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arXiv:2412.17269 · ACM Transactions on Quantum Computing

Quantum Algorithms for Discrete Log Require Precise Rotations

Jin‐Yi Cai, Ben Young

Published May 17, 2025• 0 citations
arXivHTMLPDF

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 .

DOI
10.1145/3736421

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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 .
We study the asymptotic behavior of Shor's quantum factoring and discrete log algorithms when noise affects the precise controlled rotation gates in their quantum Fourier transforms. Improving the results of Cai (2024) and Cai and Young (2025), we identify a sharp and vanishingly small noise threshold. If the noise level lies below this threshold, then the two algorithms succeed in expected polynomial time. If the noise level exceeds this threshold, then the algorithms provably fail to solve their respective problems in expected polynomial time when the underlying primes belong to a set of positive density.
Abstract We consider Shor’s quantum factoring algorithm in the setting of noisy quantum gates. Under a generic model of random noise for (controlled) rotation gates, we prove that the algorithm does not factor integers of the form pq when the noise exceeds a vanishingly small level in terms of n —the number of bits of the integer to be factored, where p and q are from a well-defined set of primes of positive density. We further prove that with probability 1 − o (1) over random prime pairs ( p, q ), Shor’s factoring algorithm does not factor numbers of the form pq , with the same level of random noise present.
The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.
Abstract In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multidimensional version of Shor’s algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in (Z/NZ)×(\mathbb {Z}/N\mathbb {Z})^{\times }(Z/NZ)× that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.
Shor's algorithm (SA) is a quantum algorithm for factoring integers. Since SA has polynomial complexity while the best classical factoring algorithms are sub-exponential, SA is cited as evidence that quantum computers are more powerful than classical computers. SA is critically dependent on the Quantum Fourier Transform (QFT) and it is known that the QFT is sensitive to errors in the quantum state input to it. In this paper, we show that the polynomial scaling of SA is destroyed by input errors to the QFT part of the algorithm. We also show that Quantum Error Correcting Codes (QECC) are not capable of suppressing errors due to operator imprecision and that propagation of operator precision errors is sufficient to severely degrade the effectiveness of SA. Additionally we show that operator imprecision in the error correction circuit for the Calderbank-Shor-Steane QECC is mathematically equivalent to decoherence on every physical qubit in a register. We conclude that, because of the effect of operator precision errors, it is likely that physically realizable quantum computers will be capable of factoring integers no more efficiently than classical computers.
Factoring large integers has long been a computationally difficult problem in classical computing, forming the foundation of widely used encryption methods like RSA. In 1994, Shor’s quantum algorithm introduced a revolutionary method for factoring integers exponentially faster than classical algorithms, laying the groundwork for quantum cryptography. Despite numerous attempts to enhance Shor’s algorithm over the last three decades, significant breakthroughs remained elusive until Regev’s discovery in 2023. Regev introduced a novel, multi-dimensional version of the quantum factoring algorithm, achieving a substantial improvement by reducing the required number of quantum gates to O(n^2 lo g⁡n )(n^(3/2) log⁡n ), compared to Shor’s originalO(n^2 lo g⁡n )original O(n^2 log⁡n ) gate complexity. Although Regev’s approach offers a significant speedup, it comes with increased qubit requirements and relies on an unproven number theory assumption. This paper presents an overview of Regev’s factoring algorithm, including a review of Shor’s work for context, followed by an examination of key recent developments and follow-up research. These include efforts to reduce the qubit count, improve error resilience, generalizegeneralise Regev’s algorithm to related problems, and validate the number theory assumption. This review serves as an accessible entry point for researchers interested in the rapidly evolving field of quantum computing and factoring algorithms.