We present a finite schematic axiomatisation of quantum circuits over d-level systems (qudits), uniform in every finite dimension d >= 2. For each d we define a PROP equipped with a family of control functors, treating control as a primitive categorical constructor. Using a translation between qudit circuits and the LOPP calculus for linear optics based on d-ary Gray codes, we obtain for each d a finite set of local axiom schemata that is sound and complete for unitary d-level circuits: two circuits denote the same unitary if and only if they are inter-derivable using axioms involving at most three wires. The generators are compatible with standard universal qudit gate families, yielding a sound equational basis for circuit rewriting and optimisation-by-rewriting. Conceptually, this extends the qubit circuit completeness results of Clément et al.\ to arbitrary finite dimension, and instantiates the control-as-constructor approach of Delorme and Perdrix in this setting, while keeping the axiom shapes uniform in d.

翻译：我们提出了一种针对d能级系统（量子比特）上量子电路的有限图式公理化方法，该方法在所有有限维度d >= 2上保持统一性。对于每个d，我们定义了一个配备控制函子族的PROP，将控制视为原始范畴构造子。通过建立量子比特电路与基于d元格雷码的线性光学LOPP演算之间的翻译，我们为每个d获得了一组有限局部公理图式，该图式对幺正d能级电路是可靠且完备的：两个电路表示相同的幺正算子当且仅当它们可通过涉及最多三条线路的公理相互推导。生成元与标准通用量子比特门族兼容，从而为电路重写和基于重写的优化提供了可靠的等式基础。从概念上讲，这项工作将Clément等人关于量子比特电路的完备性结果推广至任意有限维度，并在该框架下实例化了Delorme和Perdrix提出的"控制即构造子"方法，同时保持公理形式在维度d上的统一性。