QAVG: QPE Averaged over Variable Grids

• QAVG is a quantum-classical reconstruction technique that averages multiple low-resolution QPE runs to infer spectral features with precision beyond the nominal resolution.
• It uses vernier-type energy shifts and continuous spectral parametrization to suppress spectral leakage and manage excitation channel ambiguities.
• The method applies to both spectroscopy and finite-temperature DMFT, enabling accurate Green’s function estimation with modest quantum resources and optimized model fitting.

QPE Averaged over Variable Grids (QAVG) is a quantum-classical procedure for spectral reconstruction from low-resolution quantum phase estimation (QPE) data. In the formulation introduced for spectroscopy, it is a vernier-type approach that combines multiple QPE runs with different energy-origin shifts and a physically motivated continuous parametrization of the spectrum, so that spectral features can be inferred with deviations much smaller than the nominal QPE resolution (Kosugi et al., 28 May 2026). In the formulation developed for finite-temperature dynamical mean-field theory (DMFT), QAVG is a channel-agnostic method for reconstructing the one-particle Green’s function (GF) of an interacting many-electron system from modified QPE histograms, without resolving which excitation channel or initial thermal state contributed to each shot (Kosugi et al., 28 May 2026). Across both settings, the central idea is to average information over variable QPE grids and fit compact trial parameters to all observed histograms simultaneously.

1. Conceptual basis and motivation

Standard QPE implements controlled powers of a unitary $U=e^{-i 2\pi H t_0/N_{\mathrm{val}}}$ using an ancilla register of size $N_{\mathrm{val}}=2^n$. A single shot returns an integer $j\in\{0,\dots,N_{\mathrm{val}}-1\}$ with most likely value $j \simeq (E t_0)\bmod N_{\mathrm{val}}$, so the grid spacing in energy is $\Delta E = 1/t_0$ (Kosugi et al., 28 May 2026). The same paper states that halving $\Delta E$ requires doubling $t_0$ (or $n$), which requires roughly twice as many applications of $U$, and on noisy hardware this increase in circuit depth and ancilla count incurs large errors (Kosugi et al., 28 May 2026).

QAVG was introduced to address two related difficulties. The first is the coarse-grid limitation of practical QPE, where small ancilla registers lead to low spectral resolution and “spectral leakage,” producing oscillatory kernels and spurious local minima in fitting-based spectral extraction (Kosugi et al., 28 May 2026). The second arises in finite-temperature many-body Green’s-function estimation, where the Lehmann representation contains exponentially many excitation channels and where, for each shot, one does not know which initial state $\lambda_0$ or which operator channel was invoked (Kosugi et al., 28 May 2026).

In the DMFT setting, the target quantity is

$N_{\mathrm{val}}=2^n$0

with

$N_{\mathrm{val}}=2^n$1

and partial Lehmann contributions built from sums over channels $N_{\mathrm{val}}=2^n$2 (Kosugi et al., 28 May 2026). The paper emphasizes that thousands of such channels contribute at finite temperature, making direct histogramming of each peak via QPE impractical (Kosugi et al., 28 May 2026).

This suggests that QAVG is best understood not as a replacement for QPE, but as a reconstruction layer that compensates for limited resolution and incomplete channel information by exploiting structured redundancy across several low-resolution measurements.

2. Variable grids and averaged cost landscapes

The defining operation in QAVG is to repeat low-resolution QPE over multiple grids rather than to attempt a single high-resolution run. In the spectroscopy formulation, one fixes $N_{\mathrm{val}}=2^n$3 and performs QPE $N_{\mathrm{val}}=2^n$4 times with different origin shifts $N_{\mathrm{val}}=2^n$5, using

$N_{\mathrm{val}}=2^n$6

A typical choice is

$N_{\mathrm{val}}=2^n$7

so that the effective grids interleave (Kosugi et al., 28 May 2026).

For a single eigenstate of energy $N_{\mathrm{val}}=2^n$8, the $N_{\mathrm{val}}=2^n$9th shifted QPE kernel is

$j\in\{0,\dots,N_{\mathrm{val}}-1\}$0

or equivalently, with $j\in\{0,\dots,N_{\mathrm{val}}-1\}$1,

$j\in\{0,\dots,N_{\mathrm{val}}-1\}$2

as summarized in the CO/$j\in\{0,\dots,N_{\mathrm{val}}-1\}$3-Fe$j\in\{0,\dots,N_{\mathrm{val}}-1\}$4C$j\in\{0,\dots,N_{\mathrm{val}}-1\}$5 work (Kosugi et al., 28 May 2026).

The fitting problem is posed through per-shift histogram distances. If $j\in\{0,\dots,N_{\mathrm{val}}-1\}$6 denotes the normalized measured histogram and $j\in\{0,\dots,N_{\mathrm{val}}-1\}$7 a trial model, then a per-shift cost is defined as

$j\in\{0,\dots,N_{\mathrm{val}}-1\}$8

with, for example,

$j\in\{0,\dots,N_{\mathrm{val}}-1\}$9

QAVG then averages over all shifts,

$j \simeq (E t_0)\bmod N_{\mathrm{val}}$0

and the paper states that because the oscillatory fringes are out of phase across shifts, they largely cancel in $j \simeq (E t_0)\bmod N_{\mathrm{val}}$1, dramatically suppressing spectral leakage and the proliferation of spurious local minima (Kosugi et al., 28 May 2026).

The DMFT formulation adopts the same logic in a more general setting. A standard QPE circuit has two tunable parameters, the time scale $j \simeq (E t_0)\bmod N_{\mathrm{val}}$2 and an energy origin $j \simeq (E t_0)\bmod N_{\mathrm{val}}$3, and QAVG employs $j \simeq (E t_0)\bmod N_{\mathrm{val}}$4 different settings $j \simeq (E t_0)\bmod N_{\mathrm{val}}$5, each defining sampling points

$j \simeq (E t_0)\bmod N_{\mathrm{val}}$6

The paper states that spectral leakage is averaged out when data from all settings are combined (Kosugi et al., 28 May 2026).

A common misconception is that the method increases resolution by hardware-level refinement of the QPE grid. The formulation in both papers instead attributes the gain to joint inference across multiple low-resolution grids, coupled to a continuous model of the underlying spectrum (Kosugi et al., 28 May 2026, Kosugi et al., 28 May 2026).

3. Finite-temperature Green’s functions and the channel-agnostic formulation

The finite-temperature DMFT variant of QAVG is specialized to the one-particle Green’s function in Lehmann form. Let $j \simeq (E t_0)\bmod N_{\mathrm{val}}$7 and $j \simeq (E t_0)\bmod N_{\mathrm{val}}$8. The electron and hole parts are defined as

$j \simeq (E t_0)\bmod N_{\mathrm{val}}$9

$\Delta E = 1/t_0$0

with corresponding discrete spectral matrices

$\Delta E = 1/t_0$1

(Kosugi et al., 28 May 2026).

If the QPE ancillae resolved every excitation energy $\Delta E = 1/t_0$2, then the Green’s function could be approximated by

$\Delta E = 1/t_0$3

but the paper identifies this as impractical at finite temperature because of the large number of channels (Kosugi et al., 28 May 2026).

The channel-agnostic aspect of QAVG addresses precisely that bottleneck. In the quantum part, modified QPE circuits collect only the total excitation-energy histogram for each component of $\Delta E = 1/t_0$4, rather than labeling the contributing channel (Kosugi et al., 28 May 2026). In the classical part, the method fits a small set of trial parameters—“fictitious channels” plus smooth broadening—to reproduce all observed histograms across multiple QPE settings (Kosugi et al., 28 May 2026).

This formulation is significant because it separates what must be measured coherently from what can be inferred statistically. The data retained from QPE are aggregate histograms, while channel identity is absorbed into the fitted model. A plausible implication is that this avoids the need for mid-circuit channel labeling or direct enumeration of thermally populated initial states.

4. Reconstruction model, sampling protocol, and optimization

The DMFT paper gives an explicit step-by-step construction of QAVG (Kosugi et al., 28 May 2026). The quantum stage begins by preparing an approximation to the Gibbs state

$\Delta E = 1/t_0$5

on $\Delta E = 1/t_0$6 system qubits. For each correlated-orbital pair $\Delta E = 1/t_0$7, including both diagonal and off-diagonal cases, a channel-agnostic excitation circuit $\Delta E = 1/t_0$8 is applied so that the $\Delta E = 1/t_0$9 excitations are coherently superposed without reading out which channel occurred. This is followed by a QFT-based QPE subcircuit with $\Delta E$0 ancilla qubits implementing controlled $\Delta E$1 gates, after which all ancillae are measured (Kosugi et al., 28 May 2026).

For each setting $\Delta E$2, the protocol collects histograms $\Delta E$3 for diagonal circuits and $\Delta E$4 for off-diagonal circuits, sorted by excitation type $\Delta E$5 and, in the off-diagonal case, by $\Delta E$6 (Kosugi et al., 28 May 2026). The total number of shots is stated as

$\Delta E$7

scaling as

$\Delta E$8

to ensure accuracy $\Delta E$9 with failure probability $t_0$0 (Kosugi et al., 28 May 2026).

The reconstruction model introduces trial parameters

$t_0$1

Here $t_0$2 are $t_0$3 fictitious center energies, $t_0$4 are widths in a chosen density of states $t_0$5, and the paper specifies that a quadratic DOS is used. The vectors $t_0$6 are fictitious transition amplitudes in the natural-orbital basis, obeying

$t_0$7

(Kosugi et al., 28 May 2026).

The reconstructed spectral-weight matrices in the original orbital basis are

$t_0$8

$t_0$9

where $n$0 are occupancies from the measured one-electron matrix $n$1 (Kosugi et al., 28 May 2026).

The model measurement probabilities are then expressed by convolution with the ideal QPE lineshape

$n$2

For diagonal circuits,

$n$3

and for off-diagonal circuits,

$n$4

(Kosugi et al., 28 May 2026).

Optimization is performed using a non-uniform $n$5 distance

$n$6

which emphasizes low-energy features. The partial costs for diagonal and off-diagonal components are averaged over settings, combined with weights proportional to observed shot counts, and then summed into a total cost $n$7. The paper states that this can be minimized, for example, via replica-exchange Monte Carlo, yielding $n$8 and hence the reconstructed Green’s function

$n$9

(Kosugi et al., 28 May 2026).

The spectroscopy paper presents an analogous but simpler continuous parametrization for the CO/$U$0-Fe$U$1C$U$2 dimer, with two excited-state energies per sector and an angle $U$3 encoding orthonormal transition amplitudes via

$U$4

and

$U$5

(Kosugi et al., 28 May 2026).

5. Error behavior, resources, and hardware realization

The DMFT analysis gives explicit resource and error estimates (Kosugi et al., 28 May 2026). Statistical error is bounded through a triangle-inequality decomposition in which the term $U$6 scales as $U$7, implying that achieving accuracy $U$8 requires $U$9 shots per circuit. Summed over all $\lambda_0$0 circuits, this yields

$\lambda_0$1

(Kosugi et al., 28 May 2026).

The qubit count is given as $\lambda_0$2 system qubits, $\lambda_0$3 phase-estimation ancillae, and up to 2 excitation ancillae (Kosugi et al., 28 May 2026). Circuit depth is dominated by $\lambda_0$4 controlled Hamiltonian evolutions of length $\lambda_0$5, the inverse QFT, and the excitation-operator subcircuit; for a Trotterized Hamiltonian this scales as

$\lambda_0$6

(Kosugi et al., 28 May 2026). The paper also states that parametrization error vanishes as the number of fictitious channels approaches the number of true channels and the DOS widths approach zero, while optimization error can be driven to zero with sufficient Monte Carlo sampling (Kosugi et al., 28 May 2026).

The trapped-ion implementation provides a complementary hardware-level picture (Kosugi et al., 28 May 2026). On Quantinuum H2-2, the experiments used both physical QPE circuits and logical QPE circuits encoded in the Steane code with offline bit-flip correction. The physical realization included a four-qubit layout for “3-ancilla QPE,” where each ancilla controls $\lambda_0$7, and “1-ancilla sequential QPE,” using mid-circuit measurement and feedforward to reuse a single ancilla for three bits. Excitations were prepared by single-qubit $\lambda_0$8 rotations (Kosugi et al., 28 May 2026).

For the logical implementation, each logical qubit was encoded in seven physical qubits using a flag-augmented fault-tolerant encoder $\lambda_0$9. Logical Clifford gates $N_{\mathrm{val}}=2^n$00 were transversal, and logical $N_{\mathrm{val}}=2^n$01 rotations were implemented via a weight-3 Pauli rotation combined with offline bit-flip correction through lookup correction of measured seven-bit outcomes (Kosugi et al., 28 May 2026). The same paper reports that in the logical circuits approximately $N_{\mathrm{val}}=2^n$02 of shots were discarded by flagged error detection and another approximately $N_{\mathrm{val}}=2^n$03 of logical bits were corrected offline, while QAVG still converged to the correct parameters despite circuit depths of approximately 200 gates (Kosugi et al., 28 May 2026).

A plausible implication is that QAVG’s main hardware advantage is not the elimination of noise, but the reduction of sensitivity to coarse discretization and to leakage-induced optimization pathologies.

6. Demonstrations, performance characteristics, and limitations

Two numerical and experimental case studies anchor the current definition of QAVG. In the DMFT application to SrVO$N_{\mathrm{val}}=2^n$04, the setup used a DFT+Wannier construction with the PBE functional, a $N_{\mathrm{val}}=2^n$05 $N_{\mathrm{val}}=2^n$06-mesh, and maximally localized Wannier orbitals for the V $N_{\mathrm{val}}=2^n$07 manifold, with $N_{\mathrm{val}}=2^n$08 per spin (Kosugi et al., 28 May 2026). The impurity problem employed Kanamori interactions $N_{\mathrm{val}}=2^n$09, $N_{\mathrm{val}}=2^n$10, $N_{\mathrm{val}}=2^n$11, 3 bath sites, total $N_{\mathrm{val}}=2^n$12 spin-orbitals, temperature $N_{\mathrm{val}}=2^n$13, Boltzmann cutoff $N_{\mathrm{val}}=2^n$14, and $N_{\mathrm{val}}=2^n$15 Matsubara frequencies (Kosugi et al., 28 May 2026). The QPE settings were $N_{\mathrm{val}}=2^n$16 ancillae, grid spacing $N_{\mathrm{val}}=2^n$17, and origin shifts $N_{\mathrm{val}}=2^n$18, corresponding to $N_{\mathrm{val}}=2^n$19 (Kosugi et al., 28 May 2026).

At a one-shot reconstruction stage corresponding to iteration 10, the study used exact probability distributions without shot noise to focus on fit quality, with $N_{\mathrm{val}}=2^n$20 equally degenerate fictitious electron channels and $N_{\mathrm{val}}=2^n$21 hole channels, for a total of approximately 100 parameters (Kosugi et al., 28 May 2026). The reported result was that the QAVG-DAS reproduced the broad lower Hubbard band, quasiparticle peak, and upper Hubbard band seen in exact FCI-DMFT, while Matsubara GF traces and momentum-resolved DOS also agreed closely (Kosugi et al., 28 May 2026). In the iterative self-consistent calculation, QAVG used $N_{\mathrm{val}}=2^n$22 for electrons and $N_{\mathrm{val}}=2^n$23 for holes, each threefold degenerate, corresponding to approximately 174 parameters, and the resulting DOS and momentum-resolved spectra closely matched FCI-DMFT benchmarks, with only small dips around $N_{\mathrm{val}}=2^n$24 due to finite parametrization (Kosugi et al., 28 May 2026).

The paper identifies several performance benefits: thousands of true excitation channels were compressed into fewer than 10 fictitious channels; there was no need to resolve or label individual initial states $N_{\mathrm{val}}=2^n$25 or channels during QPE; and accuracy comparable to the exact GF was achieved with modest $N_{\mathrm{val}}=2^n$26 and $N_{\mathrm{val}}=2^n$27 (Kosugi et al., 28 May 2026).

In the CO adsorbate study, the true eigenvalues in the electron sector were $N_{\mathrm{val}}=2^n$28 and $N_{\mathrm{val}}=2^n$29, while in the hole sector they were $N_{\mathrm{val}}=2^n$30 and $N_{\mathrm{val}}=2^n$31, with nominal QPE resolution $N_{\mathrm{val}}=2^n$32 (Kosugi et al., 28 May 2026). The paper reports representative fitted values for several conditions, including noiseless QPE, physical 3-ancilla QPE, and logical 1-ancilla QPE, and states that despite $N_{\mathrm{val}}=2^n$33, all QAVG reconstructions recovered the excitation energies within approximately $N_{\mathrm{val}}=2^n$34 of the truth (Kosugi et al., 28 May 2026). It also states that direct reconstruction by placing histogram bars or summing narrow Lorentzians fails to resolve closely spaced peaks or yields excessive smoothing, whereas QAVG consistently separates both peaks (Kosugi et al., 28 May 2026).

The main limitations are also explicit. The number of trial parameters grows with the number of resolved peaks, so careful ansatz design is essential; the optimal number of shifts $N_{\mathrm{val}}=2^n$35 is not fixed, since too small a value leaves aliasing and too large a value increases measurement overhead; and extension to dense continua or finite-temperature Fermi functions requires richer interpolation kernels (Kosugi et al., 28 May 2026). For larger many-body spectra, the workflow requires selecting a manageable low-resolution grid, choosing shifts that cover $N_{\mathrm{val}}=2^n$36, collecting histograms for each shift, positing a continuous ansatz for transition amplitudes and eigenvalues, and minimizing the averaged cost (Kosugi et al., 28 May 2026).

Taken together, these results define QAVG as a family of averaged, variable-grid QPE reconstruction methods whose distinguishing features are low-resolution quantum sampling, continuous spectral parametrization, and global optimization over several shifted QPE histograms. In the trapped-ion spectroscopy setting, the method is presented as a route to hyperacuity spectral estimation on near-term and early-fault-tolerant devices (Kosugi et al., 28 May 2026). In the DMFT setting, it is presented as a practical channel-agnostic framework for finite-temperature Green’s-function reconstruction with polynomial quantum resources and a small number of classical fit parameters (Kosugi et al., 28 May 2026).