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

Yoshinori Aono·Sitong Liu·Tomoki Tanaka·Shumpei Uno·Rodney Van Meter·Naoyuki Shinohara·Ryo Nojima
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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.

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Yoshinori Aono, Sitong Liu, Tomoki Tanaka, Shumpei Uno, Rodney Van Meter, Naoyuki Shinohara, Ryo Nojima. 2021-11-11. The Present and Future of Discrete Logarithm Problems on Noisy Quantum Computers. https://doi.org/10.1109/tqe.2022.3183385

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