Depth Optimization of FLT-Based Quantum Inversion Circuit

Works on quantum computing and cryptanalysis have increased significantly in the past few years. Various constructions of arithmetic circuits as one primary elements field also been proposed. However, there only a studies finite inversion despite its essential use realizing algorithms, such Shor’s algorithm for Elliptic Curve Discrete Logarithm Problem (ECDLP). In this study, we propose to reduce depth existing Fermat’s Little Theorem (FLT)-based circuit binary field. particular, follow complete waterfall approach translate Itoh-Tsujii’s variant FLT corresponding remove inverse squaring operations employed previous work by Banegas et al., lowering number CNOT gates (i.e., count) well slightly reducing $T$ depth, which contributes reduced overall gate count. Furthermore, concretely verify our method compare it with Qiskit, computer simulation environment, constructing both from scratch performing resource analysis. Additionally, employing relative-phase Toffoli Gidney opposed standard implementation, yields lower while further depth. Our can serve an alternative time-efficient implementation.