High-dimensional quantum computation needs a native circuit-level equational theory for qudits. We give the first finite schematic equational theory that is sound and complete for exact unitary qudit circuits in every finite dimension at least two. The result is entirely circuit-level: circuits are built from local gates, sequential and parallel composition, and value-controls, and equality is derivable exactly when two circuits have the same standard unitary semantics. For each dimension, the theory is presented by a finite family of local bounded-arity axiom schemata whose diagrammatic shapes are uniform in the dimension. The key syntactic ingredient is primitive value-control, which builds control on a chosen basis value directly into the language. This gives the language a useful internal algebra of controlled operations from local rules while keeping the presentation native to qudit circuits. The result provides a finite, dimension-uniform foundation for exact equational reasoning about qudit circuits.

翻译：高维量子计算需要一种原生的电路级幺正性理论来描述量子多能级系统。我们首次给出了一个有限示意图等式理论，该理论在每一个至少为2的有限维度上对于精确幺正量子多能级电路既是可靠的也是完备的。该结果完全处于电路层面：电路由局部门、顺序与并行组合以及数值控制构成，当且仅当两个电路具有相同的标准幺正语义时，等式可推导。对于每个维度，该理论由一族有限的局部有界元公理模式呈现，这些公理模式的图解形状在不同维度下具有一致性。关键语法要素是原始数值控制，它直接将选定基矢上的控制内建到语言中。这使得该语言通过局部规则拥有一个有用的受控操作内部代数，同时保持对量子多能级电路的原生性表述。该结果为量子多能级电路的精确等式推理提供了有限且维度一致的基础。