发表机构
The University of Chicago; IBM Research（芝加哥大学; IBM研究部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对未知乘积基下至多s个非零矩阵元的稀疏量子态，提出仅用单量子比特测量即可高效学习的方法，无需纠缠门，适用于长程纠缠态，常数稀疏度下具有多项式样本与计算复杂度，多项式稀疏度下样本复杂度仍为多项式，特定基下计算复杂度为拟多项式。
AI 中文摘要
我们研究学习稀疏量子态的问题，即一个$n$量子比特的量子态，其密度矩阵在未知乘积基下至多有$s$个非零矩阵元。尽管此类态具有紧凑的经典描述，但它们可能携带长程纠缠，这阻碍了仅从局部约化密度矩阵进行重建。因此，先前的学习方法利用许多纠缠门来提取必要信息以处理此类长程纠缠态。在本工作中，我们证明稀疏态仍然可以仅使用单量子比特测量被高效学习。具体而言，当稀疏度$s$为常数时，我们的算法能够以多项式样本复杂度和经典计算复杂度从单量子比特测量中学习稀疏态。当$s$随$n$多项式增长时，稀疏态仍可从单量子比特测量中以多项式样本复杂度被学习，尽管高效经典计算在一般情况下无法保证。然而，在此情形下，当态在未知基（该基是已知固定有限集（例如泡利算符的本征基）的单量子比特基的乘积）中稀疏时，经典计算复杂度变为拟多项式。这些结果确立了无需纠缠门即可高效学习具有长程纠缠的稀疏态，并且单量子比特测量要求使我们的算法与当前量子设备兼容。
英文摘要
We study the problem of learning a sparse quantum state, an $n$-qubit quantum state whose density matrix has at most $s$ nonzero matrix entries in an unknown product basis. While such states admit compact classical descriptions, they can carry long-range entanglement that prevents reconstruction from local reduced density matrices alone. Therefore, previous learning approaches addressed such long-range-entangled states using many entangling gates to extract the necessary information. In this work, we show that sparse states can nevertheless be efficiently learned using only single-qubit measurements. Specifically, when the sparsity $s$ is constant, our algorithm can learn sparse states from single-qubit measurements with polynomial sample complexity and classical computational complexity. When $s$ grows polynomially with $n$, sparse states can still be learned from single-qubit measurements with polynomial sample complexity, although efficient classical computation is not guaranteed in general. In this regime, however, the classical computational complexity becomes quasipolynomial when the state is sparse in an unknown basis that is a product of a known fixed finite set of single-qubit bases (e.g., eigenbases of Pauli operators). These results establish efficient learning of sparse states with long-range entanglement without entangling gates, and the single-qubit measurement requirements make our algorithms compatible with current quantum devices.
