SIAM Journal on Computing, Ahead of Print.
We introduce the problem of shadow tomography: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left(\varepsilon^{-4}\cdot\log^{4}M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O(1)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Branda͂o et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.

中文翻译：

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量子态的阴影层析成像

《 SIAM计算杂志》，预印本。
我们介绍了阴影层析成像的问题：给定未知的$ D $维三维量子混合态$ \ rho $，以及已知的两次结果测量值$ E_ {1}，\ ldots，E_ {M} $，估计概率对于每个$ M $测量值，$ E_ {i} $接受$ \ rho $，且误差不超过$ \ varepsilon $。要达到这个目的，需要多少份$ \ rho $？出乎意料的是，我们给出了一个仅通过测量$ \ widetilde {O} \ left（\ varepsilon ^ {-4} \ cdot \ log ^ {4} M \ cdot \ log D \ right）$副本即可解决此问题的过程。例如，这意味着我们可以通过仅测量$ n ^ {O（1）} $个副本的任意副本，来了解某个固定多项式大小的所有接受/拒绝电路上任意$ n $ -qubit状态的行为。州。这解决了作者的一个公开问题，该问题源于他在私钥量子货币计划方面的工作，但它也可应用于受量子复制保护的软件，量子建议和量子单向通信。最近，Branda͂o等人以这项工作为基础。已经给出了使用半定编程进行阴影层析成像的另一种方法，从而节省了计算时间。