A Fourier-Label Information-Loss Barrier for Dihedral Coset Algorithms

2026-09-30

AI Summary

This study addresses the unclear effectiveness boundaries of Fourier sampling-based template algorithms for the Dihedral Coset Problem (DCP). It proposes an information loss barrier theory that integrates quantum computational analysis, information-theoretic bound derivations, and Lean 4 formal verification tools to rigorously delineate the necessary conditions for the success of Regev’s template algorithm. The primary contribution is the establishment of a no-go theorem proving that discarding a substantial number of Fourier label bits inevitably causes algorithmic failure, which consequently refutes the recently proposed DCP algorithm by Simon. Furthermore, this work releases machine-verifiable Lean 4 code that corrects existing erroneous conclusions in the literature and significantly enhances the credibility of the presented results.

Abstract

We establish a no-go theorem for a broad class of quantum algorithms for the dihedral coset problem (DCP). We consider the Fourier-sampling and subset-sum-measurement template proposed by Regev (SIAM Journal on Computing, 2004), which is one of the main approaches to solving DCP. Suppose that, after measuring the lower $n-1$ bits of the subset sum, the algorithm discards any $ω(\log n)$ bits from each of the Fourier labels. Then we prove that the algorithm cannot succeed in solving DCP. This shows that any algorithm following this template must make extensive use of the Fourier labels, and thus serves as a useful guide for developing algorithms for DCP. As a main application, we show that the recent algorithm by Simon (IACR ePrint:2026/1591, August 11 2026) does not solve DCP. We show that after the subset-sum measurement, this algorithm can be implemented (up to exponentially-small error) using only the most-significant third of the Fourier labels, and is therefore subject to our general no-go theorem. To help with verifiability, we release Lean 4 code for our results.

Problem

• Dihedral Coset Problem
• Quantum Algorithms
• Fourier Sampling
• No-go Theorem
• Subset-sum Measurement

Innovation

• Dihedral Coset Problem
• No-go Theorem
• Fourier Sampling
• Quantum Algorithms
• Formal Verification