Classical Benchmarks of a Symmetry-Adapted Variational Quantum Eigensolver for Real-Time Green's Functions in Dynamical Mean-Field Theory

TL;DR

The DMFT impurity problem is challenging due to spectral accuracy, and bath discretization limits spectral resolution. The authors address this by using a symmetry-preserving VQE combined with real-time Trotter evolution to solve a 4-site Anderson Impurity Model and extract the retarded Green's function for DMFT. Ground-state energies converge with relative error on the order of $10^{-4}$, and the real-time Green's function matches exact results well in the strongly correlated regime ($U=6,8$), though weak coupling shows deficits in low-energy features; the coarse Trotter steps used are sufficient for practical dynamical information. This work demonstrates a feasible quantum impurity solver beyond the two-site DMFT, informing ansatz design and dynamical measurement strategies and guiding future integration with real-time self-consistency loops and larger clusters.

Abstract

We present a variational quantum eigensolver (VQE) approach for solving the Anderson Impurity Model (AIM) arising in Dynamical Mean-Field Theory (DMFT). Recognizing that the minimal two-site approximation often fails to resolve essential spectral features, we investigate the efficacy of VQE for larger bath discretizations while adhering to near-term hardware constraints. We employ a symmetry-adapted ansatz enforcing conservation of particle number $(N)$, spin projection $(S_z=0)$, and total spin $(S^2=0)$ symmetry, benchmarking the performance against exact diagonalization across different interaction strengths using bath parameters extracted from the DMFT self-consistency loop. For a four-site model, the relative error in the ground state energy remains well below $0.01%$ with a compact parameter set $(N_p \le 30)$. Crucially, we demonstrate that the single-particle Green's function-the central quantity for DMFT-can be accurately extracted from VQE-prepared ground states via real-time evolution in the intermediate to strong interaction regimes. However, in the weak interaction regime, the Green's function exhibits noticeable deviations from the exact benchmark, particularly in resolving low-energy spectral features, despite the ground state energy showing excellent agreement. These findings demonstrate that VQE combined with real-time evolution can effectively extend quantum-classical hybrid DMFT beyond the two-site approximation, particularly for describing insulating phases. While this approach offers a viable pathway for simulating strongly correlated materials on near-term devices, the observation that accurate ground state energy does not guarantee accurate dynamical properties highlights a key challenge for applying such approaches to correlated metals.

Figures (6)

• Figure 1: Quantum Circuit for the variational wavefunction. $R_y$ and $R_z$ denote rotational gates along the $Y$ and $Z$ axes, respectively. $\theta[i]$ are the rotation angles, which are treated as parameters to be optimized. This schematic illustrates the specific implementation for $N=4$ sites with 3 layers; with each layer containing 10 parameters, resulting in a total of tunable parameters for the entire circuit.
• Figure 2: The gate connecting two qubits to implement the hopping term, also denoted as the Reconfigurable Beam Splitter (RBS) gate.
• Figure 3: Schematic of the 4-site one-dimensional chain for the Anderson impurity model, consisting of a single impurity site (site 0) coupled to three bath sites.
• Figure 4: Convergence of the ground state energy as a function of the number of iterations for the VQE ansatz shown in Fig. \ref{['fig:VQE_circuit']}. Results are displayed for interaction strengths $U = 4,6, \text{and } 8$.
• Figure 5: Time evolution of the imaginary part of the retarded Green's function computed using the VQE-optimized ground state and Trotter propagation with a fixed time step $\Delta t=0.2$. The upper panels compare the VQE results with exact benchmarks for interaction strengths $U=4,6, \text{and } 8$. The lower panels display the corresponding absolute error of the VQE calculation relative to the exact solution.