The study of non-collapsing measurements was initiated by Aaronson, Bouland, Fitzsimons, and Lee, who showed that BQP, when equipped with the ability to perform non-collapsing measurements (denoted as PDQP), contains both BQP and SZK, yet still requires $\Omega (N^{1/4})$ queries to find an element in an unsorted list. By formulating an alternative equivalent model of PDQP, we prove the positive weighted adversary method, obtaining a variety of new lower bounds and establishing a trade-off between queries and non-collapsing measurements. The method allows us to examine the well-studied majority and element distinctness problems, while also tightening the bound for the search problem to $\Theta (N^{1/3})$. Additionally, we explore related settings, obtaining tight bounds in BQP with the ability to copy arbitrary states (called CBQP) and PDQP with non-adaptive queries.

翻译：非坍缩测量的研究由Aaronson、Bouland、Fitzsimons和Lee开创，他们证明了当BQP具备执行非坍缩测量的能力（记为PDQP）时，该复杂性类同时包含BQP和SZK，但仍需$\Omega (N^{1/4})$次查询才能在一个无序列表中找到一个元素。通过构建PDQP的一种替代等价模型，我们证明了正权重对抗方法，从而获得了一系列新的下界，并建立了查询次数与非坍缩测量次数之间的权衡关系。该方法使我们能够深入分析已被广泛研究的多数据问题和元素互异性问题，同时将搜索问题的下界收紧至$\Theta (N^{1/3})$。此外，我们探索了相关设置，在具备任意态复制能力的BQP（称为CBQP）以及非自适应查询的PDQP中获得了紧致界。