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International Journal of Quantum Chemistry

A powerful procedure for optimizing AGP states

Hans Jørgen Aa. Jensen, Brian Weiner, J. V. Ortiz, Yngve Öhrn

Published Jun 19, 2009• 2 citations
Publisher

Abstract

We present a powerful iterative algorithm for optimizing the geminal coefficients of an antisymmetrized geminal power (AGP) wavefunction. The algorithm is based on the lowest eigenvector of a matrix closely related to the Hessian of the problem. This matrix can be derived either by using Euler's theorem or by utilizing a unitary group approach. Two important features of the scheme are that it will always converge towards a minimum and that in the neighborhood of a minimum it is comparable to a quadratically convergent Newton-Raphson method.

DOI
10.1002/qua.560220853

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We present a powerful iterative algorithm for optimizing the geminal coefficients of an antisymmetrized geminal power (AGP) wavefunction. The algorithm is based on the lowest eigenvector of a matrix closely related to the Hessian of the problem. This matrix can be derived either by using Euler's theorem or by utilizing a unitary group approach. Two important features of the scheme are that it will always converge towards a minimum and that in the neighborhood of a minimum it is comparable to a quadratically convergent Newton-Raphson method.
We investigated some variational methods to compute a wavefunction based on antisymmetric product of geminals (APG). The Waring decomposition on the APG wavefunction leads a finite sum of antisymmetrized geminal power (AGP) wavefunctions, each for which the variational principle can be applied. We call this as AGP-CI method which provides a variational solution of the APG wavefunction efficiently. However, number of AGP wavefunctions in the exact AGP-CI formalism become exponentially large in case of many-electron systems. Therefore, we also investigate the low-rank APG wavefunction, in which the coefficient matrices of geminals are factorized by the Schur decomposition. Interestingly, only a few non-zero eigenvalues (up to half number of electrons) were found from the Schur decomposition on the APG wavefunction. We developed some methods to approximate the APG wavefunction by lowering the ranks of coefficient matrices of geminals, and demonstrate their performance. Our new geminal method based on the low-rank decomposition can drastically reduce the number of variational parameters, although no efficient formalism has been elaborated so far, due to the difficulty of optimization caused by the inability to compute analytical derivatives.
We propose and implement an algorithm to calculate the norm and reduced density matrices (RDMs) of the antisymmetrized geminal power of any rank with polynomial cost. Our method scales quadratically per element of the RDMs. Numerical tests indicate that our method is very fast and capable of treating systems with a few thousand orbitals and hundreds of electrons reliably in double-precision. In addition, we present reconstruction formulas that allow one to decompose higher order RDMs in terms of linear combinations of lower order ones and geminal coefficients, thereby reducing the computational cost significantly.
Abstract For variational algorithms on the near term quantum computing hardware, it is highly desirable to use very accurate ansatze with low implementation cost. Recent studies have shown that the antisymmetrized geminal power (AGP) wavefunction can be an excellent starting point for ansatze describing systems with strong pairing correlations, as those occurring in superconductors. In this work, we show how AGP can be efficiently implemented on a quantum computer with circuit depth, number of CNOTs, and number of measurements being linear in system size. Using AGP as the initial reference, we propose and implement a unitary correlator on AGP and benchmark it on the ground state of the pairing Hamiltonian. The results show highly accurate ground state energies in all correlation regimes of this model Hamiltonian.
Abstract Electronic structure methods typically benefit from symmetry breaking and restoration, specially in the strong correlation regime. The same goes for ansätze on a quantum computer. We develop a unitary coupled cluster method based on the antisymmetrized geminal power (AGP)—a state formally equivalent to the number-projected Bardeen–Cooper–Schrieffer wavefunction. We demonstrate our method for the single-band Fermi–Hubbard Hamiltonian in one and two dimensions. We also explore post-selection as a state preparation step to obtain correlated AGP and prove that it scales no worse than O ( M ) in the number of measurements, thereby making it a less expensive alternative to gauge integration to restore particle number symmetry.