Free categorical constructions characterise quantum computing as the combination of two copies of a reversible classical model, glued by the complementarity equations of classical structures. This recipe effectively constructs a computationally universal quantum programming language from two copies of Pi, the internal language of rig groupoids. The construction consists of Hughes' arrows. Thus answer positively the question whether a computational effect exists that turns reversible classical computation into quantum computation: the quantum effect. Measurements can be added by layering a further effect on top. Our construction also enables some reasoning about quantum programs (with or without measurement) through a combination of classical reasoning and reasoning about complementarity.

翻译：摘要：自由范畴构造将量子计算刻画为可逆经典模型的两个副本通过经典结构的互补性方程黏合而成。该配方有效地从两个Pi（刚性群胚的内部语言）副本构建出计算上通用的量子编程语言。该构造采用了休斯箭头，从而肯定地回答了是否存在一种计算效应能将可逆经典计算转化为量子计算的问题——这种效应即量子效应。通过叠加进一步效应可添加测量操作。我们的构造还允许通过经典推理与互补性推理的结合，对量子程序（含或不含测量）进行一定程度的逻辑推演。