Channel-agnostic finite-temperature phase estimation averaged over variable grids: reconstruction of Green's function for dynamical mean-field theory

Abstract: For treating correlated electronic systems on quantum computers, we propose a quantum-classical hybrid scheme for dynamical mean-field theory (DMFT). In the quantum part of the scheme, we use modified quantum phase estimation (QPE) circuits suitable for the one-particle Green's function (GF) at a finite temperature so that we can extract spectral amplitudes and the excitation energies without knowing the excitation channel invoked at each measurement. In the classical part of the scheme, we adopt an approach that estimates reasonably the GF based on the data collected from the QPE sampling. We dub the approach the QPE averaged over variable grids (QAVG), that may help one to reconstruct the GF via optimization of trial parameters and modeling the probability distributions for various settings of the QPE circuits. We apply the QAVG-DMFT scheme to SrVO$_3$ to demonstrate its validity via numerical simulations.

• The paper introduces a channel-agnostic QPE method to estimate finite-temperature phases, enabling Green's function reconstruction in DMFT without prior excitation channel knowledge.
• It develops the QAVG classical algorithm to statistically infer Green's functions from ambiguous QPE data, reducing model complexity while preserving spectral accuracy.
• Numerical validation on SrVO₃ demonstrates that the hybrid approach accurately captures spectral features and achieves robust convergence in DMFT iterations.

Quantum-Classical Hybrid Green's Function Reconstruction via Variable-Grid Phase Estimation in DMFT

Introduction

The paper "Channel-agnostic finite-temperature phase estimation averaged over variable grids: reconstruction of Green's function for dynamical mean-field theory" (2605.29681) introduces an innovative quantum-classical hybrid framework that enables the reconstruction of the one-particle Green's function (GF) at finite temperature for dynamical mean-field theory (DMFT) impurity problems. The scheme efficiently leverages a modified quantum phase estimation (QPE) strategy, agnostic to excitation channels, and implements a classical post-processing algorithm—QAVG (QPE Averaged over Variable Grids)—to address the inherent ambiguity in excitation channels in the measurement statistics. The approach is validated through numerical simulations involving the prototypical correlated oxide SrVO$_3$ within a DFT+DMFT context.

DMFT and Quantum Impurity Problem Solution Paradigm

DMFT provides a controlled mapping of a correlated lattice problem onto a self-consistently coupled quantum impurity model. Within DFT+DMFT, this procedure entails extracting a correlated subspace—typically spanned by selected maximally localized Wannier orbitals (MLWOs)—and embedding it into a bath defined via hybridization fitting. Standard impurity solvers (QMC, exact diagonalization) become computational bottlenecks as the bath or subspace size increases, particularly for finite-temperature dynamics.

Recent efforts have explored quantum algorithms as alternative DMFT impurity solvers, employing techniques such as Krylov VQE, cumulant expansions, and quantum EOM. However, these previous approaches often assume knowledge of excitation channels, are tailored to ground state (zero temperature) properties, or require diagonalization at various stages, limiting their scalability or applicability to the finite-temperature regime.

Modified QPE: Channel-Agnostic, Finite-Temperature Green's Function Measurements

The core quantum component consists of a variant of QPE designed for sampling the spectral information requisite for GF reconstruction at finite temperature, without explicit knowledge of which excitation channel—initial, final state, or type (electron/hole)—is encountered in each measurement. To accomplish this, the procedure:

• Prepares the Gibbs (thermal) state of the quantum impurity Hamiltonian as the input register.
• Applies a set of excitation circuits, which, depending on the component (diagonal or off-diagonal), perform the requisite particle additions/removals (electron/hole).
• Implements controlled real-time evolution (RTE) interleaved with the excitation, controlled by ancilla registers encoding binary representations of phase evolution.
• Averages over multiple QPE circuits with variable grid parameters (resolution, offset), thus mitigating spectral leakage artifacts.

Channel-agnosticism is achieved by summing over all possible initial states in the thermal ensemble and all accessible transitions; the QPE outputs ancilla bitstrings corresponding to energy differences without explicit labeling of channel provenance.

QAVG: Statistical Modeling and Classical Parameter Inference

Because the measurement statistics from the modified QPE circuit conflate transitions from many thermally occupied initial states and obscure channel identities, a direct assignment of measured frequencies to GF matrix elements is infeasible. The QAVG technique circumvents this by:

• Defining a parametric model for the Green's function in a natural-orbital basis, where trial parameters collectively represent fictitious excitations, energies, and transition amplitudes. The parametrization leverages hyperspherical Householder construction to enforce orthonormality constraints.
• Modeling the probability distributions of QPE outcomes corresponding to these parameters (spectral matrices), with provision for finite grid effects and experimental resolution.
• Formulating a cost function as a non-uniformly weighted $L_1$ (or other) discrepancy between observed QPE histograms and the modeled probabilities, averaged across all variable QPE grid settings.
• Minimizing this cost over the trial parameters using Metropolis-type Monte Carlo sampling, yielding reconstructed GFs that are statistically consistent with the measured data.

Notably, the number of fictitious (modeled) channels is much smaller than the exponential number of physical channels at finite temperature, enabling compression and practical reconstruction even as the system size grows.

Numerical Validation: SrVO$_3$ DMFT and Green's Function Reconstruction

The authors validate the quantum-classical hybrid protocol by numerical simulation of the entire DMFT loop (excluding explicit quantum hardware simulation—using exact diagonalization for sampling the Gibbs state and computing QPE statistics). The SrVO$_3$ system is treated with a $t_{2g}$-derived impurity problem (three correlated orbitals, three bath sites per spin, 12 spin orbitals total).

Key results include:

• Successful reproduction of FCI-computed Matsubara Green's functions and spectral densities using the QAVG-reconstructed GFs, with strong agreement observed in the overall spectral features and Matsubara traces.
• Demonstration that, even when thousands of real excitation channels contribute (finite temperature), only a small number of fictitious channels in the QAVG ansatz are necessary to capture the essential spectral structure.
• Identification of discrepancies primarily in the low-energy sector, attributable to model mismatches and the limited number of trial channels, but without significant qualitative mischaracterization.
• Iterative QAVG-DMFT application, reconstructing the GF at each DMFT iteration and observing robust convergence behavior and spectral features matching traditional FCI-DMFT solutions, except for minor suppression around $\pm0.2$ eV in the reconstructed momentum-resolved DOS.

Theoretical and Practical Implications

The approach offers several important advancements and technical implications:

• Quantum impurity solvers capable of circumventing exact diagonalization at finite temperature are crucial for scalable, realistic DMFT studies of correlated materials.
• Channel-agnostic QPE allows for Green's function measurement without prior diagonalization or eigenstate preparation, which is critical as systems move beyond a few qubit regime.
• QAVG, by allowing physically plausible statistical inference from ambiguous QPE data, provides a new generic toolkit for classical post-processing in hybrid quantum algorithms.
• The demonstration that the number of effective trial parameters need not scale exponentially with system size suggests immediate applicability as quantum hardware improves, especially in the context of NISQ and early FTQC platforms.

From a technical standpoint, the explicit construction of excitation circuits for both diagonal and off-diagonal GF elements, and the derivation of statistical error bounds for measurement-based estimation, further strengthens the practical viability of the protocol.

Outlook and Future Developments

This framework positions itself as a foundational protocol for hybrid quantum simulation of many-body observables at finite temperature in condensed matter, quantum chemistry, and materials science. Anticipated directions include:

• Implementation on real quantum hardware as soon as quantum Gibbs state preparation and modest QPE depths are practical.
• Generalization to higher-order correlation functions, multiorbital and multichannel DMFT, and embedding in broader ab initio frameworks.
• Investigation of more expressive or physically tailored ansatz classes for QAVG parameterization, and the design of adaptive grid selection strategies for QPE to optimize spectral reconstruction.
• Integration with advanced error mitigation strategies and quantum error-correcting codes as quantum hardware scales [see e.g., [7441]].

Theoretical exploration into optimal statistical inference schemes for similar "indistinguishable channel" quantum measurements may also benefit algorithmic developments inside and outside many-body physics.

Conclusion

The paper delineates an effective quantum-classical hybrid protocol for finite-temperature Green's function estimation in DMFT, combining modified, channel-agnostic QPE sampling circuitry with a flexible, information-theoretic classical post-processing scheme (QAVG). Numerical reconstruction in correlated orbitals of SrVO$_3$ demonstrates that, even in highly degenerate finite-temperature scenarios, the protocol yields physically accurate spectral information with modest classical parameter counts. As quantum hardware matures, this paradigm constitutes a promising step towards practical quantum-enhanced algorithms for correlated electron materials modeling.