---
title: 'QAVG: QPE Averaged over Variable Grids'
url: https://www.emergentmind.com/topics/qpe-averaged-over-variable-grids-qavg
type: topic
---

# QAVG: QPE Averaged over Variable Grids

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 [2605.29674]. 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 [2605.29681]. 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 $\Delta E = 1/t_0$ [2605.29674]. 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 [2605.29674].

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 [2605.29674]. 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 [2605.29681].

In the DMFT setting, the target quantity is
$$
G_{mm'}(z)=G_{mm'}^{(e)}(z)+G_{mm'}^{(h)}(z),
$$
with
$$
G_{mm'}^{(\xi)}(z)=\frac{1}{\mathcal{Z}}\sum_{\lambda_0} e^{-\beta E_{\lambda_0}}\,G_{mm'}^{(\xi)}(z|\lambda_0), \qquad \xi\in\{e,h\},
$$
and partial Lehmann contributions built from sums over channels $\lambda_0\to\lambda$ [2605.29681]. The paper emphasizes that thousands of such channels contribute at finite temperature, making direct histogramming of each peak via QPE impractical [2605.29681].

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 $t_0$ and performs QPE $K$ times with different origin shifts $\delta_k$, using
$$
E_{\mathrm{orig}}^{(k)} = E_0 + \delta_k.
$$
A typical choice is
$$
\delta_k = \frac{k}{K t_0}, \qquad k=0,\dots,K-1,
$$
so that the effective grids interleave [2605.29674].

For a single eigenstate of energy $E$, the $k$th shifted QPE kernel is
$$
P_k(j|E)=\left|\frac{1}{N_{\mathrm{val}}}\sum_{m=0}^{N_{\mathrm{val}}-1}\exp\!\left[i2\pi m\frac{(E-E_{\mathrm{orig}}^{(k)})t_0-j}{N_{\mathrm{val}}}\right]\right|^2,
$$
or equivalently, with $\phi = E t_0$,
$$
P_k(j|\phi)=\left|\frac{1}{N_{\mathrm{val}}}\sum_{m=0}^{N_{\mathrm{val}}-1}e^{i2\pi m(\phi-j-\delta_k t_0)/N_{\mathrm{val}}}\right|^2
\right|^2,
$$
as summarized in the CO/$\chi$-Fe$_5$C$_2$ work [2605.29674].

The fitting problem is posed through per-shift histogram distances. If $f_{k,j}$ denotes the normalized measured histogram and $P_{k,j}(\theta)$ a trial model, then a per-shift cost is defined as
$$
C(\theta;\delta_k)=D(f_k,P_k(\theta)),
$$
with, for example,
$$
D(f,P)=\frac{1}{2}\sum_j |f_j-P_j|.
$$
QAVG then averages over all shifts,
$$
C_{\mathrm{avg}}(\theta)=\frac{1}{K}\sum_{k=1}^K C(\theta;\delta_k),
$$
and the paper states that because the oscillatory fringes are out of phase across shifts, they largely cancel in $C_{\mathrm{avg}}$, dramatically suppressing spectral leakage and the proliferation of spurious local minima [2605.29674].

The DMFT formulation adopts the same logic in a more general setting. A standard QPE circuit has two tunable parameters, the time scale $t_0$ and an energy origin $\epsilon_{\mathrm{orig}}$, and QAVG employs $n_{\mathrm{setting}}$ different settings $\bigl(t_0^{(p)},\epsilon_{\mathrm{orig}}^{(p)}\bigr)$, each defining sampling points
$$
\epsilon_j=\epsilon_{\mathrm{orig}}^{(p)}+\frac{j}{t_0^{(p)}}, \qquad j=0,\dots,N_{\mathrm{val}}-1.
$$
The paper states that spectral leakage is averaged out when data from all settings are combined [2605.29681].

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 [2605.29674, 2605.29681].

## 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 $\mathcal{H}|\Psi_\lambda\rangle = E_\lambda|\Psi_\lambda\rangle$ and $\mathcal{Z}=\mathrm{Tr}[e^{-\beta \mathcal{H}}]$. The electron and hole parts are defined as
$$
G_{mm'}^{(e)}(z)=\sum_{\lambda_0,\lambda}\frac{e^{-\beta E_{\lambda_0}}}{\mathcal{Z}}
\frac{\langle\Psi_{\lambda_0}|a_m|\Psi_\lambda\rangle
\langle\Psi_\lambda|a_{m'}^\dagger|\Psi_{\lambda_0}\rangle}
{z-(E_\lambda-E_{\lambda_0})},
$$
$$
G_{mm'}^{(h)}(z)=\sum_{\lambda_0,\lambda}\frac{e^{-\beta E_{\lambda_0}}}{\mathcal{Z}}
\frac{\langle\Psi_{\lambda_0}|a_{m'}^\dagger|\Psi_\lambda\rangle
\langle\Psi_\lambda|a_m|\Psi_{\lambda_0}\rangle}
{z-(E_{\lambda_0}-E_\lambda)},
$$
with corresponding discrete spectral matrices
$$
S_{mm'}^{(\xi)}(\epsilon)=\sum_{\lambda_0,\lambda}
B_{mm'}^{(\lambda_0\to\lambda,\xi)}\,\delta_{E_\lambda-E_{\lambda_0}},\epsilon
$$
[2605.29681].

If the QPE ancillae resolved every excitation energy $\epsilon$, then the Green’s function could be approximated by
$$
G_{mm'}^{(e)}(z)\approx \sum_\epsilon \frac{S_{mm'}^{(e)}(\epsilon)}{z-\epsilon},\qquad
G_{mm'}^{(h)}(z)\approx \sum_\epsilon \frac{S_{mm'}^{(h)}(\epsilon)}{z+\epsilon},
$$
but the paper identifies this as impractical at finite temperature because of the large number of channels [2605.29681].

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 $G$, rather than labeling the contributing channel [2605.29681]. 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 [2605.29681].

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 [2605.29681]. The quantum stage begins by preparing an approximation to the Gibbs state
$$
\rho_{\mathrm{Gibbs}}=\frac{e^{-\beta \mathcal{H}}}{\mathcal{Z}}
$$
on $n_{\mathrm{sorb}}$ system qubits. For each correlated-orbital pair $(m,m')$, including both diagonal and off-diagonal cases, a channel-agnostic excitation circuit $C_{\mathrm{exc}}$ is applied so that the $m\to m'$ excitations are coherently superposed without reading out which channel occurred. This is followed by a QFT-based QPE subcircuit with $n_{\mathrm{qval}}$ ancilla qubits implementing controlled $\exp(-i2\pi \mathcal{H} t_0^{(p)} s/N_{\mathrm{val}})$ gates, after which all ancillae are measured [2605.29681].

For each setting $p$, the protocol collects histograms $f_{\xi j}^{(p,m)}$ for diagonal circuits and $f_{\xi \sigma j}^{(p,mm')}$ for off-diagonal circuits, sorted by excitation type $\xi\in\{e,h\}$ and, in the off-diagonal case, by $\sigma=\pm$ [2605.29681]. The total number of shots is stated as
$$
M_{\mathrm{tot}}=n_{\mathrm{sorb}}M_d+n_{\mathrm{sorb}}(n_{\mathrm{sorb}}-1)M_{od},
$$
scaling as
$$
O\!\left(\frac{n_{\mathrm{sorb}}^2}{\epsilon^2 p_{\mathrm{fail}}}\right)
$$
to ensure accuracy $\epsilon$ with failure probability $p_{\mathrm{fail}}$ [2605.29681].

The reconstruction model introduces trial parameters
$$
\Lambda_\xi=\{\epsilon_{\xi\ell},\Delta E_{\xi\ell},v_{\xi\ell}^{(\nu)}\}.
$$
Here $\epsilon_{\xi\ell}$ are $n_{\mathrm{ch}}$ fictitious center energies, $\Delta E_{\xi\ell}$ are widths in a chosen density of states $\rho_{\xi\ell}(E)$, and the paper specifies that a quadratic DOS is used. The vectors $v_{\xi\ell}^{(\nu)}$ are fictitious transition amplitudes in the natural-orbital basis, obeying
$$
\sum_\ell v_{\xi\ell}^{(\nu)}v_{\xi\ell}^{(\nu')}=\delta_{\nu\nu'}
$$
[2605.29681].

The reconstructed spectral-weight matrices in the original orbital basis are
$$
W_\ell^{(e)}=
U_{\mathrm{nat}}^\dagger
\mathrm{diag}\!\bigl(\sqrt{1-n_{\mathrm{meas},\nu}}\bigr)\,
v_{e\ell}v_{e\ell}^T\,
\mathrm{diag}\!\bigl(\sqrt{1-n_{\mathrm{meas},\nu}}\bigr)\,
U_{\mathrm{nat}},
$$
$$
W_\ell^{(h)}=
U_{\mathrm{nat}}^\dagger
\mathrm{diag}\!\bigl(\sqrt{n_{\mathrm{meas},\nu}}\bigr)\,
v_{h\ell}v_{h\ell}^T\,
\mathrm{diag}\!\bigl(\sqrt{n_{\mathrm{meas},\nu}}\bigr)\,
U_{\mathrm{nat}},
$$
where $n_{\mathrm{meas},\nu}$ are occupancies from the measured one-electron matrix $\gamma_{\mathrm{meas}}$ [2605.29681].

The model measurement probabilities are then expressed by convolution with the ideal QPE lineshape
$$
P_j^{(p)}(E)=\left|
\sum_{j'}\frac{
e^{i2\pi j'((E-\epsilon_{\mathrm{orig}}^{(p)})t_0^{(p)}/N_{\mathrm{val}}-j)}
}{N_{\mathrm{val}}}
\right|^2.
$$
For diagonal circuits,
$$
P_{\xi j}^{(p,m)}(\Lambda_\xi)
=\int dE\,S_{\mathrm{rec},mm}^{(\xi)}(E;\Lambda_\xi)\,P_j^{(p)}(E),
$$
and for off-diagonal circuits,
$$
P_{\xi\sigma j}^{(p,mm')}(\Lambda_\xi)
=\int dE
\left[
\frac{S_{mm}^{(\xi)}(E)+S_{m'm'}^{(\xi)}(E)}{4}
+\sigma\cdot\frac{\mathrm{Re}\!\left(e^{i\pi/4}S_{mm'}^{(\xi)}(E)\right)}{2}
\right]P_j^{(p)}(E)
$$
[2605.29681].

Optimization is performed using a non-uniform $L_1$ distance
$$
D(h,h')=\sum_j \frac{g_j|h_j-h'_j|}{2}, \qquad g_j\propto e^{-\tau_{\mathrm{dec}}j/t_0},
$$
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 $F_e+F_h$. The paper states that this can be minimized, for example, via replica-exchange Monte Carlo, yielding $\Lambda_{\mathrm{opt}}$ and hence the reconstructed Green’s function
$$
\bar G(z)=G_{\mathrm{rec}}^{(e)}(z;\Lambda_{\mathrm{opt},e})+
G_{\mathrm{rec}}^{(h)}(z;\Lambda_{\mathrm{opt},h})
$$
[2605.29681].

The spectroscopy paper presents an analogous but simpler continuous parametrization for the CO/$\chi$-Fe$_5$C$_2$ dimer, with two excited-state energies per sector and an angle $\theta$ encoding orthonormal transition amplitudes via
$$
v_0^{(0)}=\cos\theta,\quad
v_0^{(1)}=\sin\theta,\quad
v_1^{(0)}=-\sin\theta,\quad
v_1^{(1)}=\cos\theta,
$$
and
$$
P_{k,j}(\Lambda)=\sum_{\lambda=0}^1 p_\lambda(\theta)\cdot P_k(j|\epsilon_\lambda)
$$
[2605.29674].

## 5. Error behavior, resources, and hardware realization

The DMFT analysis gives explicit resource and error estimates [2605.29681]. Statistical error is bounded through a triangle-inequality decomposition in which the term $D(f,P_{\mathrm{noisy}})$ scales as $O(1/\sqrt{M})$, implying that achieving accuracy $\epsilon$ requires $M\sim 1/\epsilon^2$ shots per circuit. Summed over all $n_{\mathrm{sorb}}^2$ circuits, this yields
$$
M_{\mathrm{tot}}=O\!\left(\frac{n_{\mathrm{sorb}}^2}{\epsilon^2}\right)
$$
[2605.29681].

The qubit count is given as $n_{\mathrm{sorb}}$ system qubits, $n_{\mathrm{qval}}$ phase-estimation ancillae, and up to 2 excitation ancillae [2605.29681]. Circuit depth is dominated by $n_{\mathrm{qval}}$ controlled Hamiltonian evolutions of length $t_0$, the inverse QFT, and the excitation-operator subcircuit; for a Trotterized Hamiltonian this scales as
$$
O\!\bigl(n_{\mathrm{qval}}\cdot(\#\ \mathrm{Trotter\ steps\ per}\ t_0)\bigr)
$$
[2605.29681]. 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 [2605.29681].

The trapped-ion implementation provides a complementary hardware-level picture [2605.29674]. 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 $U^{2^{k-1}}$, 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 $R_y$ rotations [2605.29674].

For the logical implementation, each logical qubit was encoded in seven physical qubits using a flag-augmented fault-tolerant encoder $U_{\mathrm{enc}}$. Logical Clifford gates $(H,S,\mathrm{CNOT})$ were transversal, and logical $R_z(\theta)$ rotations were implemented via a weight-3 Pauli rotation combined with offline bit-flip correction through lookup correction of measured seven-bit outcomes [2605.29674]. The same paper reports that in the logical circuits approximately $3\%$ of shots were discarded by flagged error detection and another approximately $17\%$ of logical bits were corrected offline, while QAVG still converged to the correct parameters despite circuit depths of approximately 200 gates [2605.29674].

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$_3$, the setup used a DFT+Wannier construction with the PBE functional, a $6\times 6\times 6$ $k$-mesh, and maximally localized Wannier orbitals for the V $t_{2g}$ manifold, with $n_{\mathrm{corr}}=3$ per spin [2605.29681]. The impurity problem employed Kanamori interactions $U=3.44\,\mathrm{eV}$, $U_0=2.49\,\mathrm{eV}$, $J=0.46\,\mathrm{eV}$, 3 bath sites, total $n_{\mathrm{sorb}}=12$ spin-orbitals, temperature $\beta^{-1}=0.025\,\mathrm{eV}$, Boltzmann cutoff $\epsilon_B=10^{-4}$, and $n_{\mathrm{Matsu}}=2047$ Matsubara frequencies [2605.29681]. The QPE settings were $n_{\mathrm{qval}}=7$ ancillae, grid spacing $1/t_0=1/6\,\mathrm{eV}$, and origin shifts $\Delta=0,1/(3t_0),2/(3t_0)$, corresponding to $n_{\mathrm{setting}}=3$ [2605.29681].

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 $n_{\mathrm{ch}}=6$ equally degenerate fictitious electron channels and $n_{\mathrm{ch}}=2$ hole channels, for a total of approximately 100 parameters [2605.29681]. 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 [2605.29681]. In the iterative self-consistent calculation, QAVG used $n_{\mathrm{ch}}=8$ for electrons and $n_{\mathrm{ch}}=4$ 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 $\pm 0.2\,\mathrm{eV}$ due to finite parametrization [2605.29681].

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 $\lambda_0$ or channels during QPE; and accuracy comparable to the exact GF was achieved with modest $n_{\mathrm{qval}}$ and $n_{\mathrm{setting}}$ [2605.29681].

In the CO adsorbate study, the true eigenvalues in the electron sector were $\epsilon_0^{(e)}=-0.7256\,\mathrm{eV}$ and $\epsilon_1^{(e)}=-0.1554\,\mathrm{eV}$, while in the hole sector they were $\epsilon_0^{(h)}=-1.2180\,\mathrm{eV}$ and $\epsilon_1^{(h)}=-0.4690\,\mathrm{eV}$, with nominal QPE resolution $\Delta E = 1/t_0 = 0.2\,\mathrm{eV}$ [2605.29674]. 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 $\Delta E=0.2\,\mathrm{eV}$, all QAVG reconstructions recovered the excitation energies within approximately $0.01\,\mathrm{eV}$ of the truth [2605.29674]. 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 [2605.29674].

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 $K$ 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 [2605.29674]. For larger many-body spectra, the workflow requires selecting a manageable low-resolution grid, choosing shifts that cover $[0,\Delta E)$, collecting histograms for each shift, positing a continuous ansatz for transition amplitudes and eigenvalues, and minimizing the averaged cost [2605.29674].

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 [2605.29674]. 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 [2605.29681].

Source: https://www.emergentmind.com/topics/qpe-averaged-over-variable-grids-qavg