Title: The Quantum Monadology Show Chinese title

Title: 量子单子论

Abstract: The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects. Show Chinese abstract

Abstract: 现代函数式编程语言的理论使用单子来编码计算的副作用和侧上下文，超越了基本的程序逻辑。 尽管量子计算本质上具有副作用（如量子测量）且依赖于上下文（如混合辅助态），但此前很少将这种单子范式应用于量子编程语言。 在这里，我们系统地分析由格罗滕迪克的“操作动机瑜伽”所诱导的参数化模块谱范畴上的(余)单子——针对当前目的，专门化为HC-模，并进一步到以集合为索引的复向量空间。 将索引向量空间解释为由量子测量结果参数化的替代可能的量子状态空间的集合，正如Proto-Quipper语义中所熟悉的，我们发现这些(余)单子为具有经典控制和“动态提升”量子测量结果回到经典上下文的函数式量子编程提供了一个全面的自然语言。 最后，我们指出一种领域特定的量子编程语言（QS），它以透明的do-表示法表达这些单子量子效应，可嵌入最近构建的线性同伦类型理论（LHoTT），该理论解释为参数化模块谱。 一旦嵌入LHoTT，这应该能够实现形式可验证的通用量子编程，具有线性量子类型、经典控制、动态提升，以及显著的拓扑效应。