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arXiv:1908.03902 · Physical Review A

Construction of Green's functions on a quantum computer: Quasiparticle spectra of molecules

Taichi Kosugi, Yu‐ichiro Matsushita

Published Jan 17, 2020• 40 citations
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Abstract

We propose a scheme for the construction of the one-particle Green's function (GF) of an interacting electronic system via statistical sampling on a quantum computer. Although the nonunitarity of creation and annihilation operators for the electronic spin orbitals prevents us from preparing specific states selectively, probabilistic state preparation is demonstrated to be possible for the qubits. We provide quantum circuits equipped with at most two ancillary qubits for obtaining all the components of the GF. We perform simulations of such construction of GFs for LiH and ${\mathrm{H}}_{2}\mathrm{O}$ molecules based on the unitary coupled-cluster (UCC) method to demonstrate the validity of our scheme by comparing the quasiparticle and satellite spectra exact within the UCC method and those from full configuration-interaction calculations. We also examine the accuracy of the sampling method by exploiting the Galitskii-Migdal formula, which gives the total energy only from the GF.

DOI
10.1103/physreva.101.012330

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We propose a scheme for the construction of the one-particle Green's function (GF) of an interacting electronic system via statistical sampling on a quantum computer. Although the nonunitarity of creation and annihilation operators for the electronic spin orbitals prevents us from preparing specific states selectively, probabilistic state preparation is demonstrated to be possible for the qubits. We provide quantum circuits equipped with at most two ancillary qubits for obtaining all the components of the GF. We perform simulations of such construction of GFs for LiH and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecules based on the unitary coupled-cluster (UCC) method to demonstrate the validity of our scheme by comparing the quasiparticle and satellite spectra exact within the UCC method and those from full configuration-interaction calculations. We also examine the accuracy of the sampling method by exploiting the Galitskii-Migdal formula, which gives the total energy only from the GF.
We propose a scheme for the construction of the one-particle Green's function (GF) of an interacting electronic system via statistical sampling on a quantum computer.Although the nonunitarity of creation and annihilation operators for the electronic spin orbitals prevents us from preparing specific states selectively, probabilistic state preparation is demonstrated to be possible for the qubits.We provide quantum circuits equipped with at most two ancillary qubits for obtaining all the components of the GF.We perform simulations of such construction of GFs for LiH and H 2 O molecules based on the unitary coupled-cluster (UCC) method to demonstrate the validity of our scheme by comparing the quasiparticle and satellite spectra exact within the UCC method and those from full configuration-interaction calculations.We also examine the accuracy of the sampling method by exploiting the Galitskii-Migdal formula, which gives the total energy only from the GF.
A unitary coupled cluster (UCC) form for the wave function in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on the number and type of terms are included in the wave function parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems, we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H 2 and LiH molecules with chemical accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.
We present a quantum-classical hybrid implementation of the Liouvillian recursion method to compute many-body Green's functions using a quantum computer. From an approximate ground state preparation circuit, this algorithm produces the local (r=r′r=r'r=r′) and inter-site (r≠r′r\neq r'r=r′) Green's functions Grr′(ω)G_{rr'}(ω)Grr′​(ω) by measuring observables generated recursively. We demonstrate the approach on a superconducting quantum processor for the open-boundary four-site Hubbard model. We then use the computed Green's functions as input to the Galitskii-Migdal formula to produce better ground state energy estimation than the expectation value of the Hamiltonian for the approximate circuit. Empirical results indicate exponential convergence in the number of iterations, yielding a computational complexity polynomial in the Green's-function accuracy, as measured with the Wasserstein distance. Our results also indicate significant robustness to noise and to inaccuracies of the ground state preparation, providing evidence that Liouvillian recursion is well adapted to the constraints of near-term quantum computing.
An iterative version of the qubit coupled cluster (QCC) method [I. G. Ryabinkin et al., J. Chem. Theory Comput. 2019, 14, 6317] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H 2 O, and N 2 molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of the Hamiltonian.
We present results for many-body perturbation theory for the one-body Green's function at finite temperatures using the Matsubara formalism. Our method relies on the accurate representation of the single-particle states in standard Gaussian basis sets, allowing to efficiently compute, among other observables, quasiparticle energies and Dyson orbitals of atoms and molecules. In particular, we challenge the second-order treatment of the Coulomb interaction by benchmarking its accuracy for a well-established test set of small molecules, which includes also systems where the usual Hartree-Fock treatment encounters difficulties. We discuss different schemes how to extract quasiparticle properties and assess their range of applicability. With an accurate solution and compact representation, our method is an ideal starting point to study electron dynamics in time-resolved experiments by the propagation of the Kadanoff-Baym equations.