In the past four decades, the notion of quantum polynomial-time computability has been mathematically modeled by quantum Turing machines as well as quantum circuits. This paper seeks the third model, which is a quantum analogue of the schematic (inductive or constructive) definition of (primitive) recursive functions. For quantum functions mapping finite-dimensional Hilbert spaces to themselves, we present such a schematic definition, composed of a small set of initial quantum functions and a few construction rules that dictate how to build a new quantum function from the existing ones. We prove that our schematic definition precisely characterizes all functions that can be computable with high success probabilities on well-formed quantum Turing machines in polynomial time, or equivalently uniform families of polynomial-size quantum circuits. Our new, schematic definition is quite simple and intuitive and, more importantly, it avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model. Our new approach can further open a door to the descriptional complexity of quantum functions, to the theory of higher-type quantum functionals, to the development of new first-order theories for quantum computing, and to the designing of programming languages for real-life quantum computer

中文翻译：

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量子多项式时间可计算性的示意图定义

在过去的四年里，量子多项式时间可计算性的概念已经通过量子图灵机和量子电路进行了数学建模。本文寻求第三种模型，它是（原始）递归函数的示意（归纳或构造）定义的量子模拟。对于将有限维希尔伯特空间映射到自身的量子函数，我们提出了这样一个示意图定义，由一小组初始量子函数和一些构造规则组成，这些构造规则指示如何从现有的量子函数构建新的量子函数。我们证明了我们的示意图定义精确地描述了所有可以在多项式时间内在格式良好的量子图灵机上以高成功概率计算的函数，或者等价的多项式大小的量子电路的统一族。我们新的示意图定义非常简单和直观，更重要的是，它避免了强加于量子图灵机模型的良构条件以及量子电路模型所需的均匀性条件的繁琐引入。我们的新方法可以进一步为量子函数的描述复杂性、更高类型的量子泛函理论、量子计算的新一阶理论的发展以及为现实生活设计的编程语言打开一扇门量子计算机