来自零均值Wishart分布的量子态的不相关问题特定样本
Uncorrelated problem-specific samples of quantum states from zero-mean Wishart distributions
摘要：
量子态的随机样本是量子信息科学中各种任务的重要资源，并且符合特定问题分布的样本可以是不可或缺的成分。一些算法通过遵循某些规则的彩票生成随机样本，并从彩票可以访问的分布集中产生样本。在状态空间中使用随机游走的其他算法可以针对任何分布进行定制，但代价是样本中的自相关性以及实际实现中对低维系统的限制。在本文中，我们提出了一个两步算法，从量子态空间，克服了这些限制。我们首先通过从复杂的Wishart分布族中提取不相关条目的CPU廉价的大提案样本，然后拒绝或接受提案样本中的条目，使得接受的样本严格符合目标分布。建立了量子态诱导Wishart分布的显式表达式。这使我们能够生成一个模拟目标分布的提案样本，因此，由接受率衡量的算法效率可以比作为提案的均匀样本大许多数量级。我们证明，这种采样算法是非常有效的一个量子比特和两个量子比特的状态，并合理有效的三个量子比特的状态，而它遭受的“灾难的维数”时，从结构化分布的四个量子比特状态采样。
英文摘要：
Random samples of quantum states are an important resource for various tasks in quantum information science, and samples in accordance with a problem-specific distribution can be indispensable ingredients. Some algorithms generate random samples by a lottery that follows certain rules and yield samples from the set of distributions that the lottery can access. Other algorithms, which use random walks in the state space, can be tailored to any distribution, at the price of autocorrelations in the sample and with restrictions to low-dimensional systems in practical implementations. In this paper, we present a two-step algorithm for sampling from the quantum state space that overcomes some of these limitations. We first produce a CPU-cheap large proposal sample, of uncorrelated entries, by drawing from the family of complex Wishart distributions, and then reject or accept the entries in the proposal sample such that the accepted sample is strictly in accordance with the target distribution. We establish the explicit form of the induced Wishart distribution for quantum states. This enables us to generate a proposal sample that mimics the target distribution and, therefore, the efficiency of the algorithm, measured by the acceptance rate, can be many orders of magnitude larger than that for a uniform sample as the proposal. We demonstrate that this sampling algorithm is very efficient for one-qubit and two-qubit states, and reasonably efficient for three-qubit states, while it suffers from the "curse of dimensionality" when sampling from structured distributions of four-qubit states.
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