用于模运算的量子电路的基于测量的逆计算
Measurement-based uncomputation of quantum circuits for modular arithmetic
AI总结：
本文形式化了单量子位寄存器的基于测量的逆计算（MBU）技术，将其应用于模运算电路，显著降低了Toffoli门数量和深度，并有望改进模乘法、模幂运算及量子密码分析。
摘要：
基于测量的逆计算（MBU）是一种用于执行量子电路概率性逆计算的技术。我们针对单量子位寄存器的情况对该技术进行了形式化，并展示了其在模运算中的应用。首先，我们为执行非模加法的量子电路的几种变体给出了形式化陈述：受控加法、常数加法以及受控常数加法。我们对减法和比较电路也做了同样的处理。这填补了当前文献中的空白，其中一些变体此前未被探索。然后，我们将注意力转向模运算，再次使用不同类型的普通加法器及其组合，为模加法、受控模加法、常数模加法以及受控常数模加法给出了形式化陈述。我们在单量子位寄存器的背景下引入并证明了“MBU引理”，并将其应用于上述所有模运算电路。使用MBU，对于基于[VBE96]架构的模加法器，我们将Toffoli门数量和深度减少了10%到15%；对于基于[Bea02]架构的模加法器，减少了近25%。我们的结果有潜力改进其他模运算电路，如模乘法和模幂运算，并可在量子密码分析中找到应用。
英文摘要：
Measurement-based uncomputation (MBU) is a technique used to perform probabilistic uncomputation of quantum circuits. We formalize this technique for the case of single-qubit registers, and we show applications to modular arithmetic. First, we present formal statements for several variations of quantum circuits performing non-modular addition: controlled addition, addition by a constant, and controlled addition by a constant. We do the same for subtraction and comparison circuits. This addresses gaps in the current literature, where some of these variants were previously unexplored. Then, we shift our attention to modular arithmetic, where again we present formal statements for modular addition, controlled modular addition, modular addition by a constant, and controlled modular addition by a constant, using different kinds of plain adders and combinations thereof. We introduce and prove a "MBU lemma" in the context of single-qubit registers, which we apply to all aforementioned modular arithmetic circuits. Using MBU, we reduce the Toffoli count and depth by $10\%$ to $15\%$ for modular adders based on the architecture of [VBE96], and by almost $25\%$ for modular adders based on the architecture of [Bea02]. Our results have the potential to improve other circuits for modular arithmetic, such as modular multiplication and modular exponentiation, and can find applications in quantum cryptanalysis.
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