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arXiv:2601.19896

Real-Time Iteration Scheme for Dynamical Mean-Field Theory: A Framework for Near-Term Quantum Simulation

Chakradhar Rangi, Aadi Singh, Ka-Ming Tam

cond-mat.str-elcond-mat.mtrl-sciquant-phSubmitted Jan 27, 2026• Announced Jan 28, 2026• 0 citations
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Abstract

We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.

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12 pages, 4 figures

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We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.
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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.
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)(N)(N), spin projection (Sz=0)(S_z=0)(Sz​=0), and total spin (S2=0)(S^2=0)(S2=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 0.010.01%0.01 with a compact parameter set (Np≤30)(N_p \le 30)(Np​≤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.
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 here present how a self-consistent solution of the dynamical mean field theory equations can be obtained using exact diagonalization of an Anderson impurity model with accuracies comparable to those found using renormalization group or quantum Monte Carlo methods. We show how one can solve a correlated quantum impurity coupled to several hundred uncorrelated bath sites, using a restricted active basis set. The number of bath sites determines the resolution of the obtained spectral function, which consists of peaks roughly spaced by the band width divided by the number of bath sites. The self-consistency cycles are fully performed on the real frequency axis and expressed as numerical stable matrix operations. The same impurity solver has been used on Ligand Field and finite size cluster calculations and is capable of treating involved Hamiltonians including the full rotational invariant Coulomb interaction, spin-orbit coupling and low symmetry crystal-fields. The proposed method allows for the calculation of a variety of correlation functions at little extra cost.