From Simple Sources to Quantum Advantage: Homomorphic Polynomial Transduction via Relative Decoding
Abstract
Decoded quantum interferometry (DQI) and its Hamiltonian extension (HDQI) prepare states whose amplitudes are low-degree polynomials of an objective, using Fourier transforms and coherent decoding. We recast this approach as quantum state transduction through algebra homomorphisms. We transfer an efficiently preparable operator state of a degree-D polynomial in a source Hamiltonian HA to the corresponding polynomial state of a target Hamiltonian HB=π(HA), where π is a unital *-homomorphism between the finite-dimensional source and target C*-algebras. The transduction uses operator Fourier transforms and relative decoding to recover source operators rather than individual term-selection labels. The relative distance drel is the first degree at which source and target traces disagree. We prove that 2D<drel is equivalent to preserving all inner products between degree-D polynomials. Moreover, drel≥ dord, where dord is the ordinary distance used in (H)DQI, and we give families with dord=O(1) but drel=Θ(n) and efficient decoders. Our framework replaces the complicated pilot state preparation by the more modular task of preparing a source polynomial state. DQI and HDQI arise as relation-free special cases. The framework accommodates nonuniform coefficients and noncommuting interactions and extends to fermionic, qudit, and bosonic systems, with applications to approximate optimization and Gibbs sampling. As evidence of advantages, for a nonlinear pairwise variant of optimal polynomial intersection where constant dord limits DQI-style preparations, relative decoding yields an ideal quantum score of 0.643 versus 0.606 for the best tested classical heuristic, a gap exceeding 3 percentage points.
Related
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BibTeX
@misc{shang2026from,
title = {{From Simple Sources to Quantum Advantage: Homomorphic Polynomial Transduction via Relative Decoding}},
author = {Zhong-Xia Shang and Daniel Stilck França},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35101},
eprint = {2609.35101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35101},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Simple Sources to Quantum Advantage: Homomorphic Polynomial Transduction via Relative Decoding
AU - Zhong-Xia Shang
AU - Daniel Stilck França
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35101
UR - https://arxiv.org/abs/2609.35101
AB - Decoded quantum interferometry (DQI) and its Hamiltonian extension (HDQI) prepare states whose amplitudes are low-degree polynomials of an objective, using Fourier transforms and coherent decoding. We recast this approach as quantum state transduction through algebra homomorphisms. We transfer an efficiently preparable operator state of a degree-D polynomial in a source Hamiltonian H_A to the corresponding polynomial state of a target Hamiltonian H_B=π(H_A), where π is a unital *-homomorphism between the finite-dimensional source and target C^*-algebras. The transduction uses operator Fourier transforms and relative decoding to recover source operators rather than individual term-selection labels. The relative distance d_rel is the first degree at which source and target traces disagree. We prove that 2D<d_rel is equivalent to preserving all inner products between degree-D polynomials. Moreover, d_rel≥ d_ord, where d_ord is the ordinary distance used in (H)DQI, and we give families with d_ord=O(1) but d_rel=Θ(n) and efficient decoders. Our framework replaces the complicated pilot state preparation by the more modular task of preparing a source polynomial state. DQI and HDQI arise as relation-free special cases. The framework accommodates nonuniform coefficients and noncommuting interactions and extends to fermionic, qudit, and bosonic systems, with applications to approximate optimization and Gibbs sampling. As evidence of advantages, for a nonlinear pairwise variant of optimal polynomial intersection where constant d_ord limits DQI-style preparations, relative decoding yields an ideal quantum score of 0.643 versus 0.606 for the best tested classical heuristic, a gap exceeding 3 percentage points.
ER -