发表机构
Duke Quantum Center, Duke University; Department of Electrical and Computer Engineering, Duke University; Department of Physics, Duke University（杜克大学量子中心; 杜克大学电气与计算机工程系; 杜克大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出仅用成对 SWAP 测试实现任意置换不变混合态的酉 Schur 采样，通过反复提取最大反对称子系统，将算法转化为 Young 图随机遍历，并应用于量子纯度放大以最优纯化态。
AI 中文摘要
Schur 采样是从多体量子系统中提取置换不变信息的基本原语，在量子信息科学中具有广泛的应用。基于电路的实现通常依赖于相干表示论操作，如 Clebsch-Gordan 变换、广义相位估计或对称群上的量子傅里叶变换。我们证明，对于 $n$ 个 $d$ 维量子位上的任意置换不变混合态，酉 Schur 采样可以仅通过成对 SWAP 测试来实现，即对对称和反对称子空间进行双量子位投影测量。我们的协议在金刚石距离下达到误差 $\epsilon$，使用 $O(n\min\{n,d\}^{3}\log^2(n/\epsilon))$ 次随机 SWAP 测试。核心思想是反复识别并提取最大的反对称子系统。这将酉 Schur 采样转化为对反对称性的搜索，并赋予算法作为 Young 图随机遍历的自然解释。该协议在保持 $\mathrm{SU}(d)$ 不可约表示态的同时，在多重性子系统中制备一个规范纯态。作为一个应用，我们考虑量子纯度放大（QPA），其中多个噪声副本的纯量子态被组合以产生更高纯度的态。我们证明，通过保留一个指定的输出量子位并丢弃其余部分，可以获得最优纯化态。
英文摘要
Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on $n$ $d$-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error $ε$ in diamond distance using $O(n\min\{n,d\}^{3}\log^2(n/ε))$ random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the $\mathrm{SU}(d)$-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.
Comments19 pages + 29 pages of Appendices, 2 Figures, Preliminary version
