Paper 2023/092

Estimation of Shor's Circuit for 2048-bit Integers based on Quantum Simulator

Abstract

Evaluating exact computational resources necessary for factoring large integers by Shor algorithm using an ideal quantum computer is difficult because simplified circuits were used in past experiments, in which qubits and gates were reduced as much as possible by using the features of the integers, though 15 and 21 were factored on quantum computers. In this paper, we implement Shor algorithm for general composite numbers, and factored 96 RSA-type composite numbers up to 9-bit using a quantum computer simulator. In the largest case, $N=511$ was factored within 2 hours. Then, based on these experiments, we estimate the number of gates and the depth of Shor's quantum circuits for factoring 1024-bit and 2048-bit integers. In our estimation, Shor's quantum circuit for factoring 1024-bit integers requires $2.78 \times 10^{11}$ gates, and with depth $2.24 \times 10^{11}$, while $2.23 \times 10^{12}$ gates, and with depth $1.80 \times 10^{12}$ for 2048-bit integers.