基于Reed Muller的一般二进制量子复用器优化方法
A Reed Muller-based approach for optimization of general binary quantum multiplexers
摘要：
以前的工作提供了分解酉矩阵的方法，一系列的量子复用器，但以这种方式创建的复用器是高度非最小的。本文提出了一种新的方法，用于优化具有任意单量子比特量子目标函数的量子复用器。对于量子多路复用器，我们定义了标准形式和两种新的形式：固定极性量子形式（FPQF）和克罗内克量子形式（KQF），它们分别类似于经典逻辑函数的最小乘积和形式、固定极性Reed-Muller（FPRM）形式和克罗内克Reed-Muller（KRM）形式。从FPRM和KRM形式的蝴蝶图的使用中得到灵感，我们设计了一种方法来详尽地构造所有FPQF和KQF形式。因此，新的形式可以用来优化量子线路与任意目标酉矩阵，而不是只有多控制非门，如CNOT，CCNOT，及其扩展。FPQF和KQF形式，以及FPRM和KRM经典形式，应用于各种目标门，如非，V，V+，Hadamard和泡利旋转的实验结果表明，FPQF和KQF形式大大降低了量子复用器在随机生成的数据和FPRM基准的门成本。
英文摘要：
Previous work has provided methods for decomposing unitary matrices to series of quantum multiplexers, but the multiplexers created in this way are highly non-minimal. This paper presents a new approach for optimizing quantum multiplexers with arbitrary single-qubit quantum target functions. For quantum multiplexers, we define standard forms and two types of new forms: fixed polarity quantum forms (FPQF) and Kronecker quantum forms (KQF), which are analogous to Minterm Sum of Products forms, Fixed Polarity Reed-Muller (FPRM) forms, and Kronecker Reed-Muller (KRM) forms, respectively, for classical logic functions. Drawing inspiration from the usage of butterfly diagrams for FPRM and KRM forms, we devise a method to exhaustively construct all FPQF and KQF forms. Thus, the new forms can be used to optimize quantum circuits with arbitrary target unitary matrices, rather than only multi-controlled NOT gates such as CNOT, CCNOT, and their extensions. Experimental results on FPQF and KQF forms, as well as FPRM and KRM classical forms, applied to various target gates such as NOT, V, V+, Hadamard, and Pauli rotations, demonstrate that FPQF and KQF forms greatly reduce the gate cost of quantum multiplexers in both randomly generated data and FPRM benchmarks.
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