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：自由范畴构造将量子计算刻画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。该配方有效构建了从rig群胚内部语言π的两个副本出发的计算通用量子编程语言——该构造采用休斯箭头（Hughes' arrows）实现。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：量子效应。通过叠加更高层次的效应可引入测量机制。该构造还使得通过经典推理与互补性推理相结合的方式，对量子程序（含或不含测量）的某些性质进行推理成为可能。