The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.

Modern Dynamics: Mathematical Progressions.May 25, 20248 citations

Integer Factorization by Quantum Measurements

Giuseppe Mussardo, Andrea Trombettoni

Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer $N$ is achieved in a number of steps equal to the number $k$ of its prime factors, -- e.g., if $N$ is the product of two primes, two quantum measurements are enough, regardless of the number of digits $n$ of the number $N$ . Since $k$ is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer $N$ is achieved in a number of steps equal to the number $k$ of its prime factors, -- e.g., if $N$ is the product of two primes, two quantum measurements are enough, regardless of the number of digits $n$ of the number $N$. Since $k$ is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.

arXiv:2309.10757Sep 19, 20232 citations

Algorithm for Quantum Factorization

Samuel Čarnoký

Jan 1, 20080 citations

Efficient diabatic quantum algorithm in number factorization

Anqi Shi, Haoyu Guan, Wenxian Zhang

Physics Letters AJul 22, 20204 citations

N

is achieved in a number of steps equal to the number

k

of its prime factors, -- e.g., if

N

is the product of two primes, two quantum measurements are enough, regardless of the number of digits

n

of the number

N

. Since

k

is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.