6.12. The Theoretical Structure of Shor’s Algorithm

6.12.1. General Objective of the Algorithm

Developed by Peter Shor in 1994, Shor’s Algorithm is capable of solving certain mathematical problems (such as integer factorization and discrete logarithm problems) in polynomial time using quantum computers. These problems are inefficient to solve with classical computers. The most well-known application of the algorithm is finding the prime factors of a large integer, which poses a significant threat to the security of the widely used RSA cryptosystem.

6.12.2. Integration of Classical and Quantum Stages

Shor’s Algorithm is a hybrid algorithm that combines classical and quantum computing components. Its general structure can be summarized as follows:

1. Classical Stage:
   • Preprocessing steps are performed to find the prime factors of the given integer input.
   • A random number is selected, and it is checked whether it leads to a prime factor.
   • If no prime factor is found, the process moves to the quantum stage.
2. Quantum Stage:
   • The period-finding problem is solved to determine the prime factors.
   • The Quantum Fourier Transform (QFT) is used for efficient period determination.
3. Classical Stage:
   • The results from the quantum stage are processed to calculate the prime factors.

The collaboration between these two stages ensures the efficiency of the algorithm.

6.12.3. The Period-Finding Problem

At the core of Shor’s Algorithm lies the period-finding problem. Period finding involves identifying the intervals (periods) at which a function repeats itself. Specifically, finding the period of the function f(x) = a^x mod N is a critical step in determining the prime factors. Quantum computers are capable of finding this period in polynomial time, which gives Shor’s Algorithm a significant advantage over classical algorithms.