伽罗瓦域上的量子算术
Quantum Arithmetic on Galois Fields
摘要：
本文讨论了在量子计算机上实现初等有限域运算的问题。特别感兴趣的是受控乘法运算，这是Shor算法中唯一的特定于组的运算，用于分解和解决离散对数问题。我们描述了如何建立量子线路执行这个操作的通用伽罗瓦域GF（$p^k$），以及边界情况GF（$p$）和GF（$2^k$）。我们给出了这些电路的详细大小，宽度和深度复杂性，这最终将使我们能够获得解决这些领域的DLP实例所需的量子资源量的详细上限。
英文摘要：
In this paper we discuss the problem of performing elementary finite field arithmetic on a quantum computer. Of particular interest, is the controlled-multiplication operation, which is the only group-specific operation in Shor's algorithms for factoring and solving the Discrete Log Problem. We describe how to build quantum circuits for performing this operation on the generic Galois fields GF($p^k$), as well as the boundary cases GF($p$) and GF($2^k$). We give the detailed size, width and depth complexity of such circuits, which ultimately will allow us to obtain detailed upper bounds on the amount of quantum resources needed to solve instances of the DLP on such fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
