Peter Shor's quantum factorization algorithm, also known as Shor's algorithm, is a groundbreaking quantum algorithm that efficiently solves the problem of integer factorization. It was discovered by Peter Shor in 1994 and demonstrated the potential of quantum computers to outperform classical computers for certain tasks.
The algorithm works as follows:
Input: The algorithm takes as input an integer N that is to be factored.
Quantum Fourier Transform (QFT): Shor's algorithm utilizes the quantum Fourier transform, which is a quantum analogue of the classical discrete Fourier transform. The QFT is applied to a register of qubits, which encodes the values of a periodic function.
Modular Exponentiation: The algorithm performs modular exponentiation on another register of qubits. This step involves raising a base number to a series of powers modulo N. The quantum computer exploits the inherent parallelism of quantum computation to perform these calculations efficiently.
Quantum Measurement: After applying the QFT and modular exponentiation, a quantum measurement is performed on the qubits. The measurement collapses the quantum states and provides a measurement outcome.
Classical Post-processing: The measurement outcome obtained from the quantum computation is processed classically to extract the factors of N using number theory techniques.
Shor's algorithm has significant implications because it can efficiently factor large composite numbers, which is considered a hard problem for classical computers. This poses a threat to the security of widely used cryptographic systems based on the difficulty of factoring large numbers, such as RSA encryption.
The applications of Shor's algorithm include:
Cryptanalysis: Shor's algorithm can break RSA and other commonly used public-key cryptosystems based on the difficulty of factoring large numbers. This has spurred research into post-quantum cryptography, which aims to develop new encryption algorithms resistant to quantum attacks.
Number theory: Shor's algorithm has led to advancements in the field of number theory, particularly in understanding the structure and properties of prime numbers and factoring algorithms.
It's important to note that Shor's algorithm is a theoretical breakthrough, and the practical realization of large-scale quantum computers capable of executing the algorithm remains a significant engineering challenge. Nevertheless, the algorithm's impact on cryptography and the potential implications for secure communication have been profound.