Efficient Quantum Implementation of Dynamical Mean Field Theory for Correlated Materials

TL;DR

This work addresses the bottleneck of solving impurity dynamics in dynamical mean field theory (DMFT) for strongly correlated materials by presenting a quantum-classical framework that represents the impurity ground state as a sum of fermionic Gaussian states (FGS) and uses subspace diagonalization (SGS) to keep the heavy lifting on classical hardware while implementing a low-depth, algebraically compressed quantum time-evolution circuit to compute the impurity Green's function. The SGS leverages a low-energy subspace and allows reuse across DMFT iterations (via Eigenvector Continuation), while partial circuit compression exploits the free-fermionic nature of the bath to dramatically reduce gate counts, enabling practical hardware demonstrations. Hardware experiments on IBM devices (with $N_I=1$, $N_B=3$, i.e., 8 qubits plus ancilla) show that error mitigation and PSD de-noising/extension can recover the impurity GF signals, allowing reconstruction of the Matsubara GF $\mathcal{G}_{\text{imp}}(i\omega_n)$ for DMFT self-consistency. The approach offers a near-term path toward quantum advantage in embedding theories for correlated materials and can be extended to multi-impurity (cluster) DMFT and related quantum-embedded frameworks.

Abstract

The accurate theoretical description of materials with strongly correlated electrons is a formidable challenge in condensed matter physics and computational chemistry. Dynamical Mean Field Theory (DMFT) is a successful approach that predicts behaviors of such systems by incorporating some of the correlated behavior using an impurity model, but it is limited by the need to calculate the impurity Green's function. This work proposes a framework for DMFT calculations on quantum computers, focusing on near-term applications. It leverages the structure of the impurity problem, combining a low-rank Gaussian subspace representation of the ground state and a compressed, short-depth quantum circuit that joins state preparation with time evolution to compute Green's functions. We demonstrate the convergence of the DMFT algorithm using the Gaussian subspace in a noise-free setting, and show the hardware viability of circuit compression by extracting the impurity Green's function on IBM quantum processors for a single impurity coupled to three bath orbitals (8 qubits, 1 ancilla). We discuss potential paths toward realizing this quantum computing use case in materials science.

Figures (12)

• Figure 1: Summary of this work. (a) DMFT embeds a complex lattice model into a simpler impurity model, requiring calculation of the impurity Green’s function (GF), $\mathcal{G}(i\omega_n)$. (b) The impurity ground state is approximated as a sum of fermionic Gaussian states (FGS), enabling efficient subspace diagonalization on classical hardware and retaining fidelity across DMFT steps. (c) Time evolution with SGS leverages the bath’s free-fermionic nature for circuit compression on quantum hardware, with results combined to yield $\langle \phi_i|\widehat{\mathbf{A}}(t)\widehat{\mathbf{B}}|\phi_j\rangle$. (d) Post-processing of noisy circuit outputs via PSD de-noising and extension extracts relevant frequencies, yielding the Matsubara GF $\mathcal{G}(i\omega_n)$ for subsequent DMFT cycles.
• Figure 2: Illustration of the subspace selection procedure. (a) A candidate pool, $\mathcal{M}$, of FGS (shaded region) is generated, and from this, (b) a subspace $\mathcal{S}$ is chosen which approximates the ground state of some target impurity Hamiltonian $\mathbf{\widehat{H}}_\text{imp}$. (c) When solving for $\mathbf{\widehat{H}}_\text{imp}$ with nearby Hamiltonian parameters, the same subspace $\mathcal{S}$ is used to represent the impurity ground state using SGS, a technique called Eigenvector Continuation.
• Figure 3: Faithfullness of SGS as an approximation for the true impurity model ground state (a) Fraction of the particle-selected Hilbert space $\mathcal{H}^{ps}_N$ needed for $\mathcal{E}_{GS}\leq 10^{-2}$, averaged over 10 random sets of bath parameters. The boundaries of the shaded region correspond to the standard deviation. (b, left) Here, $U=5.337$ for various system sizes computed with impurity ground states found through exact diagonalization (ED, shaded region) and those found using the SGS (dashed magenta line). (b, right) Zoomed-in view of small peaks in the full spectrum (regions on the left panel, boxed in grey).
• Figure 4: Self-consistent results for a single-impurity model with $\mathbf{N_B=3}$. (left) The density of states (DOS) of the lattice model is computed using the self-consistently determined impurity self-energy found through DMFT using impurity ground states found with exact diagonalization (ED, shaded) and with SGS (magenta dashed line). (right) The self-consistent quasiparticle weight described by \ref{['eq: quasiparticle']} for ED (black line) and the SGS (dashed magenta line with diamonds).
• Figure 5: Summary of partial compression. (a) Typically, the circuit structure of state preparation and time evolution via Trotter decomposition is done like the topmost circuit. Our compression techniques involve first (b) simplifying the controlled unitary to bring the state preparation onto the leftmost side and (c) partially compressing the Trotter evolution. (d) Finally, the state prep of $\ket{\phi_i},\ket{\phi_j}$ is compressed further, and (e) some of the Trotter evolution is absorbed into state preparation.