Observable-adapted, inversion-free quantum estimation of nuclear electroweak cross sections

The literature now contains several important pieces: Chebyshev reconstruction with rigorous finite-resolution bounds, Fourier moment computation on IBM hardware with mitigation, a 2026 O-19 quantum-classical response calculation with realistic internucleon interactions, and a 2026 proof of concept showing that some flux-averaged neutrino cross sections can be obtained from Euclidean-response moments without reconstructing the full response.

What still appears open, as of the June 18, 2026 targeted scan, is the complete combination: a realistic nuclear Hamiltonian, a physical electroweak system-probe operator, direct computation of the experimentally weighted observable, a certified algorithmic error budget, and a resource comparison against reconstruct-then-integrate baselines.

That is a sharper gap than simply finding a nucleus that has not been simulated yet. The scientific object is the whole estimator and its uncertainty contract.

For a ground state |Psi_0>, Hamiltonian H, and physical excitation operator O(q), the response function is usually written as an energy-resolved spectral object. Traditional workflows reconstruct that function and then integrate it against a kinematic or flux weight.

The observable needed by an experiment is often narrower: a known weighted integral over energy transfer. By the spectral theorem, that weighted integral can be written as an expectation value of a function of the Hamiltonian.

After rescaling the relevant Hamiltonian spectrum to [-1, 1], the experimental weight can be approximated by a Chebyshev, Fourier, piecewise-polynomial, or rational representation. A first implementation can start with Chebyshev polynomials because the response-function literature already has strong error-control tools.

The physical source state is prepared from the probed ground state. The quantum device estimates Hamiltonian moments in that source state. The classical side computes the weight coefficients and combines them into the cross-section estimator.

The identity above is useful, but it is not enough. The algorithmic contribution should be the optimization loop that chooses polynomial degree, coefficients, and measurement allocation under a target observable-level error budget.

This turns the problem from response reconstruction into cost-aware estimator design. The polynomial is judged by the cross section it produces, not by a generic pointwise fit to a response curve.

A realistic electroweak operator contains one-body and two-body pieces. For the first paper, the responsible scope is a physical one-body charge, longitudinal, or E1 operator. Two-body current implementation can be analyzed as a scaling extension rather than forced into the first benchmark.

Because the operator is generally nonunitary, the source state can be prepared through LCU or block-encoding. The metrics to report should include the LCU normalization, post-selection success probability, ancilla count, two-qubit or logical T-gate cost, and leakage from the intended symmetry sector.

The minimum viable paper should not chase fault-tolerant advantage. It should prove that the estimator produces physical, normalized, uncertainty-calibrated results and show where it is cheaper or not cheaper than reconstruction-first methods.

A sensible ladder starts with synthetic spectra and small Hermitian matrices, then moves to deuteron E1 photodisintegration or longitudinal response, then to H-3, He-3, or He-4 with a physical one-body current.

The article should not hide every error source inside a single bar. Algorithmic uncertainty can be certified or calibrated at the observable level. Physics-model uncertainty should be treated through EFT order, basis truncation, interaction ensembles, current operators, and low-energy constants.

The strongest reporting format is two-layered: one interval for algorithmic confidence and one credible band for physics modeling.

The direct estimator will not win everywhere. If many different W functions must be queried after the fact, a reconstructed response can amortize its cost. The useful output is therefore a crossover map: which weights, resolutions, and query counts make direct estimation attractive.

The minimum success criteria should be predetermined: a physical one-body probe, deuteron benchmark, noiseless total algorithmic error near or below 2 percent, 95 percent intervals with credible coverage under noisy simulations, and reproducible code plus small Hamiltonian matrices.