发表机构
Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于相对解码的量子吉布斯态制备方法，将快速混合源的制备保证转移至更广泛目标哈密顿量，利用相对距离实现高效制备，并给出新类可制备TFD示例。
AI 中文摘要
可证明的量子吉布斯态制备保证，如快速混合或带隙相干制备路径，目前仅对受限的哈密顿量族已知，且通常需要为每个新族重新推导。基于同态多项式转导（这是解码量子干涉法（DQI）和哈密顿量DQI（HDQI）的推广），我们展示了相对解码如何将这种保证从源哈密顿量$H_A$转移到目标$H_B$。将每个哈密顿量写成$m$个泡利项之和，其乘积与恒等算符成比例的项的子集（即其关系）构成一个二元线性码，源为$K_A$，目标为$K_B$，泡利标记矩阵作为奇偶校验矩阵，如同DQI中一样。从$H_A$的典型热场双重态（TFD）出发，通过贝尔变换、可逆标记映射以及商码$K_B/K_A$的相干解码器，制备$H_B$的近似TFD，从而得到其吉布斯态。由于该解码器仅解析源中不存在的目标关系，可达的逆温度由相对距离$d_{\rm rel}$决定，即任何此类关系中的最少项数。它可能远超普通距离$d_{\rm ord}$（任何目标关系中的最少项数），后者限定了DQI和HDQI的均匀精确解码半径。我们证明，线性$d_{\rm rel}$配合线性半径下的高效解码可保证恒定的逆温度。作为示例，从已知可在每个有限温度下制备TFD的最近邻自旋链出发，一个稀疏经典奇偶校验矩阵可产生有界度、几何非局域、非对易的目标，其$d_{\rm ord}=3$且$d_{\rm rel}=\Theta(m)$。据我们所知，这给出了一类新的可高效制备的TFD。
英文摘要
Provable guarantees for preparing quantum Gibbs states, such as rapid mixing or gapped coherent preparation paths, are known only for restricted Hamiltonian families and usually must be re-derived for each new family. Building on homomorphic polynomial transduction, a generalization of decoded quantum interferometry (DQI) and Hamiltonian DQI (HDQI), we show how relative decoding transfers such a guarantee from a source Hamiltonian $H_A$ to a target $H_B$. Writing each Hamiltonian as a sum of $m$ Pauli terms, the subsets of terms whose product is proportional to the identity, its relations, form a binary linear code, $K_A$ for the source and $K_B$ for the target, with the Pauli-label matrix as parity-check matrix as in DQI. Starting from the canonical thermofield double (TFD) of $H_A$, a Bell transform, a reversible label map, and a coherent decoder for the quotient code $K_B/K_A$ prepare an approximate TFD of $H_B$, and hence its Gibbs state. Because this decoder only resolves target relations absent from the source, the reachable inverse temperature is set by the relative distance $d_{\rm rel}$, the fewest terms in any such relation. It can far exceed the ordinary distance $d_{\rm ord}$, the fewest terms in any target relation, which bounds the uniform exact decoding radius of DQI and HDQI. We show that linear $d_{\rm rel}$ with efficient decoding at linear radius certifies a constant inverse temperature. As an example, starting from a nearest-neighbor spin chain whose TFD is known to be preparable at every finite temperature, a sparse classical parity-check matrix yields bounded-degree, geometrically nonlocal, noncommuting targets with $d_{\rm ord}=3$ and $d_{\rm rel}=Θ(m)$. To our knowledge, this gives a new class of efficiently preparable TFDs.
Comments60 pages, 1 figure. Update review of previous works
