发表机构
IBM Quantum; Columbia University; UC Berkeley; Princeton（IBM量子; 哥伦比亚大学; 加州大学伯克利分校; 普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种使用 $O(\log(m) \log(m/\delta) / \varepsilon^2)$ 份拷贝的阴影层析算法，通过平均情形分析和极小极大原理实现最坏情形保证，并指出测量随机性导致的 $\log(m)$ 因子障碍。
AI 中文摘要
我们给出一种阴影层析算法，该算法使用输入态的 $O(\log(m) \log(m/\delta) / \varepsilon^2)$ 份拷贝。具体而言，给定量子态 $\rho$ 和可观测量 $O_1,\ldots,O_m$，该算法输出对所有 $i\in[m]$ 的 $\mathrm{tr}(O_i\rho)$ 的估计值，这些估计值均在 $\pm\varepsilon$ 范围内准确，除概率至多为 $\delta$ 的失败情况外。关键思想是在平均情形下工作：对于从已知先验中抽取的随机态，对 $O(\log(m)/\varepsilon^2)$ 份拷贝进行的相当好的测量，除先验的一小部分外，对所有部分都给出常数误差保证。提升该保证可得到平均情形的阴影层析算法，然后极小极大原理（非构造性地）暗示存在具有相同最坏情形保证的测量。我们还确定了用这种方法进一步改进的一个障碍。虽然相当好的测量在先验的随机性上放大了成功概率，但它们并未在测量的随机性上放大成功概率。这个问题似乎是固有的，并导致了剩余的 $\log(m)$ 因子，该因子将量子问题的复杂度与其经典对应物区分开来。
英文摘要
We give a shadow tomography algorithm that uses $O(\log(m) \log(m/δ) / \varepsilon^2)$ copies of the input state. Specifically, given a quantum state $ρ$ and observables $O_1,\ldots,O_m$, the algorithm outputs estimates of $\mathrm{tr}(O_iρ)$ for all $i\in[m]$ that are all accurate to within $\pm\varepsilon$, except with probability at most $δ$. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on $O(\log(m)/\varepsilon^2)$ copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining $\log(m)$ factor separating the complexity of the quantum problem from its classical counterpart.
Comments32 pages, comments welcome
