发表机构
Joint Center for Quantum Information and Computer Science, University of Maryland and NIST; Department of Physics, University of Maryland; Department of Physics, Carnegie Mellon University; Department of Computer Science, Carnegie Mellon University; Joint Quantum Institute, University of Maryland and NIST（马里兰大学与国家标准与技术研究院量子信息与计算科学联合中心; 马里兰大学物理系; 卡内基梅隆大学物理系; 卡内基梅隆大学计算机科学系; 马里兰大学与国家标准与技术研究院联合量子研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对噪声态稳定子结构学习，提出Bell采样查找-验证算法，在弱Pauli噪声下实现多项式样本与运行时间，缩小了不可知学习与纯态算法的差距。
AI 中文摘要
Bell采样能高效学习具有少量魔性的纯态的稳定子群，进而学习稳定子零化度，这是低魔性态层析和认证的基础。然而，真实设备只能制备目标态的噪声版本，我们将其建模为作用于理想态上的Pauli信道；所得噪声态通常没有精确的Pauli稳定子。现有的混合态结构学习器依赖这类精确稳定子。不可知层析不依赖精确稳定子，但其保证随零化度$t$呈指数增长。我们证明，理想稳定子群仍编码在噪声态中，表现为使其不变的Pauli算子集合。我们给出Bell采样算法，对零化度至多$t$的态，能以精度$\varepsilon$学习该群。难点在于噪声样本破坏了纯态学习器所依赖的群结构，因此更多样本可能抹除结构而非揭示结构。我们通过一个查找-验证方案解决此问题，该方案从部分损坏的样本批次中提取稳定子结构，同时验证提议的更新。对于足够弱的全局去极化噪声，当信道总错误概率至多为$\varepsilon n/t$量级时，样本数和运行时间均为$\mathrm{poly}(n,t,1/\varepsilon)$。在恒定总错误概率（约0.16）以内，样本数保持多项式，而经典处理代价为$2^t$；对于任何低于1的错误概率，$4^t$开销即足够。任意Pauli信道表现出相同的三个区域，但阈值更严格。对于弱Pauli噪声，这大大缩小了不可知学习器（如稳定子自举）与多项式纯态算法之间的差距。
英文摘要
Bell sampling efficiently learns the stabilizer group, and hence the stabilizer nullity, of pure states with little magic, a primitive underlying tomography and certification of low-magic states. Real devices, however, only prepare noisy images of the target, which we model as a Pauli channel acting on the ideal state; the resulting noisy state generically has no exact Pauli stabilizers. Existing structure learners for mixed states rely on such exact stabilizers. Agnostic tomography does not, but its guarantees scale exponentially in the nullity $t$. We show that the ideal stabilizer group remains encoded in the noisy state as the set of Pauli operators under which it is invariant. We give Bell-sampling algorithms that learn the group to accuracy $\varepsilon$ for states of nullity at most $t$. The difficulty is that noisy samples break the group structure pure-state learners rely on, so more samples can erase structure rather than reveal it. We resolve this with a finder-verifier scheme that extracts stabilizer structure from partially corrupted sample batches while certifying proposed updates. For sufficiently weak global depolarizing noise, both samples and runtime are $\mathrm{poly}(n,t,1/\varepsilon)$ when the channel's total error probability is at most of order $\varepsilon n/t$. Up to a constant total error probability ($\approx 0.16$), samples remain polynomial while classical processing costs $2^t$, and for any error probability below one a $4^t$ overhead suffices. Arbitrary Pauli channels exhibit the same three regimes, with more restrictive thresholds. For weak Pauli noise, this closes much of the gap between agnostic learners such as stabilizer bootstrapping and polynomial pure-state algorithms.
Comments57 pages, 12 figures, 3 tables
