AI 中文总结
本文首次提出学习k-稀疏量子态的近乎最优算法，使用~O(k/ε)份副本和~O(kn/ε)时间实现高保真度，并推广到混合态，同时指出时间复杂度的开放问题。
AI 中文摘要
我们研究k-稀疏量子态层析成像问题。与经典分布学习不同，后者在支持集大小方面具有紧的样本和时间复杂度界限且已被充分理解，而此前该问题未显示出非平凡界限。我们首次给出了学习n量子比特k-稀疏纯量子态的近乎最优算法，使用~O(k/ε)份态副本和~O(kn/ε)时间，以高概率获得至少1-ε的保真度。这两个界限在多项式对数因子内均为最优。作为推论，通过随机纯化信道技术，我们还获得了学习k-稀疏秩-r混合态的近乎最优样本复杂度~O(kr/ε)的算法。对于r>1，获得与样本复杂度几乎匹配的时间复杂度仍是一个重要的开放问题。
英文摘要
We study the problem of tomography for $k$-sparse quantum states. In contrast to classical distribution learning, where tight sample and time complexity bounds in terms of support size are well understood, no non-trivial bounds were previously shown for this problem. We give the first near optimal algorithm for learning $n$-qubit $k$-sparse pure quantum states, obtaining fidelity at least $1-\varepsilon$ with high probability using $\tilde{O}(k/\varepsilon)$ copies of the state and $\tilde{O}(kn/\varepsilon)$ time. Both bounds are optimal up to polylogarithmic factors. As an implication, we also obtain an algorithm with near optimal $\tilde{O}(kr/\varepsilon)$ sample complexity for learning $k$-sparse rank-$r$ mixed states, via the random purification channel technique. Obtaining time complexity nearly matching the sample complexity, for $r>1$, remains an important open question.
Comments26 pages
