We introduce a hybrid quantum-classical algorithm to compute the Green function for strongly correlated electrons on noisy intermediate-scale quantum (NISQ) devices. The technique consists in the construction of a nonorthogonal excitation basis composed of a set of single-particle excitations generated by occupation-number operators. The excited sectors of the Hamiltonian in this basis can then be measured on the quantum device and a classical postprocessing procedure yields the Green function in the Lehmann representation. The technique allows for noise filtering, a useful feature for NISQ devices. To validate the approach, we carry out a set of proof-of-principle calculations on the single-band Hubbard model on IBM quantum hardware. For a two-site system we find good agreement between the results of quantum simulations and the exact result for the local spectral function. This proof of principle also shows that the noise filtering provides a reliable way to get rid of satellite peaks present in the spectral weight obtained from a NISQ device. A simulation of a four-site system carried out on classical hardware suggests that the approach can achieve similar accuracy for larger systems.
We introduce a hybrid quantum-classical algorithm to compute the Green function for strongly correlated electrons on noisy intermediate-scale quantum (NISQ) devices. The technique consists in the construction of a nonorthogonal excitation basis composed of a set of single-particle excitations generated by occupation-number operators. The excited sectors of the Hamiltonian in this basis can then be measured on the quantum device and a classical postprocessing procedure yields the Green function in the Lehmann representation. The technique allows for noise filtering, a useful feature for NISQ devices. To validate the approach, we carry out a set of proof-of-principle calculations on the single-band Hubbard model on IBM quantum hardware. For a two-site system we find good agreement between the results of quantum simulations and the exact result for the local spectral function. This proof of principle also shows that the noise filtering provides a reliable way to get rid of satellite peaks present in the spectral weight obtained from a NISQ device. A simulation of a four-site system carried out on classical hardware suggests that the approach can achieve similar accuracy for larger systems.
Green’s function methods lead to ab initio, systematically improvable simulations of molecules and materials while providing access to multiple experimentally observable properties such as the density of states and the spectral function. The calculation of the exact one-particle Green’s function remains a significant challenge for classical computers and was attempted only on very small systems. Here, we present a hybrid quantum-classical algorithm to calculate the imaginary-time one-particle Green’s function. The proposed algorithm combines the variational quantum eigensolver and the quantum subspace expansion methods to calculate Green’s function in Lehmann’s representation. We demonstrate the validity of this algorithm by simulating H 2 and H 4 on quantum simulators and on IBM’s quantum devices.
We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a Noisy, Intermediate Scale Quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the Variational Quantum Eigensolver algorithm. This iterative scheme, dubbed Natural Orbitalization (NOization), gradually transforms the single-particle basis to the approximate Natural-Orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density-matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) ansatz circuits faster and more accurate, and we propose an ansatz, the Multi-Reference Excitation Preserving (MREP) ansatz, that achieves great expressivity without requiring a prohibitive gate count. The one-impurity version of the ansatz has only one parameter, making the ground state preparation a trivial step, which supports the optimal character of our approach. Within a Rotationally Invariant Slave Boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP ansatz allow us to compute accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.
We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a noisy, intermediate scale quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the variational quantum eigensolver algorithm. This iterative scheme, dubbed natural orbitalization (NOization), gradually transforms the single-particle basis to the approximate natural-orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) Ansatz circuits faster and more accurate, and we propose an Ansatz, the multireference excitation-preserving (MREP) Ansatz, that achieves great expressivity without requiring a prohibitive gate count, thus bridging the gap between hardware-efficient and physically motivated strategies in variational Ansatz design. The one-impurity version of the Ansatz has only one parameter, making the ground-state preparation a trivial step, which supports the optimal character of our approach. Within a rotationally invariant slave boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP Ansatz allow us to compute accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.
Abstract We report on a quantum–classical simulation of the single-band Hubbard model using two-site dynamical mean-field theory (DMFT). Our approach uses IBM’s superconducting qubit chip to compute the zero-temperature impurity Green’s function in the time domain and a classical computer to fit the measured Green’s functions and extract their frequency domain parameters. We find that the quantum circuit synthesis (Trotter) and hardware errors lead to incorrect frequency estimates, and subsequently to an inaccurate quasiparticle weight when calculated from the frequency derivative of the self-energy. These errors produce incorrect hybridization parameters that prevent the DMFT algorithm from converging to the correct self-consistent solution. To avoid this pitfall, we compute the quasiparticle weight by integrating the quasiparticle peaks in the spectral function. This method is much less sensitive to Trotter errors and allows the algorithm to converge to self-consistency for a half-filled Mott insulating system after applying quantum error mitigation techniques to the quantum simulation data.
Quantum computers (QC) could harbor the potential to significantly advance materials simulations, particularly at the atomistic scale involving strongly correlated fermionic systems, where an accurate description of quantum many-body effects scales unfavorably with size. While a full-scale treatment of condensed matter systems with currently available noisy quantum computers remains elusive, quantum embedding schemes like dynamical mean-field theory (DMFT) allow the mapping of an effective, reduced subspace Hamiltonian to available devices to improve the accuracy of ab initio calculations such as density functional theory (DFT). Here, we report on the development of a hybrid quantum-classical DFT + DMFT simulation framework which relies on a quantum impurity solver based on the Lehmann representation of the impurity Green’s function. Hardware experiments with up to 14 qubits on the IBM Quantum system are conducted, using advanced error mitigation methods and a novel calibration scheme for an improved zero-noise extrapolation to effectively reduce adverse effects from inherent noise on current quantum devices. We showcase the utility of our quantum DFT + DMFT workflow by assessing the correlation effects on the electronic structure of a real material, Ca 2 CuO 2 Cl 2 , which is mapped to an effective single-band Hubbard Hamiltonian and the subsequently derived Anderson impurity model solved with up to 6 bath sites on available quantum hardware. Further, we carefully benchmark our quantum results with respect to exact reference solutions and experimental spectroscopy measurements. While challenges remain to scale our approach to larger, multi-orbital and multi-site systems with more bath sites, the present work marks an important milestone towards achieving utility-scale quantum computation in materials simulation.
In the framework of the hybrid quantum-classical variational cluster approach to strongly correlated electron systems one of the goals of a quantum subroutine is to find single-particle correlation functions of lattice fermions in polynomial time. Previous works suggested to use variants of the Hadamard test for this purpose, which requires an implementation of controlled single-particle fermionic operators. However, for a number of locality-preserving mappings to encode fermions into qubits, a direct construction of such operators is not possible. In this work, we propose a quantum algorithm that uses an analog of the Kubo formula adapted to a quantum circuit simulating the Hubbard model. It allows to access the Green's function of a cluster directly using only bilinears of fermionic operators and circumvents the usage of the Hadamard test. We test our algorithm in practice by using open-access simulators of noisy IBM superconducting chips.