QAVG-DMFT: Hybrid Quantum-Classical Scheme

• QAVG-DMFT is a quantum–classical hybrid scheme that combines channel-agnostic QPE with classical variable-grid reconstruction to obtain finite-temperature impurity Green’s functions.
• The method bypasses the challenges of analytic continuation by directly reconstructing spectral data from measured QPE histograms, enhancing DMFT self-consistency.
• Validation on systems like SrVO₃ shows that QAVG-DMFT reproduces key spectral features and offers advantages over traditional CT-QMC and VQE-based impurity solvers.

QAVG-DMFT is a quantum–classical hybrid scheme for dynamical mean-field theory at finite temperature in which the impurity one-particle Green’s function is reconstructed from quantum phase estimation data. In the formulation explicitly named in the literature, the acronym refers to “QPE averaged over variable grids”: modified finite-temperature, channel-agnostic QPE circuits extract excitation-energy differences, while a classical reconstruction layer infers a compact spectral representation by fitting QPE histograms collected under multiple grid settings (Kosugi et al., 28 May 2026). Within DMFT, the reconstructed impurity Green’s function is used to obtain the impurity self-energy and close the self-consistency loop. The term is not used uniformly across the broader DMFT literature: several earlier papers explicitly state that “QAVG-DMFT” is not their terminology and only discuss related GW+DMFT, QSGW+DMFT, or Dual Boson constructions (Loon et al., 2016, Tomczak, 2014, Lee et al., 2016, Held et al., 2011, Tomczak et al., 2017).

1. Definition and terminological scope

In the explicit sense introduced for finite-temperature quantum impurity simulation, QAVG-DMFT combines two components: a quantum subroutine based on modified QPE circuits for the impurity Green’s function, and a classical post-processing stage called QAVG, “QPE averaged over variable grids,” that reconstructs the Green’s function from measured probability distributions (Kosugi et al., 28 May 2026). The scheme is designed for DMFT, where the impurity problem is the computational bottleneck and where direct access to real-frequency spectral information is valuable because analytic continuation from imaginary time is ill-posed and unstable (Kosugi et al., 28 May 2026).

The term is not standardized across all uses of DMFT-related abbreviations. One paper on double occupancy in DMFT and the Dual Boson approach states that “QAVG-DMFT” does not appear in that work and contrasts a principled self-consistent Dual Boson construction with any naive averaging of impurity- and lattice-based estimates (Loon et al., 2016). Several GW+DMFT papers likewise state that they do not use the term, and instead map a possible “QAVG-DMFT” interpretation to quasiparticle-averaged or quasiparticle self-consistent GW+DMFT variants (Tomczak, 2014, Lee et al., 2016, Held et al., 2011, Tomczak et al., 2017). This suggests that the most precise encyclopedia usage is to reserve “QAVG-DMFT” for the channel-agnostic finite-temperature QPE plus variable-grid reconstruction scheme of (Kosugi et al., 28 May 2026), while separately noting the distinct quasiparticle-based reinterpretations proposed elsewhere.

2. Finite-temperature Green’s functions and channel-agnostic QPE

The finite-temperature formulation starts from standard one-particle Green’s functions. For fermionic operators $c_\alpha$ and $c_\beta$, the retarded and Matsubara Green’s functions are written as

$$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$

and

$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$

with

$$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$

The finite-temperature Lehmann representation is the basis for the QPE encoding, since the relevant spectral information is carried by energy differences between many-body eigenstates (Kosugi et al., 28 May 2026).

The central quantum innovation is a channel-agnostic finite-temperature QPE circuit. The data register is prepared in a Gibbs ensemble $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$ or an equivalent purification, and the circuit encodes difference phases $\Delta E = E_\lambda - E_{\lambda_0}$ rather than absolute many-body energies (Kosugi et al., 28 May 2026). The controlled real-time evolution is

$$U_{\mathrm{RTE}}(s) = \exp\!\left(-i \frac{2\pi H t_0 s}{N_{\mathrm{val}}}\right),$$

where $N_{\mathrm{val}} = 2^{n_{q\mathrm{val}}}$ sets the energy-register size. Because the first controlled-evolution block contributes the phase of the initial state and the second contributes the phase of the excited state, the energy register records only their difference (Kosugi et al., 28 May 2026).

The channel-agnostic property means that the protocol does not require identifying per shot whether a measurement arose from a particle-addition or particle-removal process. For diagonal excitations, one ancilla is sufficient; for off-diagonal excitations, two ancillas and linear combinations of operators $a_m$, $c_\beta$0, and their adjoints generate four outcomes labeled by $c_\beta$1 (Kosugi et al., 28 May 2026). The measured QPE probability for a bin $c_\beta$2 is not a delta function but a filtered version of the true spectrum,

$c_\beta$3

which is the QPE filter kernel for grid setting $c_\beta$4 (Kosugi et al., 28 May 2026).

A plausible implication is that the method trades explicit channel resolution and absolute-energy estimation for a representation more directly aligned with the Lehmann structure of the Green’s function. That trade is central to its finite-temperature applicability.

3. QAVG reconstruction over variable grids

QAVG is the classical inference layer that converts QPE histograms into a reconstructed spectral function and hence into impurity Green’s functions (Kosugi et al., 28 May 2026). Rather than relying on a single QPE grid, the method samples multiple grid settings

$c_\beta$5

with different $c_\beta$6, $c_\beta$7, and $c_\beta$8. The stated purpose is to suppress spectral leakage and aliasing tied to any single Dirichlet kernel and to sharpen inference through variable-grid averaging (Kosugi et al., 28 May 2026).

For diagonal spectral components, the modeled probabilities are

$c_\beta$9

and analogous expressions hold for off-diagonal matrix elements (Kosugi et al., 28 May 2026). The reconstructed spectral matrices are parameterized in the natural-orbital basis. The measured one-body density matrix $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$0 is diagonalized to obtain natural-orbital occupancies and a unitary rotation $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$1, after which the electron and hole Green’s functions are modeled as sums over fictitious channels with centers $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$2, widths $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$3, and orthonormal transition vectors $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$4 (Kosugi et al., 28 May 2026). The vectors are constrained by a hyperspherical Householder parametrization, enforcing orthonormality in the natural-orbital index.

The line shape $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$5 is chosen “quadratic” on a finite interval and normalized to unity (Kosugi et al., 28 May 2026). This is explicitly motivated by finite-temperature peak crowding, where thousands of addition and removal channels can contribute in narrow energy windows. The reconstruction therefore groups many thermally active transitions into a smaller number of fictitious broadened channels rather than attempting line-by-line recovery (Kosugi et al., 28 May 2026).

The fit is defined by minimizing a grid-averaged discrepancy between modeled and measured histograms. The basic distance is

$$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$6

and the total per-channel cost is

$$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$7

with the QAVG solution

$$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$8

These weights emphasize low-energy bins and can be chosen from excitation counts or from the measured density matrix $$G^R_{\alpha\beta}(\omega) = -i \int_0^{\infty} dt\, e^{i\omega t} \langle\{ c_\alpha(t), c_\beta^\dagger(0) \}\rangle_\beta,$$9 (Kosugi et al., 28 May 2026).

Once the spectral model is fixed, the Matsubara Green’s function is recovered from

$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$0

This yields a DMFT-compatible impurity Green’s function without analytic continuation from imaginary time (Kosugi et al., 28 May 2026).

4. DMFT embedding and self-consistency structure

Within DMFT, the impurity model is coupled to the lattice through the usual self-consistency equations. For a lattice Hamiltonian $$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$1 and local self-energy $$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$2, the lattice and local Green’s functions are

$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$3

and

$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$4

Projected onto the correlated subspace,

$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$5

For the impurity model,

$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$6

with $$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$7 (Kosugi et al., 28 May 2026).

The high-level QAVG-DMFT loop is therefore:

Step	Operation	Output
1	Build the AIM from current bath parameters	$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$8
2	Prepare a Gibbs-like input and run channel-agnostic QPE on selected circuits and grid settings	Histograms
3	Apply QAVG reconstruction	$$G_{\alpha\beta}(i\omega_n) = \int_{-\infty}^{\infty} d\omega \frac{A_{\alpha\beta}(\omega)}{i\omega_n-\omega},$$9, $$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$0
4	Use Dyson’s equation	$$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$1
5	Update lattice Green’s function, Weiss field, and bath fit	New $$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$2, $$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$3

The explicit algorithm in the paper initializes from DFT, constructs maximally localized Wannier orbitals, chooses a correlated subspace, sets an initial $$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$4, and iterates the impurity reconstruction and bath fitting inside the DMFT loop (Kosugi et al., 28 May 2026). A key point is that the quantum device is used to sample finite-temperature spectral information directly, while the classical side handles the inverse problem and DMFT bath update.

This suggests that QAVG-DMFT is best viewed not as a new lattice approximation, but as a quantum-assisted impurity-solver architecture embedded in standard DMFT self-consistency.

5. Validation on SrVO$$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$5 and relation to other quantum-assisted DMFT solvers

The explicit materials validation is carried out for cubic SrVO$$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$6 (Kosugi et al., 28 May 2026). The Anderson impurity model uses three correlated $$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$7 orbitals per spin and $$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$8 bath sites, giving

$$A_{\alpha\beta}(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R_{\alpha\beta}(\omega).$$9

spin orbitals. The Kanamori parameters are $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$0, $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$1, and $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$2, with temperature $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$3 (Kosugi et al., 28 May 2026). The numerical validation uses exact probabilities convolved with the QPE filter rather than noisy circuit simulations, thereby isolating the reconstruction problem (Kosugi et al., 28 May 2026).

For the QPE settings, the energy register has $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$4, so $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$5, the grid spacing is $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$6, and three origin shifts are used (Kosugi et al., 28 May 2026). In the one-shot reconstruction at DMFT iteration 10, the electron sector uses 6 independent fictitious excitation energies and the hole sector uses 2 (Kosugi et al., 28 May 2026). The reconstructed density of states reproduces the exact-diagonalization benchmark qualitatively, including a central quasiparticle peak and Hubbard bands, while Matsubara $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$7 agrees well except for modest discrepancies at the lowest frequencies (Kosugi et al., 28 May 2026). In iterative QAVG-DMFT, the reconstruction uses up to 8 independent electron energies and 4 hole energies, each threefold degenerate, with 123 and 51 angle parameters for the electron and hole transition systems, respectively (Kosugi et al., 28 May 2026).

A related but distinct quantum-assisted DMFT strategy uses a symmetry-adapted VQE followed by real-time evolution of a discretized Anderson impurity model (Singh et al., 3 Feb 2026). In that approach, for a four-site AIM the relative error in the ground-state energy remains well below $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$8 with $\rho_{\mathrm{Gibbs}}=e^{-\beta H}/Z$9, and the single-particle Green’s function is accurately extracted in intermediate to strong interaction regimes, while the weak interaction regime shows noticeable deviations in low-energy spectral features (Singh et al., 3 Feb 2026). The comparison is instructive: both methods target the impurity Green’s function, but the VQE approach relies on real-time evolution from a ground state, whereas QAVG-DMFT uses finite-temperature QPE and a multi-grid statistical reconstruction (Singh et al., 3 Feb 2026, Kosugi et al., 28 May 2026).

6. Relation to GW+DMFT, QSGW+DMFT, and other nonstandard uses of “QAVG”

Outside the finite-temperature QPE literature, “QAVG-DMFT” is used only as an interpretive label, not as an established scheme name. In one H$\Delta E = E_\lambda - E_{\lambda_0}$0 study, the closest match to such a label is identified as QSGW+DMFT with dynamical double-counting, where a nonlocal quasiparticle Hamiltonian is constructed from an averaged nonlocal quasiparticle renormalization matrix $\Delta E = E_\lambda - E_{\lambda_0}$1 and DMFT supplies the local dynamical self-energy (Lee et al., 2016). That work explicitly calls this mapping “the closest match to what we would call ‘QAVG-DMFT’,” but it also states that the paper itself studies fully self-consistent GW+DMFT, one-shot $\Delta E = E_\lambda - E_{\lambda_0}$2+DMFT, and QSGW+DMFT variants rather than a named QAVG-DMFT formalism (Lee et al., 2016). In its H$\Delta E = E_\lambda - E_{\lambda_0}$3 benchmarks, the dynamical-double-counting variant improves spectra near equilibrium but fails to reach the atomic limit, whereas QSGW+DMFT with static double-counting and GW+DMFT with causal double-counting recover the atomic limit (Lee et al., 2016).

A review and methodological paper on QSGW+DMFT for iron pnictides likewise states that the term “QAVG-DMFT” is not used there, and instead develops a QS GW+DMFT scheme in which nonlocal static exchange/correlation is retained in a quasiparticle Hamiltonian and DMFT supplies local dynamical correlations (Tomczak, 2014). The same terminological clarification appears in broader reviews of GW, GW+DMFT, and D$\Delta E = E_\lambda - E_{\lambda_0}$4A, where a hypothetical “QAVG-DMFT” is mapped to a quasiparticle-averaged static GW self-energy combined with a local DMFT self-energy and projector-based double-counting subtraction (Held et al., 2011, Tomczak et al., 2017).

A different ambiguity arises in finite-dimensional DMFT for two-particle observables such as double occupancy. The self-consistent Dual Boson construction resolves discrepancies between impurity- and lattice-based routes by enforcing

$\Delta E = E_\lambda - E_{\lambda_0}$5

which yields $\Delta E = E_\lambda - E_{\lambda_0}$6 by construction (Loon et al., 2016). That paper explicitly rejects any identification of “QAVG-DMFT” with a principled averaging scheme; if the label were interpreted as averaging inconsistent estimators, the paper characterizes that as ad hoc and contrasts it with a frequency-dependent two-particle self-consistency condition (Loon et al., 2016).

These uses do not define a common formalism. They instead show that the acronym can be informally attached to at least three conceptually distinct ideas: variable-grid QPE reconstruction for DMFT (Kosugi et al., 28 May 2026), quasiparticle-averaged GW+DMFT reinterpretations (Lee et al., 2016, Tomczak et al., 2017), and, negatively, ad hoc averaging procedures that are explicitly disfavored in two-particle DMFT contexts (Loon et al., 2016).

7. Limitations, error channels, and practical significance

The explicit QAVG-DMFT formulation identifies several error sources. Single-grid QPE suffers from Dirichlet-kernel sidelobes, motivating variable-grid averaging (Kosugi et al., 28 May 2026). Finite-temperature spectra contain large numbers of thermally active transitions within narrow energy windows, which motivates grouping them into fictitious broadened channels rather than attempting exact line resolution (Kosugi et al., 28 May 2026). Hardware imperfections, readout errors, and imperfect thermal-state preparation further perturb the measured histograms; the paper separates optimization, modeling, statistical, and hardware errors through a triangle-inequality bound on the distance between ideal and reconstructed distributions (Kosugi et al., 28 May 2026).

The method also assumes the availability of a Gibbs-like input state and moderate-depth controlled time evolutions (Kosugi et al., 28 May 2026). A plausible implication is that the practical frontier is set as much by thermal-state preparation and controlled-evolution quality as by the classical inverse problem. The SrVO$\Delta E = E_\lambda - E_{\lambda_0}$7 demonstration therefore validates the reconstruction concept more directly than a full noisy-hardware implementation (Kosugi et al., 28 May 2026).

In comparison with real-time VQE impurity solvers, QAVG-DMFT avoids long coherent real-time evolution as the primary spectral estimator and directly targets finite-temperature excitation energies through QPE (Singh et al., 3 Feb 2026, Kosugi et al., 28 May 2026). In comparison with CT-QMC-based DMFT, it aims to bypass analytic continuation (Kosugi et al., 28 May 2026). In comparison with ED, it still uses a finite-bath impurity Hamiltonian but distributes the computational burden between quantum sampling and compact spectral modeling (Kosugi et al., 28 May 2026).

Taken in its explicit form, QAVG-DMFT is therefore a finite-temperature quantum-assisted impurity-solver framework for DMFT in which modified QPE produces channel-agnostic spectral data and QAVG converts those data into Green’s functions suitable for self-consistent many-body calculations (Kosugi et al., 28 May 2026). The broader literature shows that the label has been used informally in other contexts, but those usages describe separate quasiparticle or averaging ideas rather than the finite-temperature QPE scheme itself (Tomczak, 2014, Lee et al., 2016, Tomczak et al., 2017).