6.13.4. Properties of QFT
6.13.4.1. Linearity
QFT is a linear transformation, meaning it preserves the principle of superposition.
6.13.4.2. Unitarity
QFT is a unitary transformation, which means its inverse is the Hermitian adjoint of QFT:
QFT‘ × QFT = I
This property ensures error-free information transfer in quantum computing.
6.13.4.3. Efficient Implementation
QFT can be implemented using O(n2) quantum gates, which allows it to be executed efficiently even for large n, maintaining polynomial time complexity.
6.13.5. Role of QFT in Shor’s Algorithm
Shor’s Algorithm enables efficient solutions to classically hard problems, such as the large integer factorization problem, on quantum computers. QFT is a fundamental component of Shor’s Algorithm and plays a critical role in the following steps:
6.13.5.1. Period Finding
Shor’s Algorithm reduces the integer factorization problem to the period-finding problem. QFT is used at this stage to determine the period of a function.
6.13.5.2. Quantum Superposition and QFT
In the quantum phase of the algorithm, the superposition of the relevant function is created, and QFT is applied to extract information about the period. QFT reveals the frequency components of the period, enabling the correct period to be identified.
6.13.5.3. Measurement and Interpretation of Results
After QFT, a measurement is performed, and the results are processed using classical algorithms to determine the prime factors.