction

Pierce, Karl; Rishi, Varun; Valeev, Edward F.

doi:10.1021/acs.jctc.0c01310

Cited by 12 publications

(26 citation statements)

References 89 publications

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“…The approximation made in eq 13 is known as the robust pseudospectral (rPS) and was introduced recently. 81 If one only includes the first line of the equation, then we get the pseudospectral (PS) method, and if one only uses the last line of the equation, then we get the tensor hypercontraction (THC) method. Recently, it was emphasized by Valeev et al 81 that the quadratic error can be obtained using rPS.…”

Section: Robust Fittingmentioning

confidence: 99%

“…If one only includes the first line of the equation, then we get the pseudospectral (PS) method, and if one only uses the last line of the equation, then we get the tensor hypercontraction (THC) method. Recently, it was emphasized by Valeev et al that the quadratic error can be obtained using rPS. In fact, this approach has been previously used in other contexts including robust density fitting , and even in quantum Monte Carlo for the evaluation of the reduced density matrices. − …”

Section: Speeding Up Exchange Evaluationmentioning

confidence: 99%

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Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Sharma

White

Beylkin

2022

J. Chem. Theory Comput.

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In this article, we present an algorithm to efficiently evaluate the exchange matrix in periodic systems when a Gaussian basis set with pseudopotentials is used. The usual algorithm for evaluating exchange matrix scales cubically with the system size because one has to perform O(N 2) fast Fourier transform (FFT). Here, we introduce an algorithm that retains the cubic scaling but reduces the prefactor significantly by eliminating the need to do FFTs during each exchange build. This is accomplished by representing the products of Gaussian basis function using a linear combination of an auxiliary basis the number of which scales linearly with the size of the system. We store the potential due to these auxiliary functions in memory, which allows us to obtain the exchange matrix without the need to do FFT, albeit at the cost of additional memory requirement. Although the basic idea of using auxiliary functions is not new, our algorithm is cheaper due to a combination of three ingredients: (a) we use a robust pseudospectral method that allows us to use a relatively small number of auxiliary basis to obtain high accuracy; (b) we use occ-RI exchange, which eliminates the need to construct the full exchange matrix; and (c) we use the (interpolative separable density fitting) ISDF algorithm to construct these auxiliary basis sets that are used in the robust pseudospectral method. The resulting algorithm is accurate, and we note that the error in the final energy decreases exponentially rapidly with the number of auxiliary functions.

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Section: Robust Fittingmentioning

confidence: 99%

Section: Speeding Up Exchange Evaluationmentioning

confidence: 99%

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Sharma

White

Beylkin

2022

J. Chem. Theory Comput.

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“…However, this could be an artifact the NEVPT2 approximation itself, the approximate NEVPT2 implementation used in this work [106], or the approximate DMRG active-space wavefunction. decomposition [110,111] but cannot be obtained through simple linear algebra relations. One strategy to obtain the THC factors is through the CP decomposition of geminal terms through alternating least-squares [110] with an iteration complexity of O(r 4 M ) where r is the size of the single particle basis.…”

Section: A Geometriesmentioning

confidence: 99%

Reliably assessing the electronic structure of cytochrome P450 on today's classical computers and tomorrow's quantum computers

Goings¹,

White²,

Tautermann³

et al. 2022

Preprint

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An accurate assessment of how quantum computers can be used for chemical simulation, especially their potential computational advantages, provides important context on how to deploy these future devices. In order to perform this assessment reliably, quantum resource estimates must be coupled with classical simulations attempting to answer relevant chemical questions and to define the classical simulation frontier. Herein, we explore the quantum and classical resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical-quantum advantage boundary. This is accomplished by analyzing the convergence of DMRG+NEVPT2 and coupled cluster singles doubles with non-iterative triples (CCSD(T)) calculations for spin-gaps in models of the CYP catalytic cycle that indicate multireference character. The quantum resources required to perform phase estimation using qubitized quantum walks are calculated for the same systems. Compilation into the surface-code provides runtime estimates to compare directly to DMRG runtimes and to evaluate potential quantum advantage. Both classical and quantum resource estimates suggest that simulation of CYP models at scales large enough to balance dynamic and multiconfigurational electron correlation has the potential to be a quantum advantage problem and emphasizes the important interplay between classical simulations and quantum algorithms development for chemical simulation.

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“…Note that although eq 8 does not impose the hermiticity of B (with respect to swapping of p and s), in practice γ r p = (β p r )* can be imposed. 17,45 The CP3 approximation can then be introduced into eq 7…”

Section: Formalismmentioning

confidence: 99%

“…Our recent experiences 17 suggest that more productive uses of the CP factorization can be found when focusing on approximating not individual tensors but on tensor networks. Thus, here we consider a problem of constructing a CP approximation for a tensor network.…”

Section: Introductionmentioning

confidence: 99%

Efficient Construction of Canonical Polyadic Approximations of Tensor Networks

Pierce

Valeev

2022

J. Chem. Theory Comput.

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We consider the problem of constructing a canonical polyadic (CP) decomposition for a tensor network, rather than a single tensor. We illustrate how it is possible to reduce the complexity of constructing an approximate CP representation of the network by leveraging its structure in the course of the CP factor optimization. The utility of this technique is demonstrated for the order-4 Coulomb interaction tensor approximated by two order-3 tensors via an approximate generalized square-root (SQ) factorization, such as density fitting or (pivoted) Cholesky. The complexity of constructing a four-way CP decomposition is reduced from scriptO ( n 4 R CP ) (for the nonapproximated Coulomb tensor) to scriptO ( n 3 R CP ) (for the SQ-factorized Coulomb tensor), where n and R CP are the basis and CP ranks, respectively. This reduces the cost of constructing the CP approximation of two-body interaction tensors of relevance to accurate many-body electronic structure by up to 2 orders of magnitude for systems with up to 36 atoms studied here. The full four-way CP approximation of the Coulomb interaction tensor is shown to be more accurate than the known approaches which utilize CP-factorizations of the SQ factors (which are also constructed with an scriptO ( n 3 R CP ) cost), such as the algebraic pseudospectral and tensor hypercontraction approaches. The CP-decomposed SQ factors can also serve as a robust initial guess for the four-way CP factors.

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Robust Approximation of Tensor Networks: Application to Grid-Free Tensor Factorization of the Coulomb Interaction

Cited by 12 publications

References 89 publications

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Reliably assessing the electronic structure of cytochrome P450 on today's classical computers and tomorrow's quantum computers

Efficient Construction of Canonical Polyadic Approximations of Tensor Networks

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