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：自由绝对的构造将量子计算定性为由古典结构的互补方程式粘合在一起的可逆古典模型的两个副本的组合。 这个配方有效地从Pi的两种副本即钻机群状体的内部语言中构建了一种计算通用量子编程语言。 构造由休斯的箭头组成。 因此可以肯定地回答这样一个问题:将可逆古典计算转换成量子计算的计算效果是否存在? 量子效应。 测量可以通过在顶部进一步分层来添加。 我们的构造还可以通过将关于互补性的经典推理和推理结合起来,对量子程序进行一些推理( 不论测量与否 ) 。