Quantum Algorithms for Discrete Log Require Precise Rotations

Abstract

Recently, Cai [3] showed that Shor’s quantum factoring algorithm fails to factor large integers when algorithm’s quantum Fourier transform (QFT) is corrupted by a vanishing level of random noise on the QFT’s precise controlled rotation gates. We show that under the same error model, Shor’s quantum discrete log algorithm, and its various modifications, fail to compute discrete logs modulo P for a positive density of primes P and a similarly vanishing level of noise. We also show that the same noise level causes Shor’s algorithm to fail with probability \(1-o(1)\) to compute discrete logs modulo P for randomly selected primes P.