Applied and Computational Engineering

An Overview of Regev’s Quantum Factoring Algorithm and Its Recent Developments

Rick Lan Chen

Published Nov 28, 2024• 1 citation

Abstract

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.

DOI: 10.54254/2755-2721/110/2024melb0122

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Applied and Computational Engineering

An Overview of Regev’s Quantum Factoring Algorithm and Its Recent Developments

Rick Lan Chen

Published Nov 28, 2024• 1 citation

Publisher

Abstract

DOI: 10.54254/2755-2721/110/2024melb0122

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O(n \log n)

qubits and

O(n^{3/2} \log n)

gates. We achieve the best of Shor and Regev (upto a logarithmic factor in the space complexity): on the one hand, Regev's circuit requires

O(n^{3/2})

qubits and

O(n^{3/2} \log n)

gates, while Shor's circuit requires

O(n^2 \log n)

gates but only

O(n \log n)

qubits. As with Regev, to factor an

n

-bit integer

N

, we run our circuit independently

O(\sqrt{n})

times and apply Regev's classical postprocessing procedure. Our optimization is achieved by implementing efficient and reversible exponentiation with Fibonacci numbers in the exponent, rather than the usual powers of 2, adapting work by Kaliski (arXiv:1711.02491) from the classical reversible setting to the quantum setting. This technique also allows us to perform quantum modular exponentiation that is efficient in both space and size without requiring significant precomputation, a result that may be useful for other quantum algorithms. A key ingredient of our exponentiation implementation is an efficient circuit for a function resembling in-place quantum-quantum modular multiplication. Our second contribution is to show that Regev's classical postprocessing procedure can be modified to tolerate a constant fraction of the quantum circuit runs being corrupted by errors. In contrast, Regev's analysis of his classical postprocessing procedure requires all

\approx \sqrt{n}

runs to be successful. In a nutshell, we achieve this using lattice reduction techniques to detect and filter out corrupt samples.

2^{n}

bits.

(\mathbb {Z}/N\mathbb {Z})^{\times }

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.