The Present and Future of Discrete Logarithm Problems on Noisy Quantum Computers

TL;DR

This paper investigates the impact of noise on quantum algorithms solving discrete logarithm problems, proposing measures to evaluate success probabilities and modifications to improve outcomes on current noisy quantum devices.

Contribution

It introduces a quantitative success measure for quantum DLP experiments and a method to enhance post-processing success probabilities under noise.

Findings

Successful experiments on IBM quantum hardware with small circuits.

Noise reduction is critical for solving larger DLP instances.

Predictions for future quantum device capabilities based on noise levels.

Abstract

The discrete logarithm problem (DLP) is the basis for several cryptographic primitives. Since Shor's work, it has been known that the DLP can be solved by combining a polynomial-size quantum circuit and a polynomial-time classical post-processing algorithm. Evaluating and predicting the instance size that quantum devices can solve is an emerging research topic. In this paper, we propose a quantitative measure based on the success probability of the post-processing algorithm to determine whether an experiment on a quantum device (or a classical simulator) succeeded. We also propose a procedure to modify bit strings observed from a Shor circuit to increase the success probability of a lattice-based post-processing algorithm. We report preliminary experiments conducted on IBM-Quantum quantum computers and near-future predictions based on noisy-device simulations. We conducted our experiments with the ibm_kawasaki device and discovered that the simplest circuit (7 qubits) from a 2-bit DLP instance achieves a sufficiently high success probability to proclaim the experiment successful. Experiments on another circuit from a slightly harder 2-bit DLP instance, on the other hand, did not succeed, and we determined that reducing the noise level by half is required to achieve a successful experiment. Finally, we give a near-term prediction based on required noise levels to solve some selected small DLP and integer factoring instances.