Quantum Algorithms for Discrete Log Require Precise Rotations

TL;DR

This paper demonstrates that quantum algorithms for discrete logarithms, including Shor's, require highly precise controlled rotations, as even minimal noise causes failure in factoring and discrete log computations.

Contribution

It extends prior work on quantum factoring noise sensitivity to discrete log algorithms, showing their failure under similar error conditions.

Findings

Shor's discrete log algorithm fails with minimal noise

Failure occurs for a positive density of primes P

Randomly selected primes P also lead to failure

Abstract

Recently, Cai showed that Shor's quantum factoring algorithm fails to factor large integers when the 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.