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arXiv:2410.07808 · Physical Review Letters

Self-Consistent Determination of Single-Impurity Anderson Model Using Hybrid Quantum-Classical Approach on a Spin Quantum Simulator

Xinfang Nie, Xuanran Zhu, Yu-ang Fan, Xinyue Long, Hongfeng Liu, Keyi Huang, Xi Cheng, Liangyu Che, Yuxuan Zheng, Yufang Feng, Xiaodong Yang, Dawei Lu

Published Oct 1, 2024• 10 citations
arXivHTMLPDF

Abstract

The accurate determination of the electronic structure of strongly correlated materials using first principle methods is of paramount importance in condensed matter physics, computational chemistry, and material science. However, due to the exponential scaling of computational resources, incorporating such materials into classical computation frameworks becomes prohibitively expensive. In 2016, Bauer et al. proposed a hybrid quantum-classical approach to correlated materials [B. Bauer et al., Hybrid quantum-classical approach to correlated materials, Phys. Rev. X 6, 031045 (2016).PRXHAE2160-330810.1103/PhysRevX.6.031045] that can efficiently tackle the electronic structure of complex correlated materials. Here, we experimentally demonstrate that approach to tackle the computational challenges associated with strongly correlated materials. By seamlessly integrating quantum computation into classical computers, we address the most computationally demanding aspect of the calculation, namely the computation of the Green's function, using a spin quantum processor. Furthermore, we realize a self-consistent determination of the single impurity Anderson model through a feedback loop between quantum and classical computations. A quantum phase transition in the Hubbard model from the metallic phase to the Mott insulator is observed as the strength of electron correlation increases. As the number of qubits with high control fidelity continues to grow, our experimental findings pave the way for solving even more complex models, such as strongly correlated crystalline materials or intricate molecules.

DOI
10.1103/physrevlett.133.140602

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The accurate determination of the electronic structure of strongly correlated materials using first principle methods is of paramount importance in condensed matter physics, computational chemistry, and material science. However, due to the exponential scaling of computational resources, incorporating such materials into classical computation frameworks becomes prohibitively expensive. In 2016, Bauer et al. proposed a hybrid quantum-classical approach to correlated materials [B. Bauer et al., Hybrid quantum-classical approach to correlated materials, Phys. Rev. X 6, 031045 (2016).PRXHAE2160-330810.1103/PhysRevX.6.031045] that can efficiently tackle the electronic structure of complex correlated materials. Here, we experimentally demonstrate that approach to tackle the computational challenges associated with strongly correlated materials. By seamlessly integrating quantum computation into classical computers, we address the most computationally demanding aspect of the calculation, namely the computation of the Green's function, using a spin quantum processor. Furthermore, we realize a self-consistent determination of the single impurity Anderson model through a feedback loop between quantum and classical computations. A quantum phase transition in the Hubbard model from the metallic phase to the Mott insulator is observed as the strength of electron correlation increases. As the number of qubits with high control fidelity continues to grow, our experimental findings pave the way for solving even more complex models, such as strongly correlated crystalline materials or intricate molecules.
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Rapid progress in noisy intermediate-scale quantum (NISQ) computing technology has led to the development of novel resource-efficient hybrid quantum-classical algorithms, such as the variational quantum eigensolver (VQE), that can address open challenges in quantum chemistry, physics, and material science.Proof-of-principle quantum chemistry simulations for small molecules have been demonstrated on NISQ devices.While several approaches have been theoretically proposed for correlated materials, NISQ simulations of interacting periodic models on current quantum devices have not yet been demonstrated.Here, we develop a hybrid quantumclassical simulation framework for correlated electron systems based on the Gutzwiller variational embedding approach.We implement this framework on Rigetti quantum processing units (QPUs) and apply it to the periodic Anderson model, which describes a correlated heavy electron band hybridizing with noninteracting conduction electrons.Our simulation results quantitatively reproduce the known ground state quantum phase diagram including metallic, Kondo and Mott insulating phases.This is the first fully self-consistent hybrid quantum-classical simulation of an infinite correlated lattice model executed on QPUs, demonstrating that the Gutzwiller hybrid quantum-classical embedding framework is a powerful approach to simulate correlated materials on NISQ hardware.This benchmark study also puts forth a concrete pathway towards practical quantum advantage on NISQ devices.
Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.
We introduce a set of quantum ansätze tailored for computing Green's functions of single-impurity models using the variational quantum eigensolver. These ansätze conserve particle number and spin, allowing for the efficient preparation of the impurity model's ground state and the Krylov bases required to express the Green's function as a continued fraction. Notably, the parameters preparing the particle and hole excitations used as initial Krylov vectors are fully determined by those optimized for the ground state, eliminating the need for secondary optimizations or alternative excitation preparation methods. We demonstrate the utility of these ansätze by computing Green's functions for single-impurity Anderson models with one bath site within a dynamical mean-field theory loop, where the impuritybath hybridization is updated self-consistently. The parameters defining the states in the Krylov bases are straightforwardly determined by orthonormality in the single-bath case, allowing for the direct computation of the relevant Hamiltonian matrix elements from these states. Future work will explore the robustness, scalability, and practical integration of these ansätze into hybrid quantum-classical simulations of strongly correlated materials.
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.
Solving the Anderson impurity model typically involves a two-step process, where one first calculates the ground state of the Hamiltonian and then computes its dynamical properties to obtain Green’s function. Here, we propose a hybrid classical/quantum algorithm where the first step is performed using a classical computer to obtain the tensor network ground state as well as its quantum circuit representation and the second step is executed on the quantum computer to obtain Green’s function. Our algorithm exploits the efficiency of tensor networks for preparing ground states on classical computers and takes advantage of quantum processors for the evaluation of the time evolution, which can become intractable on classical computers. We demonstrate the algorithm using 24 qubits on a quantum computing emulator for SrVO3 with a multi-orbital Anderson impurity model within the dynamical mean field theory. The tensor network based ground state quantum circuit preparation algorithm can also be performed for up to 60 qubits with our available computing resources, while the state vector emulation of the quantum algorithm for time evolution is beyond what is accessible with such resources. We show that, provided that the tensor network calculation is able to accurately obtain the ground state energy, this scheme does not require a perfect reproduction of the ground state wave function on the quantum circuit to give an accurate Green’s function. This hybrid approach may lead to quantum advantage in materials simulations where the ground state can be computed classically, but where the dynamical properties cannot.