通过富集范畴实现参数化量子电路语义
Parameterized Quantum Circuit Semantics Through Enriched Categories
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中文总结 AI 辅助
研究通过富集范畴实现参数化量子电路语义问题，引入基于富集范畴理论的方法，先介绍抽象范畴构造，研究特殊参数情况，最终表明幺半闭情况可统一量子控制的两种观点。
中文摘要 AI 辅助
众所周知，组合电路在幺半范畴中由弦图进行数学建模。给定门集Σ，Σ上的电路可视为由Σ生成的自由幺半范畴中的弦图。在此模型中，电路语义由该自由范畴之外的幺半函子给出。对于量子电路，此函子通常取值于酉矩阵范畴。该模型适用于具体量子电路，但无法描述参数化量子电路族，比如在分析量子态电路时出现的那些。本文引入一种基于富集范畴理论的参数化电路语义方法。首先引入抽象范畴构造，用于深入理解受控操作和量子通信。接着研究笛卡尔幺半参数和幺半闭参数的特殊情况，两者都赋予参数化语义有用性质。最后表明幺半闭情况可用于统一量子控制的两种观点。
英文摘要
It is well-known that combinatorial circuits are modeled mathematically by string diagrams in monoidal categories. Given a gate set $Σ$, the circuits over $Σ$ can be thought of as string diagrams in the free monoidal category generated by $Σ$. In this model, circuit semantics are then given by monoidal functors out of this free category. For quantum circuits, this functor is often valued in the category of unitary matrices. This model suffices for concrete quantum circuits, but fails to describe parameterized families of quantum circuits, such as those which arise in the analysis of ansatz circuits. In this paper, we introduce an approach to parameterized circuit semantics, which is based on enriched category theory. We first introduce an abstract categorical construction, and use this to gain new insights on controlled operations and quantum communication. We then study the special cases of Cartesian monoidal parameters and monoidal closed parameters, both endowing the parameterized semantics with useful constructions. We conclude by showing that the monoidal closed case can be used to unify two perspectives on quantum control.
