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arXiv:2002.05987 · Physical Review Research

Eliminating the wave-function singularity for ultracold atoms by a similarity transformation

Péter Jeszenszki, Ulrich Ebling, Hongjun Luo, Ali Alavi, Joachim Brand

Published Nov 20, 2020• 25 citations
arXivHTMLPDF

Abstract

A hyperbolic singularity in the wave function of s-wave interacting atoms is the root problem for any accurate numerical simulation.Here, we apply the transcorrelated method, whereby the wave-function singularity is explicitly described by a two-body Jastrow factor, and then folded into the Hamiltonian via a similarity transformation.The resulting nonsingular eigenfunctions are approximated by stochastic Fock-space diagonalization with energy errors scaling with 1/M in the number M of single-particle basis functions.The performance of the transcorrelated method is demonstrated on the example of strongly correlated fermions with unitary interactions.The current method provides the most accurate ground-state energies so far for three and four fermions in a rectangular box with periodic boundary conditions.

DOI
10.1103/physrevresearch.2.043270

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A hyperbolic singularity in the wave function of s-wave interacting atoms is the root problem for any accurate numerical simulation.Here, we apply the transcorrelated method, whereby the wave-function singularity is explicitly described by a two-body Jastrow factor, and then folded into the Hamiltonian via a similarity transformation.The resulting nonsingular eigenfunctions are approximated by stochastic Fock-space diagonalization with energy errors scaling with 1/M in the number M of single-particle basis functions.The performance of the transcorrelated method is demonstrated on the example of strongly correlated fermions with unitary interactions.The current method provides the most accurate ground-state energies so far for three and four fermions in a rectangular box with periodic boundary conditions.
Exact diagonalization expansions of Bose or Fermi gases with contact interactions converge very slowly due to a nonanalytic cusp in the wave function. Here we develop a transcorrelated approach where the cusp is treated exactly and folded into the many-body Hamiltonian with a similarity transformation that removes the leading-order singularity. The resulting transcorrelated Hamiltonian is not Hermitian but can be treated numerically with a standard projection approach. The smoothness of the wave function improves by at least one order and thus the convergence rate for the ground-state energy improves. By numerical investigation of a one-dimensional gas of spin-12\frac{1}{2}21​ fermions we find the error in the transcorrelated energy to scale as ${M}^{\ensuremath{-}3}$ with a single-particle basis of MMM plane waves compared to ${M}^{\ensuremath{-}1}$ for the expansion of the original Hamiltonian and ${M}^{\ensuremath{-}2}$ using conventional lattice renormalization.
Quantum simulations of electronic structure with a transformed Hamiltonian that includes some electron correlation effects are demonstrated. The transcorrelated Hamiltonian used in this work is efficiently constructed classically, at polynomial cost, by an approximate similarity transformation with an explicitly correlated two-body unitary operator. This Hamiltonian is Hermitian, includes no more than two-particle interactions, and is free of electron-electron singularities. We investigate the effect of such a transformed Hamiltonian on the accuracy and computational cost of quantum simulations by focusing on a widely used solver for the Schrödinger equation, namely the variational quantum eigensolver method, based on the unitary coupled cluster with singles and doubles (q-UCCSD) Ansatz. Nevertheless, the formalism presented here translates straightforwardly to other quantum algorithms for chemistry. Our results demonstrate that a transcorrelated Hamiltonian, paired with extremely compact bases, produces explicitly correlated energies comparable to those from much larger bases. For the chemical species studied here, explicitly correlated energies based on an underlying 6-31G basis had cc-pVTZ quality. The use of the very compact transcorrelated Hamiltonian reduces the number of CNOT gates required to achieve cc-pVTZ quality by up to two orders of magnitude, and the number of qubits by a factor of three.
Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground-state wave function directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B 99, 075119 (2019)] have demonstrated that the use of momentum-space representation, combined with a nonunitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more ``compact'' ground-state wave function, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the Ansatz-based quantum imaginary-time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to four orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength (U/t=4U/t=4U/t=4), enabling the use of shallower quantum circuits for wave-function Ans\"atzes. In addition, we propose a more efficient implementation of the quantum imaginary-time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.
In this work, we present the first implementation of the transcorrelated electronic Hamiltonian in an optimization procedure for matrix product states by the density matrix renormalization group (DMRG) algorithm. In the transcorrelation ansatz, the electronic Hamiltonian is similarity-transformed with a Jastrow factor to describe the cusp in the wave function at electron–electron coalescence. As a result, the wave function is easier to approximate accurately with the conventional expansion in terms of one-particle basis functions and Slater determinants. The transcorrelated Hamiltonian in first quantization comprises up to three-body interactions, which we deal with in the standard way by applying robust density fitting to two- and three-body integrals entering the second-quantized representation of this Hamiltonian. The lack of hermiticity of the transcorrelated Hamiltonian is taken care of along the lines of the first work on transcorrelated DMRG [ J. Chem. Phys. 2020, 153, 164115] by encoding it as a matrix product operator and optimizing the corresponding ground state wave function with imaginary-time time-dependent DMRG. We demonstrate our quantum chemical transcorrelated DMRG approach at the example of several atoms and first-row diatomic molecules. We show that transcorrelation improves the convergence rate to the complete basis set limit in comparison to conventional DMRG. Moreover, we study extensions of our approach that aim at reducing the cost of handling the matrix product operator representation of the transcorrelated Hamiltonian.
The transcorrelated (TC) method is one of the wave-function-based methods used for first-principles electronic structure calculations, and in terms of the computational cost is applicable to solid-state calculation. In this method, a many-body wave function of electrons is approximated as a product of the Jastrow factor and the Slater determinant, and the first-principles Hamiltonian is similarity-transformed by the Jastrow factor. The Schrödinger equation is rewritten as an eigenvalue problem for this similarity-transformed Hamiltonian, from which one obtains a self-consistent field (SCF) equation for optimizing one-electron orbitals in the Slater determinant at low computational cost. In contrast, optimization of the Jastrow factor is computationally much more expensive and has not been performed for solid-state calculation of the TC method before. In this study, we develop a new method for optimizing the Jastrow factor at a reasonable computational cost using the random-phase approximation (RPA) and pseudo-variance minimization. We apply this method to some simple solids, and find that the band gap of a wide-band-gap insulator is much improved by RPA.
The transcorrelated (TC) method dramatically accelerates the convergence of correlated calculations toward the complete-basis-set (CBS) limit by folding a Jastrow correlator into the Hamiltonian via a similarity transformation, incorporating the electron--electron cusp into the effective interaction. We make the TC framework practical for large systems and flexible, multi-center correlators by compressing the grid-evaluated TC integrals with the interpolative separable density-fitting (ISDF) approximation, combined with the effective two-body (xTC) treatment of the three-body operator. This low-rank representation reduces storage and integration costs by orders of magnitude, and a multi-GPU implementation with automatic differentiation of the correlator makes the construction routine for large basis sets. We demonstrate the resulting ISDF-xTC-CCSD method on the linear hydrogen chain, reaching the joint thermodynamic and CBS limits with basis sets up to cc-pV5Z in agreement with state-of-the-art many-body references to within about 1~mHa/atom, and on the benzene ground-state energy with up to 1200 orbitals (cc-pCV5Z), where the method attains state-of-the-art accuracy at the coupled cluster singles and doubles level and its CBS extrapolation is markedly more robust than that of conventional coupled-cluster methods.