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：自由范畴构造刻画了量子计算作为两个可逆经典模型副本的组合，通过经典结构的互补性方程粘合而成。这一配方从两个Pi副本（即环群胚的内部语言）有效构造出计算上通用的量子编程语言，其构造过程基于Hughes箭头。由此正面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——即量子效应。通过进一步叠加效应层还可加入测量操作。该构造还使我们能通过经典推理与互补性推理的结合，对含测量或不含测量的量子程序进行某种程度的逻辑推理。