Unconditional Correctness of Recent Quantum Algorithms for Factoring and Computing Discrete Logarithms

Core Concepts

Recent quantum algorithms by Regev and others can factor integers and compute discrete logarithms more efficiently than Shor's original algorithm, but their correctness relied on an unproven number-theoretic conjecture. This paper provides an unconditional proof of the correctness of these improved quantum algorithms.

Abstract

The content discusses recent advancements in quantum algorithms for factoring integers and computing discrete logarithms.
Key highlights:

In 1994, Shor introduced a quantum algorithm that can factor integers and compute discrete logarithms in polynomial time.
In 2023, Regev proposed a multidimensional version of Shor's algorithm that requires fewer quantum gates, but its correctness relied on an unproven number-theoretic conjecture.
This paper provides an unconditional proof of the correctness of Regev's algorithm and subsequent variants, by proving a version of Regev's conjecture using tools from analytic number theory.
The authors show that there exists a quantum circuit with O(n^(3/2) log^3 n) gates and O(n log^3 n) qubits that can solve the factoring and discrete logarithm problems with high probability, without relying on any unproven assumptions.
The key technical result is Theorem 2.18, which shows that certain lattices have a basis of short vectors with high probability, by using zero-density estimates for Dirichlet characters.

Stats

There are O(n^(3/2) log^3 n) quantum gates and O(n log^3 n) qubits in the quantum circuit.
The quantum circuit is called O(√n) times to solve the factoring or discrete logarithm problem.

Quotes

"There is a quantum circuit having O(n^3/2 log^3 n) quantum gates and O(n log^3 n) qubits with the following property. There is a classical randomised polynomial-time algorithm that solves the factoring problem using O(√n) calls to this quantum circuit, and succeeds with probability Θ(1)."
"There is a quantum circuit having O(n^3/2 log^3 n) quantum gates and O(n log^3 n) qubits with the following property. There is a classical randomised polynomial-time algorithm that solves the discrete logarithm problem using O(√n) calls to this quantum circuit, and succeeds with probability Θ(1)."

Key Insights Distilled From

Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

by Cédr... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16450.pdf

Deeper Inquiries

What other number-theoretic conjectures or assumptions could be used to obtain similar unconditional results for quantum algorithms

To obtain similar unconditional results for quantum algorithms, other number-theoretic conjectures or assumptions could be explored. One potential avenue is leveraging the properties of primitive roots modulo N. The existence of primitive roots is a fundamental concept in number theory and is closely related to the multiplicative structure of the group of units modulo N. By formulating conjectures or assumptions related to the distribution and properties of primitive roots, it may be possible to establish conditions under which quantum algorithms for factoring and computing discrete logarithms can be unconditionally correct.

How might the techniques in this paper be extended to analyze the performance of quantum algorithms for other computational problems beyond factoring and discrete logarithms

The techniques employed in the paper can be extended to analyze the performance of quantum algorithms for a wide range of computational problems beyond factoring and discrete logarithms. One promising direction is to apply similar lattice-based methods to problems in cryptography, such as cryptographic protocols based on lattice problems. Lattice-based cryptography is a rapidly growing field that offers post-quantum security guarantees. By adapting the techniques from the paper to analyze the efficiency and correctness of quantum algorithms for lattice-based cryptographic schemes, researchers can gain insights into the practical implications of these algorithms in real-world applications.

Are there any connections between the lattice-based techniques used in this paper and the development of post-quantum cryptography based on lattice problems

There are significant connections between the lattice-based techniques used in the paper and the development of post-quantum cryptography based on lattice problems. Lattices play a central role in both quantum algorithms and lattice-based cryptography. The efficient manipulation of lattices is essential for the success of quantum algorithms for factoring and discrete logarithms, as demonstrated in the paper. Similarly, in lattice-based cryptography, the hardness of certain lattice problems forms the basis for cryptographic schemes that are resilient against quantum attacks. By further exploring the connections between quantum algorithms and lattice-based cryptography, researchers can advance the development of secure cryptographic protocols that are resistant to quantum threats.