Task-Directed Spectral Design for Local Quantum Tomography
Abstract
Many tomography tasks require only a low-dimensional function of the state. We study local pure-state tomography under projective-fidelity access when a parametric model, a credible center, and a differentiable downstream task are known before acquisition. The pullback Fubini--Study metric and the task metric define coordinate-invariant generalized eigendirections, which we realize as a fixed library of geodesic fidelity probes. We prove two optimality results: the probe aligned with a task coordinate is its unique variance-minimizing location up to orientation, and, when prior and measurement information are diagonal in this basis, a diminishing-return greedy rule exactly minimizes the integer-budget local task-variance surrogate over the spectral library. Under common per-shot information and the stated large-budget condition, the absolute and relative gaps to arbitrary probe directions are respectively O(B−3) and O(B−2). A closed-form local estimator is followed by a task-metric localization projection that cannot increase quadratic task loss when localization holds. In six-qubit numerical experiments, the Schmidt-spectrum design activates only one or two settings and has lower mean task error than the post hoc best adaptive results across all tested dimensions and copy budgets. For full-state tomography, it likewise has lower mean infidelity throughout the tested local regime. The fixed batch uses no feedback loops and far fewer settings; under the illustrative latency model, its acquisition cost is roughly three to four orders of magnitude lower than that of the adaptive methods.
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BibTeX
@misc{hay2026taskdirected,
title = {{Task-Directed Spectral Design for Local Quantum Tomography}},
author = {Guy Hay and Nir Sharon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33723},
eprint = {2610.33723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33723},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Task-Directed Spectral Design for Local Quantum Tomography
AU - Guy Hay
AU - Nir Sharon
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33723
UR - https://arxiv.org/abs/2610.33723
AB - Many tomography tasks require only a low-dimensional function of the state. We study local pure-state tomography under projective-fidelity access when a parametric model, a credible center, and a differentiable downstream task are known before acquisition. The pullback Fubini--Study metric and the task metric define coordinate-invariant generalized eigendirections, which we realize as a fixed library of geodesic fidelity probes. We prove two optimality results: the probe aligned with a task coordinate is its unique variance-minimizing location up to orientation, and, when prior and measurement information are diagonal in this basis, a diminishing-return greedy rule exactly minimizes the integer-budget local task-variance surrogate over the spectral library. Under common per-shot information and the stated large-budget condition, the absolute and relative gaps to arbitrary probe directions are respectively O(B^−3) and O(B^−2). A closed-form local estimator is followed by a task-metric localization projection that cannot increase quadratic task loss when localization holds. In six-qubit numerical experiments, the Schmidt-spectrum design activates only one or two settings and has lower mean task error than the post hoc best adaptive results across all tested dimensions and copy budgets. For full-state tomography, it likewise has lower mean infidelity throughout the tested local regime. The fixed batch uses no feedback loops and far fewer settings; under the illustrative latency model, its acquisition cost is roughly three to four orders of magnitude lower than that of the adaptive methods.
ER -