Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many-body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as ${\sum }_{\mathit{a},\mathit{b}}{\left|\mathrm{\Delta }{\rho }_{\mathit{a}\mathit{b}}\right|}^2\sim 5^N/N_{\mathrm{tot}}$ for full tomography, and ${\mathrm{\Delta }\mu }^2\sim 7^N/N_{\mathrm{tot}}^2$ for purity estimation, for $N$ qubits and $N_{\mathrm{tot}}$ measurements performed.

中文翻译：

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具有随机单脉冲旋转的多量子位系统的量子态断层扫描和纯度测量

量子态断层扫描和量子态全局特性的其他测量是通过量子模拟器理解多体物理学不可或缺的工具。不幸的是，估计这些全局量所需的系统实验测量数量与系统大小呈指数关系。在这里，我们考虑使用随机轴测量进行量子态断层扫描和状态纯度估计。我们对任何给定算法的此类方法中的统计偏差进行一般分析。然后我们提出了一个简单的协议，它只依赖于单脉冲旋转。我们发现它降低了指数增长的基础，计算统计方差以缩放为${\sum }_{\mathit{一种},\mathit{乙}}{\left|\mathrm{\Delta }{\rho }_{\mathit{一种}\mathit{乙}}\right|}^2～5^N/N_{总}$ 用于全断层扫描，和 ${\mathrm{\Delta }\mu }^2～7^N/N_{总}^2$ 用于纯度估计，用于 $N$ 量子位和 $N_{总}$ 进行的测量。