Quantum Readout Complexity for Normalized Linear Functionals
Abstract
Quantum numerical algorithms often encode a positive solution q in the amplitudes of a quantum state, while the required scientific output is a normalized linear functional of q. Basis measurements produce probabilities determined by squared solution values, while the functional uses the solution values themselves. We study the complexity of this output-interface problem for Fϕ(q)=∫ϕq dν/∫ q dν and identify a target-dependent susceptibility χϕ that governs the amount of copies of the solution state needed. For a fixed positive reference state and a neighborhood around it, the optimal number of unknown-state copies is Θ (χ0ε−2log(1/β)), where ε is the absolute error tolerance and β is the allowed failure probability. We show that two fixed single-copy measurements attain the same rate. We then analyze how this baseline changes when the interface retains classical information, uses adaptive allocation, supplies coherent access, or restricts the allowed solution family. Therefore χϕ quantifies a central cost of using a solution encoded in an L2-normalized quantum state to estimate an L1-normalized linear observable.
Related
- Linear combination of Schrödingerization for quantum linear systems with optimal matrix-query complexity
- Quantum Complexity of Solving Linear Equations on Higher-Order Networks
- Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians
- Multimodal Visual Sensing of Temperature and Pressure: From Spectroscopic Readout to Multiple Linear Regression-Enhanced RGB Analysis
- Multiple Linear Regression-Enhanced RGB-based Luminescence Thermometry for High-sensitivity Thermal Readout
BibTeX
@misc{vencenogueira2026quantum,
title = {{Quantum Readout Complexity for Normalized Linear Functionals}},
author = {Roi Vence-Nogueira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08890},
eprint = {2610.08890},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08890},
}
RIS
TY - UNPB
TI - Quantum Readout Complexity for Normalized Linear Functionals
AU - Roi Vence-Nogueira
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08890
UR - https://arxiv.org/abs/2610.08890
AB - Quantum numerical algorithms often encode a positive solution q in the amplitudes of a quantum state, while the required scientific output is a normalized linear functional of q. Basis measurements produce probabilities determined by squared solution values, while the functional uses the solution values themselves. We study the complexity of this output-interface problem for F_ϕ(q)=∫ϕq dν/∫ q dν and identify a target-dependent susceptibility χ_ϕ that governs the amount of copies of the solution state needed. For a fixed positive reference state and a neighborhood around it, the optimal number of unknown-state copies is Θ (χ_0ε^−2log(1/β)), where ε is the absolute error tolerance and β is the allowed failure probability. We show that two fixed single-copy measurements attain the same rate. We then analyze how this baseline changes when the interface retains classical information, uses adaptive allocation, supplies coherent access, or restricts the allowed solution family. Therefore χ_ϕ quantifies a central cost of using a solution encoded in an L^2-normalized quantum state to estimate an L^1-normalized linear observable.
ER -