[论文解读] Alternation in Quantum Programming: From Superposition of Data to Superposition of Programs

Mingsheng Ying, Nengkun Yu+1 more|arXiv (Cornell University)|Feb 20, 2014

Quantum Computing Algorithms and Architecture参考文献 40被引用 10

一句话总结

本文通过将经典交替推广至量子领域，提出了一种新颖的量子编程范式——程序的叠加态，从而发展出Dijkstra的GCL的量子扩展——QGCL。它定义了两种不同的交替形式：经典（量子程序的）交替与量子交替，后者实现了程序的真正叠加态，构成基于算子值函数受控复合的新语义的核心。

ABSTRACT

We extract a novel quantum programming paradigm - superposition of programs - from the design idea of a popular class of quantum algorithms, namely quantum walk-based algorithms. The generality of this paradigm is guaranteed by the universality of quantum walks as a computational model. A new quantum programming language QGCL is then proposed to support the paradigm of superposition of programs. This language can be seen as a quantum extension of Dijkstra's GCL (Guarded Command Language). Surprisingly, alternation in GCL splits into two different notions in the quantum setting: classical alternation (of quantum programs) and quantum alternation, with the latter being introduced in QGCL for the first time. Quantum alternation is the key program construct for realizing the paradigm of superposition of programs. The denotational semantics of QGCL are defined by introducing a new mathematical tool called the guarded composition of operator-valued functions. Then the weakest precondition semantics of QGCL can straightforwardly derived. Another very useful program construct in realizing the quantum programming paradigm of superposition of programs, called quantum choice, can be easily defined in terms of quantum alternation. The relation between quantum choices and probabilistic choices is clarified through defining the notion of local variables. We derive a family of algebraic laws for QGCL programs that can be used in program verification, transformations and compilation. The expressive power of QGCL is illustrated by several examples where various variants and generalizations of quantum walks are conveniently expressed using quantum alternation and quantum choice. We believe that quantum programming with quantum alternation and choice will play an important role in further exploiting the power of quantum computing.

研究动机与目标

• 识别并形式化一种新的量子编程范式——程序的叠加态，其灵感来源于基于量子行走的算法。
• 通过引入量子交替作为基本构造，解决经典控制流在量子编程中的局限性。
• 为支持量子控制与数据叠加的量子程序，建立形式化语义。
• 通过代数定律与最弱前置条件语义，为量子程序的验证、转换与编译提供基础。

提出的方法

• 提出QGCL作为Dijkstra的GCL的量子扩展，将量子交替与量子选择作为一等构造引入。
• 引入一种新的数学工具——算子值函数的受控复合，以定义QGCL的指称语义。
• 从指称框架推导出最弱前置条件语义，实现对量子程序的正式验证。
• 通过局部变量的概念，将量子选择定义为量子交替的衍生形式，阐明其与概率选择的关系。
• 建立一组QGCL的代数定律，支持程序转换、优化与验证。
• 通过多量子比特与纠缠控制的量子行走示例，展示其表达能力，这些示例使用量子交替与选择实现。

实验结果

研究问题

• RQ1 经典编程中的交替如何推广以支持量子程序的叠加态？
• RQ2 经典交替的量子对应物有哪些？它们在计算行为上如何不同？
• RQ3 如何设计一种量子编程语言，以同时支持量子数据与量子控制流？
• RQ4 对于包含量子交替与选择的程序，需要何种形式化语义与验证技术？
• RQ5 与经典控制流相比，量子交替能否支持更高效或更新颖的量子算法？

主要发现

• 提出量子交替作为新型程序构造，可实现程序的叠加态，其与经典量子程序交替有本质区别。
• QGCL的指称语义通过算子值函数的受控复合定义，提供了严谨的数学基础。
• 从指称模型推导出最弱前置条件语义，支持对量子程序的正式验证。
• 量子选择通过量子交替定义，其与概率选择的关系通过局部变量概念得以澄清。
• 建立了一组QGCL的代数定律，支持程序转换、优化与验证。
• 示例表明，QGCL能够优雅地表达具有多量子比特与纠缠控制的复杂量子行走，充分展现其表达能力。