发表机构
University of Copenhagen（哥本哈根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出基于相对解码的同态多项式转导框架，将量子干涉测量推广至更一般代数同态，证明相对距离条件可保留内积，并在非线性多项式交集任务中实现超越经典启发式的量子优势。
AI 中文摘要
解码量子干涉测量（DQI）及其哈密顿量扩展（HDQI）利用傅里叶变换和相干解码制备振幅为目标低次多项式的量子态。我们将此方法重新表述为通过代数同态的量子态转导。我们将源哈密顿量 $H_A$ 中一个次数为 $D$ 的多项式的可高效制备的算符态，转移到目标哈密顿量 $H_B=\pi(H_A)$ 的对应多项式态，其中 $\pi$ 是有限维源与目标 $C^*$-代数之间的幺正 $*$-同态。该转导使用算符傅里叶变换和相对解码来恢复源算符，而非单个项选择标签。相对距离 $d_{\mathrm{rel}}$ 是源与目标迹首次不一致的次数。我们证明 $2D<d_{\mathrm{rel}}$ 等价于保留所有次数不超过 $D$ 的多项式之间的内积。此外，$d_{\mathrm{rel}}\ge d_{\mathrm{ord}}$，其中 $d_{\mathrm{ord}}$ 是 (H)DQI 中使用的普通距离，我们给出了 $d_{\mathrm{ord}}=O(1)$ 但 $d_{\mathrm{rel}}=\Theta(n)$ 且具有高效解码器的族。我们的框架将复杂的导引态制备替换为更模块化的源多项式态制备任务。DQI 和 HDQI 作为无关系特例出现。该框架可处理非均匀系数和非对易相互作用，并扩展到费米子、量子比特（qudit）和玻色子系统，应用于近似优化和吉布斯采样。作为优势的证据，对于非线性成对变体的最优多项式交集，其中常数 $d_{\mathrm{ord}}$ 限制了 DQI 式制备，相对解码产生的理想量子得分为 0.643，而最佳测试经典启发式为 0.606，差距超过 3 个百分点。
英文摘要
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 $d_{\mathrm{rel}}$ is the first degree at which source and target traces disagree. We prove that $2D<d_{\mathrm{rel}}$ is equivalent to preserving all inner products between degree-$D$ polynomials. Moreover, $d_{\mathrm{rel}}\ge d_{\mathrm{ord}}$, where $d_{\mathrm{ord}}$ is the ordinary distance used in (H)DQI, and we give families with $d_{\mathrm{ord}}=O(1)$ but $d_{\mathrm{rel}}=Θ(n)$ 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 $d_{\mathrm{ord}}$ 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.
Comments84 pages, 1 figure, comments are welcome!
