Quantum query complexity has several nice properties with respect to composition. First, bounded-error quantum query algorithms can be composed without incurring log factors through error reduction $exactness$. Second, through careful accounting $thriftiness$, the total query complexity is smaller if subroutines are mostly run on cheaper inputs -- a property that is much less obvious in quantum algorithms than in their classical counterparts. While these properties were previously seen through the model of span programs (alternatively, the dual adversary bound), a recent work by two of the authors (Belovs, Yolcu 2023) showed how to achieve these benefits without converting to span programs, by defining $\textit{quantum Las Vegas query complexity}$. Independently, recent works, including by one of the authors (Jeffery 2022), have worked towards bringing thriftiness to the more practically significant setting of quantum $time$ complexity.

In this work, we show how to achieve both exactness and thriftiness in the setting of time complexity. We generalize the quantum subroutine composition results of Jeffery 2022 so that, in particular, no error reduction is needed. We give a time complexity version of the well-known result in quantum query complexity, $Q(f\circ g)=\mathcal{O}(Q(f)\cdot Q(g))$, without log factors.

We achieve this by employing a novel approach to the design of quantum algorithms based on what we call $transducers$, and which we think is of large independent interest. While a span program is a completely different computational model, a transducer is a direct generalisation of a quantum algorithm, which allows for much greater transparency and control. Transducers naturally characterize general state conversion, rather than only decision problems; provide a very simple treatment of other quantum primitives such as quantum walks; and lend themselves well to time complexity analysis.

中文翻译：

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驯服量子时间复杂性


量子查询复杂性在组合方面有几个很好的特性。首先，可以通过减少错误$精确性$来构建有界错误量子查询算法，而不会产生对数因子。其次，通过仔细计算“节俭性”，如果子例程主要在更便宜的输入上运行，则总查询复杂性会更小——这一特性在量子算法中比在经典算法中不那么明显。虽然这些属性之前是通过跨度程序模型（或者双重对手界限）看到的，但两位作者最近的一项工作（Belovs，Yolcu 2023）展示了如何在不转换为跨度程序的情况下实现这些好处，通过定义 $ \textit{量子拉斯维加斯查询复杂度}$。独立地，最近的工作，包括其中一位作者的工作（Jeffery 2022），致力于将节俭带入更具有实际意义的量子“时间”复杂性设置。


在这项工作中，我们展示了如何在时间复杂度的设置下实现精确性和节俭性。我们概括了 Jeffery 2022 的量子子程序组合结果，因此特别不需要减少误差。我们给出了众所周知的量子查询复杂度结果的时间复杂度版本 $Q(f\circ g)=\mathcal{O}(Q(f)\cdot Q(g))$，没有对数因子。


我们通过采用一种新颖的量子算法设计方法来实现这一目标，该方法基于我们所谓的“传感器”，我们认为这种方法具有很大的独立意义。虽然跨度程序是一种完全不同的计算模型，但传感器是量子算法的直接概括，它允许更大的透明度和控制。传感器自然地表征一般状态转换，而不仅仅是决策问题；提供对其他量子原语（例如量子行走）的非常简单的处理；并且非常适合时间复杂度分析。