OPTIMIZATION FOR THE QUANTUM CIRCUIT OF ELLIPTIC CURVE DISCRETE LOGARITHMS

With the help of techniques such as windowed arithmetic and the coset representation of modular integers, the overall optimization and resource estimation for the quantum circuit of elliptic curve discrete logarithms Shor's algorithm was shown. The simulation experiment of the designed quantum circuit was carried out. The T gate and the depth of the overall circuit could be reduced by techniques such as windowed arithmetic and the coset representation of modular integers. The T count was 32n³ + O(n²log n) and the measurement depth was 12n³ + O(n²log n). Due to the windowed semiclassical Fourier transform, the space usage included 8n + O(log n) logical qubits. This paper achieved a trade-off between time, space, and resource costs while introducing only a small approximation error (which could exponentially decrease with the increase in the number of padding elements).